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Adversarial training injects such examples into training data to increase ro-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 294, + 470, + 308 + ], + "spans": [ + { + "bbox": [ + 141, + 294, + 470, + 308 + ], + "score": 1.0, + "content": "bustness. To scale this technique to large datasets, perturbations are crafted using", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 306, + 470, + 318 + ], + "spans": [ + { + "bbox": [ + 141, + 306, + 470, + 318 + ], + "score": 1.0, + "content": "fast single-step methods that maximize a linear approximation of the model’s loss.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 141, + 273, + 470, + 318 + ] + }, + { + "type": "text", + "bbox": [ + 143, + 320, + 468, + 396 + ], + "lines": [ + { + "bbox": [ + 142, + 320, + 470, + 333 + ], + "spans": [ + { + "bbox": [ + 142, + 320, + 470, + 333 + ], + "score": 1.0, + "content": "We show that this form of adversarial training converges to a degenerate global", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 331, + 469, + 342 + ], + "spans": [ + { + "bbox": [ + 142, + 331, + 469, + 342 + ], + "score": 1.0, + "content": "minimum, wherein small curvature artifacts near the data points obfuscate a lin-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 342, + 469, + 354 + ], + "spans": [ + { + "bbox": [ + 141, + 342, + 469, + 354 + ], + "score": 1.0, + "content": "ear approximation of the loss. The model thus learns to generate weak perturba-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 353, + 469, + 364 + ], + "spans": [ + { + "bbox": [ + 141, + 353, + 469, + 364 + ], + "score": 1.0, + "content": "tions, rather than defend against strong ones. As a result, we find that adversarial", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 363, + 469, + 375 + ], + "spans": [ + { + "bbox": [ + 141, + 363, + 469, + 375 + ], + "score": 1.0, + "content": "training remains vulnerable to black-box attacks, where we transfer perturbations", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 142, + 375, + 469, + 387 + ], + "spans": [ + { + "bbox": [ + 142, + 375, + 469, + 387 + ], + "score": 1.0, + "content": "computed on undefended models, as well as to a powerful novel single-step attack", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 385, + 457, + 398 + ], + "spans": [ + { + "bbox": [ + 141, + 385, + 457, + 398 + ], + "score": 1.0, + "content": "that escapes the non-smooth vicinity of the input data via a small random step.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 141, + 320, + 470, + 398 + ] + }, + { + "type": "text", + "bbox": [ + 143, + 398, + 469, + 453 + ], + "lines": [ + { + "bbox": [ + 142, + 398, + 469, + 411 + ], + "spans": [ + { + "bbox": [ + 142, + 398, + 469, + 411 + ], + "score": 1.0, + "content": "We further introduce Ensemble Adversarial Training, a technique that augments", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 409, + 469, + 422 + ], + "spans": [ + { + "bbox": [ + 141, + 409, + 469, + 422 + ], + "score": 1.0, + "content": "training data with perturbations transferred from other models. On ImageNet,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 420, + 469, + 432 + ], + "spans": [ + { + "bbox": [ + 141, + 420, + 469, + 432 + ], + "score": 1.0, + "content": "Ensemble Adversarial Training yields models with strong robustness to black-box", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 432, + 470, + 443 + ], + "spans": [ + { + "bbox": [ + 141, + 432, + 470, + 443 + ], + "score": 1.0, + "content": "attacks. In particular, our most robust model won the first round of the NIPS 2017", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 443, + 451, + 455 + ], + "spans": [ + { + "bbox": [ + 141, + 443, + 451, + 455 + ], + "score": 1.0, + "content": "competition on Defenses against Adversarial Attacks (Kurakin et al., 2017c).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 141, + 398, + 470, + 455 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 477, + 206, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 208, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 208, + 492 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 502, + 505, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "Machine learning (ML) models are often vulnerable to adversarial examples, maliciously perturbed", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "inputs designed to mislead a model at test time (Biggio et al., 2013; Szegedy et al., 2013; Goodfellow", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "et al., 2014b; Papernot et al., 2016a). Furthermore, Szegedy et al. (2013) showed that these inputs", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "transfer across models: the same adversarial example is often misclassified by different models,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "thus enabling simple black-box attacks on deployed models (Papernot et al., 2017; Liu et al., 2017).", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 502, + 506, + 559 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 563, + 505, + 629 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "Adversarial training (Szegedy et al., 2013) increases robustness by augmenting training data with", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "adversarial examples. Madry et al. (2017) showed that adversarially trained models can be made", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "robust to white-box attacks (i.e., with knowledge of the model parameters) if the perturbations com-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "puted during training closely maximize the model’s loss. However, prior attempts at scaling this", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 104, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "approach to ImageNet-scale tasks (Deng et al., 2009) have proven unsuccessful (Kurakin et al.,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 618, + 140, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 140, + 630 + ], + "score": 1.0, + "content": "2017b).", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45.5, + "bbox_fs": [ + 104, + 563, + 506, + 630 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 635, + 505, + 712 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 506, + 647 + ], + "score": 1.0, + "content": "It is thus natural to ask whether it is possible, at scale, to achieve robustness against the class of", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "black-box adversaries Towards this goal, Kurakin et al. (2017b) adversarially trained an Inception", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 657, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 669 + ], + "score": 1.0, + "content": "v3 model (Szegedy et al., 2016b) on ImageNet using a “single-step” attack based on a linearization", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "of the model’s loss (Goodfellow et al., 2014b). Their trained model is robust to single-step perturba-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "tions but remains vulnerable to more costly “multi-step” attacks. Yet, Kurakin et al. (2017b) found", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 690, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 701 + ], + "score": 1.0, + "content": "that these attacks fail to reliably transfer between models, and thus concluded that the robustness of", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 700, + 503, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 503, + 713 + ], + "score": 1.0, + "content": "their model should extend to black-box adversaries. Surprisingly, we show that this is not the case.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 52, + "bbox_fs": [ + 105, + 635, + 506, + 713 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 148 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "We demonstrate, formally and empirically, that adversarial training with single-step methods admits", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "a degenerate global minimum, wherein the model’s loss can not be reliably approximated by a linear", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "function. Specifically, we find that the model’s decision surface exhibits sharp curvature near the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "score": 1.0, + "content": "data points, thus degrading attacks based on a single gradient computation. In addition to the model", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "of Kurakin et al. (2017b), we reveal similar overfitting in an adversarially trained Inception ResNet", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 500, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 500, + 149 + ], + "score": 1.0, + "content": "v2 model (Szegedy et al., 2016a), and a variety of models trained on MNIST (LeCun et al., 1998).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 152, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 152, + 505, + 168 + ], + "score": 1.0, + "content": "We harness this result in two ways. First, we show that adversarially trained models using single-step", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "methods remain vulnerable to simple attacks. For black-box adversaries, we find that perturbations", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "crafted on an undefended model often transfer to an adversarially trained one. We also introduce a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 504, + 199 + ], + "score": 1.0, + "content": "simple yet powerful single-step attack that applies a small random perturbation—to escape the non-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "smooth vicinity of the data point—before linearizing the model’s loss. While seemingly weaker than", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "the Fast Gradient Sign Method of Goodfellow et al. (2014b), our attack significantly outperforms it", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 452, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 452, + 233 + ], + "score": 1.0, + "content": "for a same perturbation norm, for models trained with or without adversarial training.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 236, + 504, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "score": 1.0, + "content": "Second, we propose Ensemble Adversarial Training, a training methodology that incorporates per-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "turbed inputs transferred from other pre-trained models. Our approach decouples adversarial exam-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "ple generation from the parameters of the trained model, and increases the diversity of perturbations", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 283 + ], + "score": 1.0, + "content": "seen during training. We train Inception v3 and Inception ResNet v2 models on ImageNet that ex-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "hibit increased robustness to adversarial examples transferred from other holdout models, using var-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "ious single-step and multi-step attacks (Goodfellow et al., 2014b; Carlini & Wagner, 2017a; Kurakin", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "et al., 2017a; Madry et al., 2017). We also show that our methods globally reduce the dimensionality", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "of the space of adversarial examples (Tramer et al., 2017). Our Inception ResNet v2 model won the `", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "first round of the NIPS 2017 competition on Defenses Against Adversarial Attacks (Kurakin et al.,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 336, + 442, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 442, + 351 + ], + "score": 1.0, + "content": "2017c), where it was evaluated on other competitors’ attacks in a black-box setting.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 108, + 367, + 211, + 380 + ], + "lines": [ + { + "bbox": [ + 104, + 366, + 213, + 383 + ], + "spans": [ + { + "bbox": [ + 104, + 366, + 213, + 383 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 394, + 505, + 450 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "Various defensive techniques against adversarial examples in deep neural networks have been pro-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 404, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 418 + ], + "score": 1.0, + "content": "posed (Gu & Rigazio, 2014; Luo et al., 2015; Papernot et al., 2016c; Nayebi & Ganguli, 2017; Cisse", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 415, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 104, + 415, + 506, + 430 + ], + "score": 1.0, + "content": "et al., 2017) and many remain vulnerable to adaptive attackers (Carlini & Wagner, 2017a;b; Baluja", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "score": 1.0, + "content": "& Fischer, 2017). Adversarial training (Szegedy et al., 2013; Goodfellow et al., 2014b; Kurakin", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 438, + 497, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 497, + 451 + ], + "score": 1.0, + "content": "et al., 2017b; Madry et al., 2017) appears to hold the greatest promise for learning robust models.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "score": 1.0, + "content": "Madry et al. (2017) show that adversarial training on MNIST yields models that are robust to white-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "box attacks, if the adversarial examples used in training closely maximize the model’s loss. More-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "over, recent works by Sinha et al. (2018), Raghunathan et al. (2018) and Kolter & Wong (2017)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "even succeed in providing certifiable robustness for small perturbations on MNIST. As we argue in", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 498, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 512 + ], + "score": 1.0, + "content": "Appendix C, the MNIST dataset is peculiar in that there exists a simple “closed-form” denoising", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 104, + 509, + 506, + 523 + ], + "score": 1.0, + "content": "procedure (namely feature binarization) which leads to similarly robust models without adversarial", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 521, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 505, + 533 + ], + "score": 1.0, + "content": "training. This may explain why robustness to white-box attacks is hard to scale to tasks such as Im-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 531, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 544 + ], + "score": 1.0, + "content": "ageNet (Kurakin et al., 2017b). We believe that the existence of a simple robust baseline for MNIST", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 542, + 440, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 440, + 556 + ], + "score": 1.0, + "content": "can be useful for understanding some limitations of adversarial training techniques.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 559, + 505, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "Szegedy et al. (2013) found that adversarial examples transfer between models, thus enabling black-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 572, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 582 + ], + "score": 1.0, + "content": "box attacks on deployed models. Papernot et al. (2017) showed that black-box attacks could succeed", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "with no access to training data, by exploiting the target model’s predictions to extract (Tramer et al., `", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 591, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 606 + ], + "score": 1.0, + "content": "2016) a surrogate model. Some prior works have hinted that adversarially trained models may", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 604, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 615 + ], + "score": 1.0, + "content": "remain vulnerable to black-box attacks: Goodfellow et al. (2014b) found that an adversarial maxout", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 615, + 504, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 504, + 626 + ], + "score": 1.0, + "content": "network on MNIST has slightly higher error on transferred examples than on white-box examples.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 626, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 638 + ], + "score": 1.0, + "content": "Papernot et al. (2017) further showed that a model trained on small perturbations can be evaded", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "score": 1.0, + "content": "by transferring perturbations of larger magnitude. Our finding that adversarial training degrades", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 647, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 505, + 661 + ], + "score": 1.0, + "content": "the accuracy of linear approximations of the model’s loss is as an instance of a gradient-masking", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 657, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 671 + ], + "score": 1.0, + "content": "phenomenon (Papernot et al., 2016b), which affects other defensive techniques (Papernot et al.,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 668, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 104, + 668, + 505, + 682 + ], + "score": 1.0, + "content": "2016c; Carlini & Wagner, 2017a; Nayebi & Ganguli, 2017; Brendel & Bethge, 2017; Athalye et al.,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 680, + 136, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 136, + 693 + ], + "score": 1.0, + "content": "2018).", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 43.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 701, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 119, + 699, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 119, + 699, + 506, + 714 + ], + "score": 1.0, + "content": "1We publicly released our model after the first round, and it could thereafter be targeted using white-box", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "attacks. Nevertheless, a majority of the top submissions in the final round, e.g. (Xie et al., 2018) built upon our", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 720, + 165, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 165, + 732 + ], + "score": 1.0, + "content": "released model.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 148 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "We demonstrate, formally and empirically, that adversarial training with single-step methods admits", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "a degenerate global minimum, wherein the model’s loss can not be reliably approximated by a linear", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "function. Specifically, we find that the model’s decision surface exhibits sharp curvature near the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 127 + ], + "score": 1.0, + "content": "data points, thus degrading attacks based on a single gradient computation. In addition to the model", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "of Kurakin et al. (2017b), we reveal similar overfitting in an adversarially trained Inception ResNet", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 500, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 500, + 149 + ], + "score": 1.0, + "content": "v2 model (Szegedy et al., 2016a), and a variety of models trained on MNIST (LeCun et al., 1998).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 82, + 506, + 149 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 152, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 152, + 505, + 168 + ], + "score": 1.0, + "content": "We harness this result in two ways. First, we show that adversarially trained models using single-step", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "methods remain vulnerable to simple attacks. For black-box adversaries, we find that perturbations", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "crafted on an undefended model often transfer to an adversarially trained one. We also introduce a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 504, + 199 + ], + "score": 1.0, + "content": "simple yet powerful single-step attack that applies a small random perturbation—to escape the non-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "smooth vicinity of the data point—before linearizing the model’s loss. While seemingly weaker than", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "the Fast Gradient Sign Method of Goodfellow et al. (2014b), our attack significantly outperforms it", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 452, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 452, + 233 + ], + "score": 1.0, + "content": "for a same perturbation norm, for models trained with or without adversarial training.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 152, + 506, + 233 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 236, + 504, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "score": 1.0, + "content": "Second, we propose Ensemble Adversarial Training, a training methodology that incorporates per-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "turbed inputs transferred from other pre-trained models. Our approach decouples adversarial exam-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "ple generation from the parameters of the trained model, and increases the diversity of perturbations", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 283 + ], + "score": 1.0, + "content": "seen during training. We train Inception v3 and Inception ResNet v2 models on ImageNet that ex-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "hibit increased robustness to adversarial examples transferred from other holdout models, using var-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "ious single-step and multi-step attacks (Goodfellow et al., 2014b; Carlini & Wagner, 2017a; Kurakin", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "et al., 2017a; Madry et al., 2017). We also show that our methods globally reduce the dimensionality", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "of the space of adversarial examples (Tramer et al., 2017). Our Inception ResNet v2 model won the `", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "first round of the NIPS 2017 competition on Defenses Against Adversarial Attacks (Kurakin et al.,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 336, + 442, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 442, + 351 + ], + "score": 1.0, + "content": "2017c), where it was evaluated on other competitors’ attacks in a black-box setting.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 236, + 506, + 351 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 367, + 211, + 380 + ], + "lines": [ + { + "bbox": [ + 104, + 366, + 213, + 383 + ], + "spans": [ + { + "bbox": [ + 104, + 366, + 213, + 383 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 394, + 505, + 450 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "Various defensive techniques against adversarial examples in deep neural networks have been pro-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 404, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 418 + ], + "score": 1.0, + "content": "posed (Gu & Rigazio, 2014; Luo et al., 2015; Papernot et al., 2016c; Nayebi & Ganguli, 2017; Cisse", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 415, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 104, + 415, + 506, + 430 + ], + "score": 1.0, + "content": "et al., 2017) and many remain vulnerable to adaptive attackers (Carlini & Wagner, 2017a;b; Baluja", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "score": 1.0, + "content": "& Fischer, 2017). Adversarial training (Szegedy et al., 2013; Goodfellow et al., 2014b; Kurakin", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 438, + 497, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 497, + 451 + ], + "score": 1.0, + "content": "et al., 2017b; Madry et al., 2017) appears to hold the greatest promise for learning robust models.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 394, + 506, + 451 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "score": 1.0, + "content": "Madry et al. (2017) show that adversarial training on MNIST yields models that are robust to white-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "box attacks, if the adversarial examples used in training closely maximize the model’s loss. More-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "over, recent works by Sinha et al. (2018), Raghunathan et al. (2018) and Kolter & Wong (2017)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "even succeed in providing certifiable robustness for small perturbations on MNIST. As we argue in", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 498, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 512 + ], + "score": 1.0, + "content": "Appendix C, the MNIST dataset is peculiar in that there exists a simple “closed-form” denoising", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 104, + 509, + 506, + 523 + ], + "score": 1.0, + "content": "procedure (namely feature binarization) which leads to similarly robust models without adversarial", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 521, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 505, + 533 + ], + "score": 1.0, + "content": "training. This may explain why robustness to white-box attacks is hard to scale to tasks such as Im-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 531, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 544 + ], + "score": 1.0, + "content": "ageNet (Kurakin et al., 2017b). We believe that the existence of a simple robust baseline for MNIST", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 542, + 440, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 440, + 556 + ], + "score": 1.0, + "content": "can be useful for understanding some limitations of adversarial training techniques.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33, + "bbox_fs": [ + 104, + 455, + 506, + 556 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 559, + 505, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "Szegedy et al. (2013) found that adversarial examples transfer between models, thus enabling black-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 572, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 582 + ], + "score": 1.0, + "content": "box attacks on deployed models. Papernot et al. (2017) showed that black-box attacks could succeed", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "with no access to training data, by exploiting the target model’s predictions to extract (Tramer et al., `", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 591, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 606 + ], + "score": 1.0, + "content": "2016) a surrogate model. Some prior works have hinted that adversarially trained models may", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 604, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 615 + ], + "score": 1.0, + "content": "remain vulnerable to black-box attacks: Goodfellow et al. (2014b) found that an adversarial maxout", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 615, + 504, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 504, + 626 + ], + "score": 1.0, + "content": "network on MNIST has slightly higher error on transferred examples than on white-box examples.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 626, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 638 + ], + "score": 1.0, + "content": "Papernot et al. (2017) further showed that a model trained on small perturbations can be evaded", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "score": 1.0, + "content": "by transferring perturbations of larger magnitude. 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Although security against white-box", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 283, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 505, + 297 + ], + "score": 1.0, + "content": "attacks is the stronger notion (and the one we ideally want ML models to achieve), black-box security", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 295, + 368, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 368, + 307 + ], + "score": 1.0, + "content": "is a reasonable and more tractable goal for deployed ML models.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 107, + 319, + 241, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 242, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 242, + 332 + ], + "score": 1.0, + "content": "3.2 ADVERSARIAL TRAINING", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 105, + 340, + 503, + 363 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "Following Madry et al. (2017), we consider an adversarial variant of standard Empirical Risk Mini-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 351, + 437, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 437, + 364 + ], + "score": 1.0, + "content": "mization (ERM), where our aim is to minimize the risk over adversarial examples:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 367, + 426, + 393 + ], + "lines": [ + { + "bbox": [ + 185, + 367, + 426, + 393 + ], + "spans": [ + { + "bbox": [ + 185, + 367, + 426, + 393 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { h ^ { * } = \\underset { h \\in \\mathcal { H } } { \\arg \\operatorname* { m i n } } } & { \\underset { ( x , y _ { \\mathrm { t u e } } ) \\sim \\mathcal { D } } { \\mathbb { E } } \\left[ \\underset { \\| x ^ { \\mathrm { a d v } } - x \\| _ { \\infty } \\leq \\epsilon } { \\operatorname* { m a x } } L ( h ( x ^ { \\mathrm { a d v } } ) , y _ { \\mathrm { t r u e } } ) \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "bb598728c43a71396f18ddba779ad2fedec19442160579402efc68a250b7ebb2.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 185, + 367, + 426, + 393 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 398, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 106, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "Madry et al. (2017) argue that adversarial training has a natural interpretation in this context, where", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 409, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 505, + 422 + ], + "score": 1.0, + "content": "a given attack (see below) is used to approximate solutions to the inner maximization problem, and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "the outer minimization problem corresponds to training over these examples. Note that the original", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "formulation of adversarial training (Szegedy et al., 2013; Goodfellow et al., 2014b), which we use", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 103, + 440, + 463, + 460 + ], + "spans": [ + { + "bbox": [ + 103, + 440, + 463, + 460 + ], + "score": 1.0, + "content": "in our experiments, trains on both the “clean” examples x and adversarial examples xadv.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 460, + 505, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 411, + 473 + ], + "score": 1.0, + "content": "We consider three algorithms to generate adversarial examples with bounded", + "type": "text" + }, + { + "bbox": [ + 411, + 461, + 425, + 473 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "norm. 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We enforce", + "type": "text" + }, + { + "bbox": [ + 261, + 483, + 315, + 497 + ], + "score": 0.94, + "content": "x ^ { \\mathrm { a d v } } \\in [ 0 , 1 ] ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 482, + 438, + 498 + ], + "score": 1.0, + "content": "by clipping all components of", + "type": "text" + }, + { + "bbox": [ + 439, + 483, + 455, + 495 + ], + "score": 0.9, + "content": "x ^ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 482, + 459, + 498 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 501, + 504, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 515 + ], + "score": 1.0, + "content": "Fast Gradient Sign Method (FGSM). This method (Goodfellow et al., 2014b) linearizes the inner", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 513, + 227, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 227, + 525 + ], + "score": 1.0, + "content": "maximization problem in (1):", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 529, + 394, + 545 + ], + "lines": [ + { + "bbox": [ + 216, + 529, + 394, + 545 + ], + "spans": [ + { + "bbox": [ + 216, + 529, + 394, + 545 + ], + "score": 0.9, + "content": "x _ { \\mathrm { F G S M } } ^ { \\mathrm { a d v } } : = x + \\varepsilon \\cdot \\mathrm { s i g n } \\left( \\nabla _ { x } L ( h ( x ) , y _ { \\mathrm { t r u e } } ) \\right) \\ .", + "type": "interline_equation", + "image_path": "c985f58e7b152bf6835de024c641b20075bd33f15d6b58c10d550c3be6c5e20d.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 216, + 529, + 394, + 545 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 554, + 506, + 578 + ], + "lines": [ + { + "bbox": [ + 106, + 554, + 504, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 504, + 567 + ], + "score": 1.0, + "content": "Single-Step Least-Likely Class Method (Step-LL). This variant of FGSM introduced by Kurakin", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 565, + 379, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 284, + 580 + ], + "score": 1.0, + "content": "et al. (2017a;b) targets the least-likely class,", + "type": "text" + }, + { + "bbox": [ + 284, + 566, + 374, + 579 + ], + "score": 0.92, + "content": "y _ { \\mathrm { L L } } = \\arg \\operatorname* { m i n } \\{ h ( x ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 565, + 379, + 580 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 583, + 388, + 599 + ], + "lines": [ + { + "bbox": [ + 222, + 583, + 388, + 599 + ], + "spans": [ + { + "bbox": [ + 222, + 583, + 388, + 599 + ], + "score": 0.92, + "content": "x _ { \\mathrm { L L } } ^ { \\mathrm { a d v } } : = x - \\varepsilon \\cdot \\mathrm { s i g n } \\left( \\nabla _ { x } L ( h ( x ) , y _ { \\mathrm { L L } } ) \\right) .", + "type": "interline_equation", + "image_path": "78115ccdc8738100b4286bfb16132885a2adce33cb9d8c6acf798bc0c8604d96.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 222, + 583, + 388, + 599 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "Although this attack only indirectly tackles the inner maximization in (1), Kurakin et al. (2017b)", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 614, + 378, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 378, + 627 + ], + "score": 1.0, + "content": "find it to be the most effective for adversarial training on ImageNet.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 504, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 496, + 644 + ], + "score": 1.0, + "content": "Iterative Attack (I-FGSM or Iter-LL). This method iteratively applies the FGSM or Step-LL", + "type": "text" + }, + { + "bbox": [ + 497, + 631, + 504, + 641 + ], + "score": 0.72, + "content": "k", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 189, + 655 + ], + "score": 1.0, + "content": "times with step-size", + "type": "text" + }, + { + "bbox": [ + 190, + 642, + 227, + 654 + ], + "score": 0.93, + "content": "\\alpha \\geq \\epsilon / k", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 643, + 356, + 655 + ], + "score": 1.0, + "content": "and projects each step onto the", + "type": "text" + }, + { + "bbox": [ + 357, + 642, + 371, + 654 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 643, + 425, + 655 + ], + "score": 1.0, + "content": "ball of norm", + "type": "text" + }, + { + "bbox": [ + 426, + 645, + 432, + 652 + ], + "score": 0.53, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 643, + 463, + 655 + ], + "score": 1.0, + "content": "around", + "type": "text" + }, + { + "bbox": [ + 464, + 644, + 470, + 652 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 643, + 505, + 655 + ], + "score": 1.0, + "content": ". It uses", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 653, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 398, + 666 + ], + "score": 1.0, + "content": "projected gradient descent to solve the maximization in (1). For fixed", + "type": "text" + }, + { + "bbox": [ + 398, + 655, + 404, + 663 + ], + "score": 0.5, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 653, + 506, + 666 + ], + "score": 1.0, + "content": ", iterative attacks induce", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 664, + 484, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 484, + 677 + ], + "score": 1.0, + "content": "higher error rates than single-step attacks, but transfer at lower rates (Kurakin et al., 2017a;b).", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "title", + "bbox": [ + 106, + 688, + 476, + 700 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 478, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 478, + 701 + ], + "score": 1.0, + "content": "3.3 A DEGENERATE GLOBAL MINIMUM FOR SINGLE-STEP ADVERSARIAL TRAINING", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 502, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "score": 1.0, + "content": "When performing adversarial training with a single-step attack (e.g., the FGSM or Step-LL methods", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "score": 1.0, + "content": "above), we approximate Equation (1) by replacing the solution to the inner maximization problem", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "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": "title", + "bbox": [ + 106, + 81, + 354, + 94 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 355, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 355, + 96 + ], + "score": 1.0, + "content": "3 THE ADVERSARIAL TRAINING FRAMEWORK", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 104, + 505, + 142 + ], + "lines": [ + { + "bbox": [ + 105, + 103, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 279, + 120 + ], + "score": 1.0, + "content": "We consider a classification task with data", + "type": "text" + }, + { + "bbox": [ + 280, + 104, + 325, + 118 + ], + "score": 0.94, + "content": "x \\in [ 0 , 1 ] ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 103, + 369, + 120 + ], + "score": 1.0, + "content": "and labels", + "type": "text" + }, + { + "bbox": [ + 369, + 106, + 412, + 118 + ], + "score": 0.92, + "content": "y _ { \\mathrm { t r u e } } \\in \\mathbb { Z } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 103, + 505, + 120 + ], + "score": 1.0, + "content": "sampled from a distri-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 134, + 129 + ], + "score": 1.0, + "content": "bution", + "type": "text" + }, + { + "bbox": [ + 134, + 118, + 143, + 127 + ], + "score": 0.76, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 117, + 307, + 129 + ], + "score": 1.0, + "content": ". 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We consider the well-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 198, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 171, + 213 + ], + "score": 1.0, + "content": "studied class of", + "type": "text" + }, + { + "bbox": [ + 171, + 200, + 185, + 211 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 198, + 506, + 213 + ], + "score": 1.0, + "content": "bounded adversaries (Goodfellow et al., 2014b; Madry et al., 2017) that, given", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 209, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 104, + 209, + 160, + 226 + ], + "score": 1.0, + "content": "some budget", + "type": "text" + }, + { + "bbox": [ + 160, + 214, + 166, + 222 + ], + "score": 0.51, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 209, + 239, + 226 + ], + "score": 1.0, + "content": ", output examples", + "type": "text" + }, + { + "bbox": [ + 239, + 211, + 256, + 222 + ], + "score": 0.87, + "content": "x ^ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 209, + 285, + 226 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 286, + 211, + 360, + 224 + ], + "score": 0.92, + "content": "\\| x ^ { \\mathrm { a d v } } - x \\| _ { \\infty } \\leq \\epsilon .", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 209, + 506, + 226 + ], + "score": 1.0, + "content": ". As we comment in Appendix C.1,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 222, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 107, + 223, + 120, + 235 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 222, + 506, + 236 + ], + "score": 1.0, + "content": "robustness is of course not an end-goal for secure ML. We use this standard model to showcase", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 234, + 471, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 471, + 247 + ], + "score": 1.0, + "content": "limitations of prior adversarial training methods, and evaluate our proposed improvements.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5, + "bbox_fs": [ + 104, + 175, + 506, + 247 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 250, + 504, + 307 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 505, + 263 + ], + "score": 1.0, + "content": "We distinguish between white-box adversaries that have access to the target model’s parameters", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 261, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 127, + 275 + ], + "score": 1.0, + "content": "(i.e.,", + "type": "text" + }, + { + "bbox": [ + 127, + 263, + 135, + 272 + ], + "score": 0.66, + "content": "h _ { . }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 261, + 505, + 275 + ], + "score": 1.0, + "content": "), and black-box adversaries with only partial information about the model’s inner workings.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 272, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 506, + 286 + ], + "score": 1.0, + "content": "Formal definitions for these adversaries are in Appendix A. Although security against white-box", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 283, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 505, + 297 + ], + "score": 1.0, + "content": "attacks is the stronger notion (and the one we ideally want ML models to achieve), black-box security", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 295, + 368, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 368, + 307 + ], + "score": 1.0, + "content": "is a reasonable and more tractable goal for deployed ML models.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 250, + 506, + 307 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 319, + 241, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 242, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 242, + 332 + ], + "score": 1.0, + "content": "3.2 ADVERSARIAL TRAINING", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 105, + 340, + 503, + 363 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "Following Madry et al. (2017), we consider an adversarial variant of standard Empirical Risk Mini-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 351, + 437, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 437, + 364 + ], + "score": 1.0, + "content": "mization (ERM), where our aim is to minimize the risk over adversarial examples:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 106, + 340, + 505, + 364 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 367, + 426, + 393 + ], + "lines": [ + { + "bbox": [ + 185, + 367, + 426, + 393 + ], + "spans": [ + { + "bbox": [ + 185, + 367, + 426, + 393 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { h ^ { * } = \\underset { h \\in \\mathcal { H } } { \\arg \\operatorname* { m i n } } } & { \\underset { ( x , y _ { \\mathrm { t u e } } ) \\sim \\mathcal { D } } { \\mathbb { E } } \\left[ \\underset { \\| x ^ { \\mathrm { a d v } } - x \\| _ { \\infty } \\leq \\epsilon } { \\operatorname* { m a x } } L ( h ( x ^ { \\mathrm { a d v } } ) , y _ { \\mathrm { t r u e } } ) \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "bb598728c43a71396f18ddba779ad2fedec19442160579402efc68a250b7ebb2.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 185, + 367, + 426, + 393 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 398, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 106, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "Madry et al. (2017) argue that adversarial training has a natural interpretation in this context, where", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 409, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 505, + 422 + ], + "score": 1.0, + "content": "a given attack (see below) is used to approximate solutions to the inner maximization problem, and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "the outer minimization problem corresponds to training over these examples. Note that the original", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "formulation of adversarial training (Szegedy et al., 2013; Goodfellow et al., 2014b), which we use", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 103, + 440, + 463, + 460 + ], + "spans": [ + { + "bbox": [ + 103, + 440, + 463, + 460 + ], + "score": 1.0, + "content": "in our experiments, trains on both the “clean” examples x and adversarial examples xadv.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22, + "bbox_fs": [ + 103, + 398, + 506, + 460 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 460, + 505, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 411, + 473 + ], + "score": 1.0, + "content": "We consider three algorithms to generate adversarial examples with bounded", + "type": "text" + }, + { + "bbox": [ + 411, + 461, + 425, + 473 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "norm. The first two", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 472, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 505, + 485 + ], + "score": 1.0, + "content": "are single-step (i.e., they require a single gradient computation); the third is iterative—it computes", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 482, + 459, + 498 + ], + "spans": [ + { + "bbox": [ + 104, + 482, + 261, + 498 + ], + "score": 1.0, + "content": "multiple gradient updates. We enforce", + "type": "text" + }, + { + "bbox": [ + 261, + 483, + 315, + 497 + ], + "score": 0.94, + "content": "x ^ { \\mathrm { a d v } } \\in [ 0 , 1 ] ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 482, + 438, + 498 + ], + "score": 1.0, + "content": "by clipping all components of", + "type": "text" + }, + { + "bbox": [ + 439, + 483, + 455, + 495 + ], + "score": 0.9, + "content": "x ^ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 482, + 459, + 498 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 460, + 505, + 498 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 501, + 504, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 515 + ], + "score": 1.0, + "content": "Fast Gradient Sign Method (FGSM). This method (Goodfellow et al., 2014b) linearizes the inner", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 513, + 227, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 227, + 525 + ], + "score": 1.0, + "content": "maximization problem in (1):", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 500, + 506, + 525 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 529, + 394, + 545 + ], + "lines": [ + { + "bbox": [ + 216, + 529, + 394, + 545 + ], + "spans": [ + { + "bbox": [ + 216, + 529, + 394, + 545 + ], + "score": 0.9, + "content": "x _ { \\mathrm { F G S M } } ^ { \\mathrm { a d v } } : = x + \\varepsilon \\cdot \\mathrm { s i g n } \\left( \\nabla _ { x } L ( h ( x ) , y _ { \\mathrm { t r u e } } ) \\right) \\ .", + "type": "interline_equation", + "image_path": "c985f58e7b152bf6835de024c641b20075bd33f15d6b58c10d550c3be6c5e20d.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 216, + 529, + 394, + 545 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 554, + 506, + 578 + ], + "lines": [ + { + "bbox": [ + 106, + 554, + 504, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 504, + 567 + ], + "score": 1.0, + "content": "Single-Step Least-Likely Class Method (Step-LL). This variant of FGSM introduced by Kurakin", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 565, + 379, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 284, + 580 + ], + "score": 1.0, + "content": "et al. (2017a;b) targets the least-likely class,", + "type": "text" + }, + { + "bbox": [ + 284, + 566, + 374, + 579 + ], + "score": 0.92, + "content": "y _ { \\mathrm { L L } } = \\arg \\operatorname* { m i n } \\{ h ( x ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 565, + 379, + 580 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 554, + 504, + 580 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 583, + 388, + 599 + ], + "lines": [ + { + "bbox": [ + 222, + 583, + 388, + 599 + ], + "spans": [ + { + "bbox": [ + 222, + 583, + 388, + 599 + ], + "score": 0.92, + "content": "x _ { \\mathrm { L L } } ^ { \\mathrm { a d v } } : = x - \\varepsilon \\cdot \\mathrm { s i g n } \\left( \\nabla _ { x } L ( h ( x ) , y _ { \\mathrm { L L } } ) \\right) .", + "type": "interline_equation", + "image_path": "78115ccdc8738100b4286bfb16132885a2adce33cb9d8c6acf798bc0c8604d96.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 222, + 583, + 388, + 599 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "Although this attack only indirectly tackles the inner maximization in (1), Kurakin et al. (2017b)", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 614, + 378, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 378, + 627 + ], + "score": 1.0, + "content": "find it to be the most effective for adversarial training on ImageNet.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 106, + 603, + 505, + 627 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 631, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 504, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 496, + 644 + ], + "score": 1.0, + "content": "Iterative Attack (I-FGSM or Iter-LL). This method iteratively applies the FGSM or Step-LL", + "type": "text" + }, + { + "bbox": [ + 497, + 631, + 504, + 641 + ], + "score": 0.72, + "content": "k", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 189, + 655 + ], + "score": 1.0, + "content": "times with step-size", + "type": "text" + }, + { + "bbox": [ + 190, + 642, + 227, + 654 + ], + "score": 0.93, + "content": "\\alpha \\geq \\epsilon / k", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 643, + 356, + 655 + ], + "score": 1.0, + "content": "and projects each step onto the", + "type": "text" + }, + { + "bbox": [ + 357, + 642, + 371, + 654 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 643, + 425, + 655 + ], + "score": 1.0, + "content": "ball of norm", + "type": "text" + }, + { + "bbox": [ + 426, + 645, + 432, + 652 + ], + "score": 0.53, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 643, + 463, + 655 + ], + "score": 1.0, + "content": "around", + "type": "text" + }, + { + "bbox": [ + 464, + 644, + 470, + 652 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 643, + 505, + 655 + ], + "score": 1.0, + "content": ". It uses", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 653, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 398, + 666 + ], + "score": 1.0, + "content": "projected gradient descent to solve the maximization in (1). For fixed", + "type": "text" + }, + { + "bbox": [ + 398, + 655, + 404, + 663 + ], + "score": 0.5, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 653, + 506, + 666 + ], + "score": 1.0, + "content": ", iterative attacks induce", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 664, + 484, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 484, + 677 + ], + "score": 1.0, + "content": "higher error rates than single-step attacks, but transfer at lower rates (Kurakin et al., 2017a;b).", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 630, + 506, + 677 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 688, + 476, + 700 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 478, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 478, + 701 + ], + "score": 1.0, + "content": "3.3 A DEGENERATE GLOBAL MINIMUM FOR SINGLE-STEP ADVERSARIAL TRAINING", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 502, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "score": 1.0, + "content": "When performing adversarial training with a single-step attack (e.g., the FGSM or Step-LL methods", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "score": 1.0, + "content": "above), we approximate Equation (1) by replacing the solution to the inner maximization problem", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5, + "bbox_fs": [ + 106, + 709, + 504, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 428, + 95 + ], + "lines": [ + { + "bbox": [ + 103, + 77, + 430, + 99 + ], + "spans": [ + { + "bbox": [ + 103, + 77, + 299, + 99 + ], + "score": 1.0, + "content": "in with the output of the single-step attack (e.g.,", + "type": "text" + }, + { + "bbox": [ + 300, + 81, + 326, + 95 + ], + "score": 0.89, + "content": "x _ { \\mathrm { F G S M } } ^ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 77, + 430, + 99 + ], + "score": 1.0, + "content": "in (2)). That is, we solve", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 96, + 404, + 120 + ], + "lines": [ + { + "bbox": [ + 206, + 96, + 404, + 120 + ], + "spans": [ + { + "bbox": [ + 206, + 96, + 404, + 120 + ], + "score": 0.92, + "content": "h ^ { * } = \\underset { h \\in \\mathcal { H } } { \\arg \\operatorname* { m i n } } \\quad \\underset { ( x , y _ { \\mathrm { t u c } } ) \\sim \\mathcal { D } } { \\mathbb { E } } \\left[ L \\big ( h ( x _ { \\mathrm { F G S M } } ^ { \\mathrm { a d v } } ) , y _ { \\mathrm { t r u e } } \\big ) \\right] .", + "type": "interline_equation", + "image_path": "4820e53bab307eb015554db789d1a935d2dd34d2e5e80b5fc3db9786195099bd.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 206, + 96, + 404, + 120 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 122, + 504, + 145 + ], + "lines": [ + { + "bbox": [ + 105, + 121, + 506, + 136 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 186, + 136 + ], + "score": 1.0, + "content": "For model families", + "type": "text" + }, + { + "bbox": [ + 186, + 123, + 196, + 133 + ], + "score": 0.81, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 121, + 506, + 136 + ], + "score": 1.0, + "content": "with high expressive power, this alternative optimization problem admits at", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 133, + 310, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 294, + 145 + ], + "score": 1.0, + "content": "least two substantially different global minima", + "type": "text" + }, + { + "bbox": [ + 294, + 133, + 306, + 144 + ], + "score": 0.88, + "content": "h ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 133, + 310, + 145 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 498, + 167 + ], + "lines": [ + { + "bbox": [ + 102, + 150, + 500, + 171 + ], + "spans": [ + { + "bbox": [ + 102, + 150, + 165, + 171 + ], + "score": 1.0, + "content": "• For an input", + "type": "text" + }, + { + "bbox": [ + 165, + 158, + 172, + 165 + ], + "score": 0.77, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 150, + 195, + 171 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 195, + 156, + 204, + 165 + ], + "score": 0.83, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 150, + 252, + 171 + ], + "score": 1.0, + "content": ", there is no", + "type": "text" + }, + { + "bbox": [ + 252, + 154, + 269, + 165 + ], + "score": 0.87, + "content": "x ^ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 150, + 303, + 171 + ], + "score": 1.0, + "content": "close to", + "type": "text" + }, + { + "bbox": [ + 304, + 158, + 311, + 165 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 150, + 325, + 171 + ], + "score": 1.0, + "content": "(in", + "type": "text" + }, + { + "bbox": [ + 325, + 156, + 339, + 167 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 150, + 500, + 171 + ], + "score": 1.0, + "content": "norm) that induces a high loss. That is,", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 170, + 429, + 193 + ], + "lines": [ + { + "bbox": [ + 189, + 170, + 429, + 193 + ], + "spans": [ + { + "bbox": [ + 189, + 170, + 429, + 193 + ], + "score": 0.92, + "content": "L ( h ^ { * } ( x _ { \\mathrm { F G S M } } ^ { \\mathrm { a d v } } ) , y _ { \\mathrm { t r u e } } ) \\approx \\operatorname* { m a x } _ { \\| x ^ { \\mathrm { a d v } } - x \\| _ { \\infty } \\leq \\epsilon } L ( h ^ { * } ( x ^ { \\mathrm { a d v } } ) , y _ { \\mathrm { t r u e } } ) ] \\approx 0 .", + "type": "interline_equation", + "image_path": "164425822e9912f819ca2d3e3b69e9e9185ac2681b7dd7636c65d3b1f8703f55.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 189, + 170, + 429, + 193 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 112, + 196, + 357, + 208 + ], + "lines": [ + { + "bbox": [ + 113, + 195, + 358, + 210 + ], + "spans": [ + { + "bbox": [ + 113, + 195, + 176, + 210 + ], + "score": 1.0, + "content": "In other words,", + "type": "text" + }, + { + "bbox": [ + 177, + 196, + 188, + 207 + ], + "score": 0.88, + "content": "h ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 195, + 248, + 210 + ], + "score": 1.0, + "content": "is robust to all", + "type": "text" + }, + { + "bbox": [ + 249, + 197, + 262, + 208 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 195, + 358, + 210 + ], + "score": 1.0, + "content": "bounded perturbations.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 211, + 504, + 234 + ], + "lines": [ + { + "bbox": [ + 103, + 208, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 103, + 208, + 177, + 225 + ], + "score": 1.0, + "content": "• The minimizer", + "type": "text" + }, + { + "bbox": [ + 178, + 210, + 189, + 221 + ], + "score": 0.87, + "content": "h ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 208, + 505, + 225 + ], + "score": 1.0, + "content": "is a model for which the approximation method underlying the attack (i.e.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 113, + 222, + 396, + 235 + ], + "spans": [ + { + "bbox": [ + 113, + 222, + 396, + 235 + ], + "score": 1.0, + "content": "linearization in our case) poorly fits the model’s loss function. That is,", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 236, + 421, + 259 + ], + "lines": [ + { + "bbox": [ + 197, + 236, + 421, + 259 + ], + "spans": [ + { + "bbox": [ + 197, + 236, + 421, + 259 + ], + "score": 0.92, + "content": "L ( h ^ { * } ( x _ { \\mathrm { F G S M } } ^ { \\mathrm { a d v } } ) , y _ { \\mathrm { t r u e } } ) \\ll \\operatorname* { m a x } _ { \\| x ^ { \\mathrm { a d v } } - x \\| _ { \\infty } \\le \\epsilon } L ( h ^ { * } ( x ^ { \\mathrm { a d v } } ) , y _ { \\mathrm { t r u e } } ) ] .", + "type": "interline_equation", + "image_path": "439ee11b1fc08260022f56149735ad065009d2f1d712f0090738556183705254.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 197, + 236, + 421, + 259 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 115, + 263, + 450, + 276 + ], + "lines": [ + { + "bbox": [ + 111, + 261, + 452, + 279 + ], + "spans": [ + { + "bbox": [ + 111, + 261, + 243, + 279 + ], + "score": 1.0, + "content": "Thus the attack when applied to", + "type": "text" + }, + { + "bbox": [ + 243, + 264, + 254, + 275 + ], + "score": 0.87, + "content": "h ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 261, + 329, + 279 + ], + "score": 1.0, + "content": "produces samples", + "type": "text" + }, + { + "bbox": [ + 329, + 263, + 347, + 275 + ], + "score": 0.89, + "content": "x ^ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 261, + 452, + 279 + ], + "score": 1.0, + "content": "that are far from optimal.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 285, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 284, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 299 + ], + "score": 1.0, + "content": "Note that this second “degenerate” minimum can be more subtle than a simple case of overfitting to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 297, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 308 + ], + "score": 1.0, + "content": "samples produced from single-step attacks. 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This phenomenon", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "relates to the notion of Reward Hacking (Amodei et al., 2016) wherein an agent maximizes its", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 350, + 494, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 494, + 363 + ], + "score": 1.0, + "content": "formal objective function via unintended behavior that fails to captures the designer’s true intent.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 375, + 290, + 386 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 291, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 291, + 388 + ], + "score": 1.0, + "content": "3.4 ENSEMBLE ADVERSARIAL TRAINING", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 395, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "The degenerate minimum described in Section 3.3 is attainable because the learned model’s param-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "eters influence the quality of both the minimization and maximization in (1). One solution is to use", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "a stronger adversarial example generation process, at a high performance cost (Madry et al., 2017).", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "Alternatively, Baluja & Fischer (2017) suggest training an adversarial generator model as in the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "GAN framework (Goodfellow et al., 2014a). The power of this generator is likely to require careful", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 451, + 504, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 504, + 462 + ], + "score": 1.0, + "content": "tuning, to avoid similar degenerate minima (where the generator or classifier overpowers the other).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 467, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "We propose a conceptually simpler approach to decouple the generation of adversarial examples", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "from the model being trained, while simultaneously drawing an explicit connection with robustness", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "to black-box adversaries. Our method, which we call Ensemble Adversarial Training, augments a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 501, + 504, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 504, + 512 + ], + "score": 1.0, + "content": "model’s training data with adversarial examples crafted on other static pre-trained models. Intu-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "itively, as adversarial examples transfer between models, perturbations crafted on an external model", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "are good approximations for the maximization problem in (1). Moreover, the learned model can", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "not influence the “strength” of these adversarial examples. As a result, minimizing the training loss", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 545, + 409, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 409, + 556 + ], + "score": 1.0, + "content": "implies increased robustness to black-box attacks from some set of models.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 567, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "Domain Adaptation with multiple sources. We can draw a connection between Ensemble Adver-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "sarial Training and multiple-source Domain Adaptation (Mansour et al., 2009; Zhang et al., 2012).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 590, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 601 + ], + "score": 1.0, + "content": "In Domain Adaptation, a model trained on data sampled from one or more source distributions", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 600, + 410, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 151, + 612 + ], + "score": 0.93, + "content": "\\boldsymbol { S } _ { 1 } , \\ldots , \\boldsymbol { S } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 600, + 249, + 613 + ], + "score": 1.0, + "content": "is evaluated on samples", + "type": "text" + }, + { + "bbox": [ + 249, + 602, + 256, + 610 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 600, + 397, + 613 + ], + "score": 1.0, + "content": "from a different target distribution", + "type": "text" + }, + { + "bbox": [ + 397, + 601, + 406, + 610 + ], + "score": 0.8, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 600, + 410, + 613 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 106, + 616, + 505, + 685 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 123, + 631 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 617, + 135, + 629 + ], + "score": 0.89, + "content": "A _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 616, + 347, + 631 + ], + "score": 1.0, + "content": "be an adversarial distribution obtained by sampling", + "type": "text" + }, + { + "bbox": [ + 347, + 617, + 381, + 629 + ], + "score": 0.9, + "content": "( x , y _ { \\mathrm { t r u e } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 616, + 406, + 631 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 406, + 618, + 415, + 627 + ], + "score": 0.75, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 616, + 506, + 631 + ], + "score": 1.0, + "content": ", computing an adver-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 102, + 623, + 508, + 646 + ], + "spans": [ + { + "bbox": [ + 102, + 623, + 168, + 646 + ], + "score": 1.0, + "content": "sarial example", + "type": "text" + }, + { + "bbox": [ + 169, + 628, + 186, + 640 + ], + "score": 0.88, + "content": "x ^ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 623, + 297, + 646 + ], + "score": 1.0, + "content": "for some model such that", + "type": "text" + }, + { + "bbox": [ + 298, + 629, + 374, + 642 + ], + "score": 0.92, + "content": "\\| x ^ { \\mathrm { a d v } } - x \\| _ { \\infty } \\leq \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 623, + 441, + 646 + ], + "score": 1.0, + "content": ", and outputting", + "type": "text" + }, + { + "bbox": [ + 442, + 629, + 487, + 642 + ], + "score": 0.89, + "content": "( x ^ { \\mathrm { a d v } } , y _ { \\mathrm { t r u e } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 623, + 508, + 646 + ], + "score": 1.0, + "content": ". In", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 346, + 654 + ], + "score": 1.0, + "content": "Ensemble Adversarial Training, the source distributions are", + "type": "text" + }, + { + "bbox": [ + 346, + 642, + 355, + 651 + ], + "score": 0.78, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 641, + 437, + 654 + ], + "score": 1.0, + "content": "(the clean data) and", + "type": "text" + }, + { + "bbox": [ + 437, + 642, + 486, + 653 + ], + "score": 0.9, + "content": "\\mathcal { A } _ { 1 } , \\ldots , \\mathcal { A } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "(the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 652, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 505, + 664 + ], + "score": 1.0, + "content": "attacks overs the currently trained model and the static pre-trained models). The target distribution", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 663, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 663, + 306, + 676 + ], + "score": 1.0, + "content": "takes the form of an unseen black-box adversary", + "type": "text" + }, + { + "bbox": [ + 307, + 663, + 320, + 673 + ], + "score": 0.88, + "content": "\\mathcal { A } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 663, + 505, + 676 + ], + "score": 1.0, + "content": ". Standard generalization bounds for Domain", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 674, + 427, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 674, + 427, + 687 + ], + "score": 1.0, + "content": "Adaptation (Mansour et al., 2009; Zhang et al., 2012) yield the following result.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 686, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 220, + 699 + ], + "score": 1.0, + "content": "Theorem 1 (informal). Let", + "type": "text" + }, + { + "bbox": [ + 221, + 687, + 255, + 698 + ], + "score": 0.9, + "content": "h ^ { \\ast } \\in \\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 686, + 505, + 699 + ], + "score": 1.0, + "content": "be a model learned with Ensemble Adversarial Training and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 223, + 711 + ], + "score": 1.0, + "content": "static black-box adversaries", + "type": "text" + }, + { + "bbox": [ + 224, + 699, + 272, + 710 + ], + "score": 0.9, + "content": "\\mathcal { A } _ { 1 } , \\ldots , \\mathcal { A } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 699, + 313, + 711 + ], + "score": 1.0, + "content": ". 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That is,", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 102, + 150, + 500, + 171 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 170, + 429, + 193 + ], + "lines": [ + { + "bbox": [ + 189, + 170, + 429, + 193 + ], + "spans": [ + { + "bbox": [ + 189, + 170, + 429, + 193 + ], + "score": 0.92, + "content": "L ( h ^ { * } ( x _ { \\mathrm { F G S M } } ^ { \\mathrm { a d v } } ) , y _ { \\mathrm { t r u e } } ) \\approx \\operatorname* { m a x } _ { \\| x ^ { \\mathrm { a d v } } - x \\| _ { \\infty } \\leq \\epsilon } L ( h ^ { * } ( x ^ { \\mathrm { a d v } } ) , y _ { \\mathrm { t r u e } } ) ] \\approx 0 .", + "type": "interline_equation", + "image_path": "164425822e9912f819ca2d3e3b69e9e9185ac2681b7dd7636c65d3b1f8703f55.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 189, + 170, + 429, + 193 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 112, + 196, + 357, + 208 + ], + "lines": [ + { + "bbox": [ + 113, + 195, + 358, + 210 + ], + "spans": [ + { + "bbox": [ + 113, + 195, + 176, + 210 + ], + "score": 1.0, + "content": "In other words,", + "type": "text" + }, + { + "bbox": [ + 177, + 196, + 188, + 207 + ], + "score": 0.88, + "content": "h ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 195, + 248, + 210 + ], + "score": 1.0, + "content": "is robust to all", + "type": "text" + }, + { + "bbox": [ + 249, + 197, + 262, + 208 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 195, + 358, + 210 + ], + "score": 1.0, + "content": "bounded perturbations.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 113, + 195, + 358, + 210 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 211, + 504, + 234 + ], + "lines": [ + { + "bbox": [ + 103, + 208, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 103, + 208, + 177, + 225 + ], + "score": 1.0, + "content": "• The minimizer", + "type": "text" + }, + { + "bbox": [ + 178, + 210, + 189, + 221 + ], + "score": 0.87, + "content": "h ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 208, + 505, + 225 + ], + "score": 1.0, + "content": "is a model for which the approximation method underlying the attack (i.e.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 113, + 222, + 396, + 235 + ], + "spans": [ + { + "bbox": [ + 113, + 222, + 396, + 235 + ], + "score": 1.0, + "content": "linearization in our case) poorly fits the model’s loss function. 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Indeed, we show in Section 4.1 that single-step attacks", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 306, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 506, + 320 + ], + "score": 1.0, + "content": "applied to adversarially trained models create “adversarial” examples that are easy to classify even", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 318, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 104, + 318, + 506, + 331 + ], + "score": 1.0, + "content": "for undefended models. Thus, adversarial training does not simply learn to resist the particular attack", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "used during training, but actually to make that attack perform worse overall. This phenomenon", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "relates to the notion of Reward Hacking (Amodei et al., 2016) wherein an agent maximizes its", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 350, + 494, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 494, + 363 + ], + "score": 1.0, + "content": "formal objective function via unintended behavior that fails to captures the designer’s true intent.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14, + "bbox_fs": [ + 104, + 284, + 506, + 363 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 375, + 290, + 386 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 291, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 291, + 388 + ], + "score": 1.0, + "content": "3.4 ENSEMBLE ADVERSARIAL TRAINING", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 395, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "The degenerate minimum described in Section 3.3 is attainable because the learned model’s param-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "eters influence the quality of both the minimization and maximization in (1). One solution is to use", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "a stronger adversarial example generation process, at a high performance cost (Madry et al., 2017).", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "Alternatively, Baluja & Fischer (2017) suggest training an adversarial generator model as in the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "GAN framework (Goodfellow et al., 2014a). The power of this generator is likely to require careful", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 451, + 504, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 504, + 462 + ], + "score": 1.0, + "content": "tuning, to avoid similar degenerate minima (where the generator or classifier overpowers the other).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 395, + 506, + 462 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 467, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "We propose a conceptually simpler approach to decouple the generation of adversarial examples", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "from the model being trained, while simultaneously drawing an explicit connection with robustness", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "to black-box adversaries. Our method, which we call Ensemble Adversarial Training, augments a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 501, + 504, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 504, + 512 + ], + "score": 1.0, + "content": "model’s training data with adversarial examples crafted on other static pre-trained models. Intu-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "itively, as adversarial examples transfer between models, perturbations crafted on an external model", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "are good approximations for the maximization problem in (1). Moreover, the learned model can", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "not influence the “strength” of these adversarial examples. As a result, minimizing the training loss", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 545, + 409, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 409, + 556 + ], + "score": 1.0, + "content": "implies increased robustness to black-box attacks from some set of models.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 467, + 506, + 556 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 567, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "Domain Adaptation with multiple sources. We can draw a connection between Ensemble Adver-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "sarial Training and multiple-source Domain Adaptation (Mansour et al., 2009; Zhang et al., 2012).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 590, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 601 + ], + "score": 1.0, + "content": "In Domain Adaptation, a model trained on data sampled from one or more source distributions", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 600, + 410, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 151, + 612 + ], + "score": 0.93, + "content": "\\boldsymbol { S } _ { 1 } , \\ldots , \\boldsymbol { S } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 600, + 249, + 613 + ], + "score": 1.0, + "content": "is evaluated on samples", + "type": "text" + }, + { + "bbox": [ + 249, + 602, + 256, + 610 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 600, + 397, + 613 + ], + "score": 1.0, + "content": "from a different target distribution", + "type": "text" + }, + { + "bbox": [ + 397, + 601, + 406, + 610 + ], + "score": 0.8, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 600, + 410, + 613 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 567, + 505, + 613 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 616, + 505, + 685 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 123, + 631 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 617, + 135, + 629 + ], + "score": 0.89, + "content": "A _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 616, + 347, + 631 + ], + "score": 1.0, + "content": "be an adversarial distribution obtained by sampling", + "type": "text" + }, + { + "bbox": [ + 347, + 617, + 381, + 629 + ], + "score": 0.9, + "content": "( x , y _ { \\mathrm { t r u e } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 616, + 406, + 631 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 406, + 618, + 415, + 627 + ], + "score": 0.75, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 616, + 506, + 631 + ], + "score": 1.0, + "content": ", computing an adver-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 102, + 623, + 508, + 646 + ], + "spans": [ + { + "bbox": [ + 102, + 623, + 168, + 646 + ], + "score": 1.0, + "content": "sarial example", + "type": "text" + }, + { + "bbox": [ + 169, + 628, + 186, + 640 + ], + "score": 0.88, + "content": "x ^ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 623, + 297, + 646 + ], + "score": 1.0, + "content": "for some model such that", + "type": "text" + }, + { + "bbox": [ + 298, + 629, + 374, + 642 + ], + "score": 0.92, + "content": "\\| x ^ { \\mathrm { a d v } } - x \\| _ { \\infty } \\leq \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 623, + 441, + 646 + ], + "score": 1.0, + "content": ", and outputting", + "type": "text" + }, + { + "bbox": [ + 442, + 629, + 487, + 642 + ], + "score": 0.89, + "content": "( x ^ { \\mathrm { a d v } } , y _ { \\mathrm { t r u e } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 623, + 508, + 646 + ], + "score": 1.0, + "content": ". In", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 346, + 654 + ], + "score": 1.0, + "content": "Ensemble Adversarial Training, the source distributions are", + "type": "text" + }, + { + "bbox": [ + 346, + 642, + 355, + 651 + ], + "score": 0.78, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 641, + 437, + 654 + ], + "score": 1.0, + "content": "(the clean data) and", + "type": "text" + }, + { + "bbox": [ + 437, + 642, + 486, + 653 + ], + "score": 0.9, + "content": "\\mathcal { A } _ { 1 } , \\ldots , \\mathcal { A } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "(the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 652, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 505, + 664 + ], + "score": 1.0, + "content": "attacks overs the currently trained model and the static pre-trained models). The target distribution", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 663, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 663, + 306, + 676 + ], + "score": 1.0, + "content": "takes the form of an unseen black-box adversary", + "type": "text" + }, + { + "bbox": [ + 307, + 663, + 320, + 673 + ], + "score": 0.88, + "content": "\\mathcal { A } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 663, + 505, + 676 + ], + "score": 1.0, + "content": ". Standard generalization bounds for Domain", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 674, + 427, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 674, + 427, + 687 + ], + "score": 1.0, + "content": "Adaptation (Mansour et al., 2009; Zhang et al., 2012) yield the following result.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5, + "bbox_fs": [ + 102, + 616, + 508, + 687 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 686, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 220, + 699 + ], + "score": 1.0, + "content": "Theorem 1 (informal). Let", + "type": "text" + }, + { + "bbox": [ + 221, + 687, + 255, + 698 + ], + "score": 0.9, + "content": "h ^ { \\ast } \\in \\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 686, + 505, + 699 + ], + "score": 1.0, + "content": "be a model learned with Ensemble Adversarial Training and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 223, + 711 + ], + "score": 1.0, + "content": "static black-box adversaries", + "type": "text" + }, + { + "bbox": [ + 224, + 699, + 272, + 710 + ], + "score": 0.9, + "content": "\\mathcal { A } _ { 1 } , \\ldots , \\mathcal { A } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 699, + 313, + 711 + ], + "score": 1.0, + "content": ". 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We plot the loss of model", + "type": "text" + }, + { + "bbox": [ + 482, + 237, + 504, + 249 + ], + "score": 0.88, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 248, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 145, + 262 + ], + "score": 1.0, + "content": "on points", + "type": "text" + }, + { + "bbox": [ + 145, + 248, + 239, + 261 + ], + "score": 0.91, + "content": "x ^ { * } = x + \\epsilon _ { 1 } \\cdot g + \\epsilon _ { 2 } \\cdot g ^ { \\perp }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 249, + 269, + 262 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 269, + 251, + 276, + 261 + ], + "score": 0.78, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 249, + 379, + 262 + ], + "score": 1.0, + "content": "is the signed gradient and", + "type": "text" + }, + { + "bbox": [ + 379, + 248, + 392, + 261 + ], + "score": 0.91, + "content": "g ^ { \\perp }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 249, + 506, + 262 + ], + "score": 1.0, + "content": "is an orthogonal adversarial", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 259, + 476, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 264, + 273 + ], + "score": 1.0, + "content": "direction. Plot (b) is a zoom of (a) near", + "type": "text" + }, + { + "bbox": [ + 265, + 263, + 272, + 270 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 259, + 476, + 273 + ], + "score": 1.0, + "content": ". The gradient poorly approximates the global loss.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 285, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 381, + 299 + ], + "score": 1.0, + "content": "We give a formal statement of this result and of the assumptions on", + "type": "text" + }, + { + "bbox": [ + 381, + 286, + 394, + 296 + ], + "score": 0.88, + "content": "\\mathcal { A } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "in Appendix B. Of course,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "ideally we would like guarantees against arbitrary future adversaries. For very low-dimensional", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "score": 1.0, + "content": "tasks (e.g., MNIST), stronger guarantees are within reach for specific classes of adversaries (e.g.,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 319, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 120, + 330 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 319, + 506, + 331 + ], + "score": 1.0, + "content": "bounded perturbations (Madry et al., 2017; Sinha et al., 2018; Raghunathan et al., 2018; Kolter", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "& Wong, 2017)), yet they also fail to extend to other adversaries not considered at training time (see", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "score": 1.0, + "content": "Appendix C.1 for a discussion). For ImageNet-scale tasks, stronger formal guarantees appear out of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "reach, and we thus resort to an experimental assessment of the robustness of Ensemble Adversarially", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 363, + 424, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 424, + 374 + ], + "score": 1.0, + "content": "Trained models to various non-interactive black-box adversaries in Section 4.2.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 108, + 392, + 200, + 404 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 201, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 201, + 406 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 417, + 504, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "We show the existence of a degenerate minimum, as described in Section 3.3, for the adversar-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 458, + 442 + ], + "score": 1.0, + "content": "ially trained Inception v3 model of Kurakin et al. (2017b). Their model (denoted", + "type": "text" + }, + { + "bbox": [ + 459, + 429, + 482, + 440 + ], + "score": 0.88, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } , }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 428, + 506, + 442 + ], + "score": 1.0, + "content": ") was", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 245, + 452 + ], + "score": 1.0, + "content": "trained on a Step-LL attack with", + "type": "text" + }, + { + "bbox": [ + 245, + 440, + 298, + 451 + ], + "score": 0.86, + "content": "\\epsilon \\leq 1 6 / 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 439, + 505, + 452 + ], + "score": 1.0, + "content": ". We also adversarially train an Inception ResNet", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 449, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 431, + 465 + ], + "score": 1.0, + "content": "v2 model (Szegedy et al., 2016a) using the same setup. We denote this model by", + "type": "text" + }, + { + "bbox": [ + 432, + 451, + 463, + 462 + ], + "score": 0.89, + "content": "\\mathrm { I R v } 2 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 449, + 506, + 465 + ], + "score": 1.0, + "content": ". We refer", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 462, + 446, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 446, + 474 + ], + "score": 1.0, + "content": "the reader to (Kurakin et al., 2017b) for details on the adversarial training procedure.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 505, + 600 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "We first measure the approximation-ratio of the Step-LL attack for the inner maximization in (1).", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "As we do not know the true maximum, we lower-bound it using an iterative attack. For 1,000", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 500, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 472, + 512 + ], + "score": 1.0, + "content": "random test points, we find that for a standard Inception v3 model, step-LL gets within", + "type": "text" + }, + { + "bbox": [ + 472, + 500, + 492, + 511 + ], + "score": 0.87, + "content": "1 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 501, + 506, + 512 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "score": 1.0, + "content": "the optimum loss on average. This attack is thus a good candidate for adversarial training. Yet,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 104, + 522, + 135, + 535 + ], + "score": 1.0, + "content": "for the", + "type": "text" + }, + { + "bbox": [ + 136, + 523, + 158, + 534 + ], + "score": 0.9, + "content": "{ \\bf V } { \\boldsymbol 3 } _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 522, + 321, + 535 + ], + "score": 1.0, + "content": "model, the approximation ratio drops to", + "type": "text" + }, + { + "bbox": [ + 321, + 522, + 335, + 533 + ], + "score": 0.86, + "content": "7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 522, + 506, + 535 + ], + "score": 1.0, + "content": ", confirming that the learned model is less", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "amenable to linearization. We obtain similar results for Inception ResNet v2 models. The ratio is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 543, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 126, + 555 + ], + "score": 0.88, + "content": "1 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 543, + 232, + 557 + ], + "score": 1.0, + "content": "for a standard model, and", + "type": "text" + }, + { + "bbox": [ + 232, + 544, + 247, + 555 + ], + "score": 0.87, + "content": "8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 543, + 262, + 557 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 263, + 545, + 294, + 556 + ], + "score": 0.9, + "content": "\\mathrm { I R v } 2 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 543, + 506, + 557 + ], + "score": 1.0, + "content": ". Similarly, we look at the cosine similarity between", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "the perturbations given by a single-step and multi-step attack. The more linear the model, the more", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "similar we expect both perturbations to be. The average similarity drops from 0.13 for Inception v3", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 576, + 507, + 591 + ], + "spans": [ + { + "bbox": [ + 104, + 576, + 150, + 591 + ], + "score": 1.0, + "content": "to 0.02 for", + "type": "text" + }, + { + "bbox": [ + 151, + 577, + 172, + 589 + ], + "score": 0.89, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 576, + 365, + 591 + ], + "score": 1.0, + "content": ". This effect is not due to the decision surface of", + "type": "text" + }, + { + "bbox": [ + 365, + 578, + 387, + 588 + ], + "score": 0.9, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 576, + 507, + 591 + ], + "score": 1.0, + "content": "being “too flat” near the data", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 588, + 495, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 291, + 601 + ], + "score": 1.0, + "content": "points: the average gradient norm is larger for", + "type": "text" + }, + { + "bbox": [ + 292, + 588, + 314, + 600 + ], + "score": 0.9, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 588, + 495, + 601 + ], + "score": 1.0, + "content": "(0.17) than for the standard v3 model (0.10).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 504, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 482, + 619 + ], + "score": 1.0, + "content": "We visualize this “gradient-masking” effect (Papernot et al., 2016b) by plotting the loss of", + "type": "text" + }, + { + "bbox": [ + 482, + 605, + 504, + 616 + ], + "score": 0.86, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 615, + 507, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 159, + 630 + ], + "score": 1.0, + "content": "on examples", + "type": "text" + }, + { + "bbox": [ + 160, + 615, + 266, + 628 + ], + "score": 0.93, + "content": "x ^ { * } = \\bar { x } + \\epsilon _ { 1 } \\cdot g + \\epsilon _ { 2 } \\cdot \\bar { g } ^ { ^ { \\perp } }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 615, + 297, + 630 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 298, + 618, + 304, + 628 + ], + "score": 0.8, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 615, + 434, + 630 + ], + "score": 1.0, + "content": "is the signed gradient of model", + "type": "text" + }, + { + "bbox": [ + 434, + 616, + 456, + 627 + ], + "score": 0.9, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 615, + 474, + 630 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 474, + 616, + 487, + 627 + ], + "score": 0.89, + "content": "g ^ { \\bot }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 615, + 507, + 630 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 216, + 639 + ], + "score": 1.0, + "content": "signed vector orthogonal to", + "type": "text" + }, + { + "bbox": [ + 217, + 629, + 223, + 639 + ], + "score": 0.78, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 627, + 428, + 639 + ], + "score": 1.0, + "content": ". Looking forward to Section 4.1, we actually chose", + "type": "text" + }, + { + "bbox": [ + 428, + 627, + 440, + 639 + ], + "score": 0.89, + "content": "g ^ { \\bot }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "to be the signed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 440, + 651 + ], + "score": 1.0, + "content": "gradient of another Inception model, from which adversarial examples transfer to", + "type": "text" + }, + { + "bbox": [ + 440, + 639, + 462, + 649 + ], + "score": 0.9, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 637, + 506, + 651 + ], + "score": 1.0, + "content": ". Figure 1", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 380, + 662 + ], + "score": 1.0, + "content": "shows that the loss is highly curved in the vicinity of the data point", + "type": "text" + }, + { + "bbox": [ + 380, + 651, + 387, + 659 + ], + "score": 0.72, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 649, + 505, + 662 + ], + "score": 1.0, + "content": ", and that the gradient poorly", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 660, + 462, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 462, + 672 + ], + "score": 1.0, + "content": "reflects the global loss landscape. Similar plots for additional data points are in Figure 4.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 107, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 107, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "We show similar results for adversarially trained MNIST models in Appendix C.2. On this task,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "input dropout (Srivastava et al., 2014) mitigates adversarial training’s overfitting problem, in some", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "cases. Presumably, the random input mask diversifies the perturbations seen during training (dropout", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "at intermediate layers does not mitigate the overfitting effect). Mishkin et al. (2017) find that input", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 720, + 499, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 499, + 734 + ], + "score": 1.0, + "content": "dropout significantly degrades accuracy on ImageNet, so we did not include it in our experiments.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 118, + 89, + 468, + 230 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 118, + 89, + 468, + 230 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 118, + 89, + 468, + 230 + ], + "spans": [ + { + "bbox": [ + 118, + 89, + 468, + 230 + ], + "score": 0.969, + "type": "image", + "image_path": "364615ab0fa795bce9de81e64d675fc54bea01c98991140721a6039e35625055.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 118, + 89, + 468, + 136.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 118, + 136.0, + 468, + 183.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 118, + 183.0, + 468, + 230.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 236, + 504, + 272 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 234, + 504, + 251 + ], + "spans": [ + { + "bbox": [ + 104, + 234, + 482, + 251 + ], + "score": 1.0, + "content": "Figure 1: Gradient masking in single-step adversarial training. We plot the loss of model", + "type": "text" + }, + { + "bbox": [ + 482, + 237, + 504, + 249 + ], + "score": 0.88, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 248, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 145, + 262 + ], + "score": 1.0, + "content": "on points", + "type": "text" + }, + { + "bbox": [ + 145, + 248, + 239, + 261 + ], + "score": 0.91, + "content": "x ^ { * } = x + \\epsilon _ { 1 } \\cdot g + \\epsilon _ { 2 } \\cdot g ^ { \\perp }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 249, + 269, + 262 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 269, + 251, + 276, + 261 + ], + "score": 0.78, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 249, + 379, + 262 + ], + "score": 1.0, + "content": "is the signed gradient and", + "type": "text" + }, + { + "bbox": [ + 379, + 248, + 392, + 261 + ], + "score": 0.91, + "content": "g ^ { \\perp }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 249, + 506, + 262 + ], + "score": 1.0, + "content": "is an orthogonal adversarial", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 259, + 476, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 264, + 273 + ], + "score": 1.0, + "content": "direction. Plot (b) is a zoom of (a) near", + "type": "text" + }, + { + "bbox": [ + 265, + 263, + 272, + 270 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 259, + 476, + 273 + ], + "score": 1.0, + "content": ". The gradient poorly approximates the global loss.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 285, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 381, + 299 + ], + "score": 1.0, + "content": "We give a formal statement of this result and of the assumptions on", + "type": "text" + }, + { + "bbox": [ + 381, + 286, + 394, + 296 + ], + "score": 0.88, + "content": "\\mathcal { A } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "in Appendix B. Of course,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "ideally we would like guarantees against arbitrary future adversaries. For very low-dimensional", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "score": 1.0, + "content": "tasks (e.g., MNIST), stronger guarantees are within reach for specific classes of adversaries (e.g.,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 319, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 120, + 330 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 319, + 506, + 331 + ], + "score": 1.0, + "content": "bounded perturbations (Madry et al., 2017; Sinha et al., 2018; Raghunathan et al., 2018; Kolter", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "& Wong, 2017)), yet they also fail to extend to other adversaries not considered at training time (see", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "score": 1.0, + "content": "Appendix C.1 for a discussion). For ImageNet-scale tasks, stronger formal guarantees appear out of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "reach, and we thus resort to an experimental assessment of the robustness of Ensemble Adversarially", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 363, + 424, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 424, + 374 + ], + "score": 1.0, + "content": "Trained models to various non-interactive black-box adversaries in Section 4.2.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 285, + 506, + 374 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 392, + 200, + 404 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 201, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 201, + 406 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 417, + 504, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "We show the existence of a degenerate minimum, as described in Section 3.3, for the adversar-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 458, + 442 + ], + "score": 1.0, + "content": "ially trained Inception v3 model of Kurakin et al. (2017b). Their model (denoted", + "type": "text" + }, + { + "bbox": [ + 459, + 429, + 482, + 440 + ], + "score": 0.88, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } , }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 428, + 506, + 442 + ], + "score": 1.0, + "content": ") was", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 245, + 452 + ], + "score": 1.0, + "content": "trained on a Step-LL attack with", + "type": "text" + }, + { + "bbox": [ + 245, + 440, + 298, + 451 + ], + "score": 0.86, + "content": "\\epsilon \\leq 1 6 / 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 439, + 505, + 452 + ], + "score": 1.0, + "content": ". We also adversarially train an Inception ResNet", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 449, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 431, + 465 + ], + "score": 1.0, + "content": "v2 model (Szegedy et al., 2016a) using the same setup. We denote this model by", + "type": "text" + }, + { + "bbox": [ + 432, + 451, + 463, + 462 + ], + "score": 0.89, + "content": "\\mathrm { I R v } 2 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 449, + 506, + 465 + ], + "score": 1.0, + "content": ". We refer", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 462, + 446, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 446, + 474 + ], + "score": 1.0, + "content": "the reader to (Kurakin et al., 2017b) for details on the adversarial training procedure.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 418, + 506, + 474 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 505, + 600 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "We first measure the approximation-ratio of the Step-LL attack for the inner maximization in (1).", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "As we do not know the true maximum, we lower-bound it using an iterative attack. For 1,000", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 500, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 472, + 512 + ], + "score": 1.0, + "content": "random test points, we find that for a standard Inception v3 model, step-LL gets within", + "type": "text" + }, + { + "bbox": [ + 472, + 500, + 492, + 511 + ], + "score": 0.87, + "content": "1 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 501, + 506, + 512 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "score": 1.0, + "content": "the optimum loss on average. This attack is thus a good candidate for adversarial training. Yet,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 104, + 522, + 135, + 535 + ], + "score": 1.0, + "content": "for the", + "type": "text" + }, + { + "bbox": [ + 136, + 523, + 158, + 534 + ], + "score": 0.9, + "content": "{ \\bf V } { \\boldsymbol 3 } _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 522, + 321, + 535 + ], + "score": 1.0, + "content": "model, the approximation ratio drops to", + "type": "text" + }, + { + "bbox": [ + 321, + 522, + 335, + 533 + ], + "score": 0.86, + "content": "7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 522, + 506, + 535 + ], + "score": 1.0, + "content": ", confirming that the learned model is less", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "amenable to linearization. We obtain similar results for Inception ResNet v2 models. The ratio is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 543, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 126, + 555 + ], + "score": 0.88, + "content": "1 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 543, + 232, + 557 + ], + "score": 1.0, + "content": "for a standard model, and", + "type": "text" + }, + { + "bbox": [ + 232, + 544, + 247, + 555 + ], + "score": 0.87, + "content": "8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 543, + 262, + 557 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 263, + 545, + 294, + 556 + ], + "score": 0.9, + "content": "\\mathrm { I R v } 2 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 543, + 506, + 557 + ], + "score": 1.0, + "content": ". Similarly, we look at the cosine similarity between", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "the perturbations given by a single-step and multi-step attack. The more linear the model, the more", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "similar we expect both perturbations to be. The average similarity drops from 0.13 for Inception v3", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 576, + 507, + 591 + ], + "spans": [ + { + "bbox": [ + 104, + 576, + 150, + 591 + ], + "score": 1.0, + "content": "to 0.02 for", + "type": "text" + }, + { + "bbox": [ + 151, + 577, + 172, + 589 + ], + "score": 0.89, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 576, + 365, + 591 + ], + "score": 1.0, + "content": ". This effect is not due to the decision surface of", + "type": "text" + }, + { + "bbox": [ + 365, + 578, + 387, + 588 + ], + "score": 0.9, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 576, + 507, + 591 + ], + "score": 1.0, + "content": "being “too flat” near the data", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 588, + 495, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 291, + 601 + ], + "score": 1.0, + "content": "points: the average gradient norm is larger for", + "type": "text" + }, + { + "bbox": [ + 292, + 588, + 314, + 600 + ], + "score": 0.9, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 588, + 495, + 601 + ], + "score": 1.0, + "content": "(0.17) than for the standard v3 model (0.10).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 478, + 507, + 601 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 504, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 482, + 619 + ], + "score": 1.0, + "content": "We visualize this “gradient-masking” effect (Papernot et al., 2016b) by plotting the loss of", + "type": "text" + }, + { + "bbox": [ + 482, + 605, + 504, + 616 + ], + "score": 0.86, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 615, + 507, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 159, + 630 + ], + "score": 1.0, + "content": "on examples", + "type": "text" + }, + { + "bbox": [ + 160, + 615, + 266, + 628 + ], + "score": 0.93, + "content": "x ^ { * } = \\bar { x } + \\epsilon _ { 1 } \\cdot g + \\epsilon _ { 2 } \\cdot \\bar { g } ^ { ^ { \\perp } }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 615, + 297, + 630 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 298, + 618, + 304, + 628 + ], + "score": 0.8, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 615, + 434, + 630 + ], + "score": 1.0, + "content": "is the signed gradient of model", + "type": "text" + }, + { + "bbox": [ + 434, + 616, + 456, + 627 + ], + "score": 0.9, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 615, + 474, + 630 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 474, + 616, + 487, + 627 + ], + "score": 0.89, + "content": "g ^ { \\bot }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 615, + 507, + 630 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 216, + 639 + ], + "score": 1.0, + "content": "signed vector orthogonal to", + "type": "text" + }, + { + "bbox": [ + 217, + 629, + 223, + 639 + ], + "score": 0.78, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 627, + 428, + 639 + ], + "score": 1.0, + "content": ". Looking forward to Section 4.1, we actually chose", + "type": "text" + }, + { + "bbox": [ + 428, + 627, + 440, + 639 + ], + "score": 0.89, + "content": "g ^ { \\bot }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "to be the signed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 440, + 651 + ], + "score": 1.0, + "content": "gradient of another Inception model, from which adversarial examples transfer to", + "type": "text" + }, + { + "bbox": [ + 440, + 639, + 462, + 649 + ], + "score": 0.9, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 637, + 506, + 651 + ], + "score": 1.0, + "content": ". Figure 1", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 380, + 662 + ], + "score": 1.0, + "content": "shows that the loss is highly curved in the vicinity of the data point", + "type": "text" + }, + { + "bbox": [ + 380, + 651, + 387, + 659 + ], + "score": 0.72, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 649, + 505, + 662 + ], + "score": 1.0, + "content": ", and that the gradient poorly", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 660, + 462, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 462, + 672 + ], + "score": 1.0, + "content": "reflects the global loss landscape. Similar plots for additional data points are in Figure 4.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 602, + 507, + 672 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 107, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 107, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "We show similar results for adversarially trained MNIST models in Appendix C.2. On this task,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "input dropout (Srivastava et al., 2014) mitigates adversarial training’s overfitting problem, in some", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "cases. Presumably, the random input mask diversifies the perturbations seen during training (dropout", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "at intermediate layers does not mitigate the overfitting effect). Mishkin et al. (2017) find that input", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 720, + 499, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 499, + 734 + ], + "score": 1.0, + "content": "dropout significantly degrades accuracy on ImageNet, so we did not include it in our experiments.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 677, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 119, + 303, + 203 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 506, + 114 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 194, + 93 + ], + "score": 1.0, + "content": "Table 1: Error rates", + "type": "text" + }, + { + "bbox": [ + 194, + 80, + 219, + 91 + ], + "score": 0.82, + "content": "( \\mathbf { i n \\% } )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 79, + 505, + 93 + ], + "score": 1.0, + "content": "of adversarial examples transferred between models. We use Step-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 141, + 104 + ], + "score": 1.0, + "content": "LL with", + "type": "text" + }, + { + "bbox": [ + 141, + 91, + 184, + 103 + ], + "score": 0.92, + "content": "\\epsilon = { ^ { 1 6 } } / { 2 5 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 90, + 505, + 104 + ], + "score": 1.0, + "content": "for 10,000 random test inputs. Diagonal elements represent a white-box attack.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 101, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 505, + 115 + ], + "score": 1.0, + "content": "The best attack for each target appears in bold. Similar results for MNIST models appear in Table 7.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 106, + 119, + 303, + 203 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 119, + 303, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 303, + 203 + ], + "score": 0.68, + "html": "
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", + "type": "table", + "image_path": "b2d72ba58e3626f5c7f75d14e26c350d203259ef4972300b3acbc3a2a3b22637.jpg" + } + ] + } + ], + "index": 11.5, + "virtual_lines": [ + { + "bbox": [ + 309, + 119, + 506, + 133.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 309, + 133.0, + 506, + 147.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 309, + 147.0, + 506, + 161.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 309, + 161.0, + 506, + 175.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 309, + 175.0, + 506, + 189.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 309, + 189.0, + 506, + 203.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 395, + 204, + 417, + 213 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 395, + 204, + 417, + 213 + ], + "spans": [], + "index": 16 + } + ], + "index": 16 + } + ], + "index": 13.75 + }, + { + "type": "title", + "bbox": [ + 106, + 232, + 384, + 244 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 385, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 385, + 245 + ], + "score": 1.0, + "content": "4.1 ATTACKS AGAINST ADVERSARIALLY TRAINED NETWORKS", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 320 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 267 + ], + "score": 1.0, + "content": "Kurakin et al. (2017b) found their adversarially trained model to be robust to various single-step", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "attacks. They conclude that this robustness should translate to attacks transferred from other models.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "As we have shown, the robustness to single-step attacks is actually misleading, as the model has", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "learned to degrade the information contained in the model’s gradient. As a consequence, we find that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 297, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 121, + 310 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 122, + 298, + 144, + 309 + ], + "score": 0.91, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 297, + 505, + 310 + ], + "score": 1.0, + "content": "model is substantially more vulnerable to single-step attacks than Kurakin et al. (2017b)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 308, + 477, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 414, + 321 + ], + "score": 1.0, + "content": "predicted, both in a white-box and black-box setting. The same holds for the", + "type": "text" + }, + { + "bbox": [ + 414, + 308, + 446, + 320 + ], + "score": 0.89, + "content": "\\mathrm { I R v } 2 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 308, + 477, + 321 + ], + "score": 1.0, + "content": "model.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 179, + 339 + ], + "score": 1.0, + "content": "In addition to the", + "type": "text" + }, + { + "bbox": [ + 180, + 326, + 201, + 337 + ], + "score": 0.9, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 324, + 221, + 339 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 221, + 326, + 253, + 337 + ], + "score": 0.9, + "content": "\\mathrm { I R v } 2 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 324, + 506, + 339 + ], + "score": 1.0, + "content": "models, we consider standard Inception v3, Inception v4 and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "Inception ResNet v2 models. These models are available in the TensorFlow-Slim library (Abadi", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "et al., 2015). We describe similar results for a variety of models trained on MNIST in Appendix C.2.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 365, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 504, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 504, + 377 + ], + "score": 1.0, + "content": "Black-box attacks. Table 1 shows error rates for single-step attacks transferred between models.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "score": 1.0, + "content": "We compute perturbations on one model (the source) and transfer them to all others (the targets).", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 386, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 400 + ], + "score": 1.0, + "content": "When the source and target are the same, the attack is white-box. Adversarial training greatly", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 399, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 410 + ], + "score": 1.0, + "content": "increases robustness to white-box single-step attacks, but incurs a higher error rate in a black-box", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 409, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 506, + 421 + ], + "score": 1.0, + "content": "setting. Thus, the robustness gain observed when evaluating defended models in isolation is mislead-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "ing. Given the ubiquity of this pitfall among proposed defenses against adversarial examples (Carlini", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 430, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 151, + 444 + ], + "score": 1.0, + "content": "& Wagner,", + "type": "text" + }, + { + "bbox": [ + 152, + 431, + 177, + 442 + ], + "score": 0.33, + "content": "2 0 1 7 \\mathrm { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 430, + 505, + 444 + ], + "score": 1.0, + "content": "; Brendel & Bethge, 2017; Papernot et al., 2016b), we advise researchers to always", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 441, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 455 + ], + "score": 1.0, + "content": "consider both white-box and black-box adversaries when evaluating defensive strategies. Notably,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 104, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "a similar discrepancy between white-box and black-box attacks was recently observed in Buckman", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 464, + 160, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 160, + 476 + ], + "score": 1.0, + "content": "et al. (2018).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 480, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "Attacks crafted on adversarial models are found to be weaker even against undefended models (i.e.,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 491, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 155, + 506 + ], + "score": 1.0, + "content": "when using", + "type": "text" + }, + { + "bbox": [ + 156, + 492, + 178, + 504 + ], + "score": 0.9, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 491, + 190, + 506 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 191, + 492, + 222, + 503 + ], + "score": 0.92, + "content": "\\mathrm { I R v } 2 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 491, + 506, + 506 + ], + "score": 1.0, + "content": "as source, the attack transfers with lower probability). This confirms", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "our intuition from Section 3.3: adversarial training does not just overfit to perturbations that affect", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 514, + 498, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 498, + 527 + ], + "score": 1.0, + "content": "standard models, but actively degrades the linear approximation underlying the single-step attack.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 107, + 531, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "A new randomized single-step attack. The loss function visualization in Figure 1 shows that", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 543, + 504, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 504, + 555 + ], + "score": 1.0, + "content": "sharp curvature artifacts localized near the data points can mask the true direction of steepest ascent.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "We thus suggest to prepend single-step attacks by a small random step, in order to “escape” the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "non-smooth vicinity of the data point before linearizing the model’s loss. Our new attack, called", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 576, + 501, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 147, + 586 + ], + "score": 0.69, + "content": "\\mathrm { R + F G S M }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 576, + 205, + 589 + ], + "score": 1.0, + "content": "(alternatively,", + "type": "text" + }, + { + "bbox": [ + 205, + 576, + 219, + 586 + ], + "score": 0.35, + "content": "\\mathrm { R } { + } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 576, + 406, + 589 + ], + "score": 1.0, + "content": "Step-LL), is defined as follows, for parameters", + "type": "text" + }, + { + "bbox": [ + 406, + 578, + 412, + 585 + ], + "score": 0.73, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 576, + 429, + 589 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 429, + 577, + 437, + 585 + ], + "score": 0.79, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 576, + 468, + 589 + ], + "score": 1.0, + "content": "(where", + "type": "text" + }, + { + "bbox": [ + 469, + 576, + 494, + 586 + ], + "score": 0.87, + "content": "\\alpha < \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 576, + 501, + 589 + ], + "score": 1.0, + "content": "):", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43 + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 594, + 480, + 610 + ], + "lines": [ + { + "bbox": [ + 118, + 594, + 480, + 610 + ], + "spans": [ + { + "bbox": [ + 118, + 594, + 480, + 610 + ], + "score": 0.9, + "content": "x ^ { \\mathrm { a d v } } = x ^ { \\prime } + ( \\varepsilon - \\alpha ) \\cdot \\mathsf { s i g n } \\left( \\nabla _ { x ^ { \\prime } } J ( x ^ { \\prime } , y _ { \\mathrm { t r u e } } ) \\right) , \\quad \\mathrm { w h e r e } \\quad x ^ { \\prime } = x + \\alpha \\cdot \\mathsf { s i g n } ( \\mathcal { N } ( \\mathbf { 0 } ^ { d } , \\mathbf { I } ^ { d } ) ) ~ .", + "type": "interline_equation", + "image_path": "a5a69dd287ac528c510c51f907431e520dfa67fc81620a099cda74fec32341a5.jpg" + } + ] + } + ], + "index": 46, + "virtual_lines": [ + { + "bbox": [ + 118, + 594, + 480, + 610 + ], + "spans": [], + "index": 46 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 615, + 506, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 376, + 629 + ], + "score": 1.0, + "content": "Note that the attack requires a single gradient computation. The", + "type": "text" + }, + { + "bbox": [ + 376, + 616, + 418, + 627 + ], + "score": 0.8, + "content": "\\mathrm { R + F G S M }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "is a computationally", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "efficient alternative to iterative methods that have high success rates in a white-box setting. Our", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 638, + 492, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 492, + 650 + ], + "score": 1.0, + "content": "attack can be seen as a single-step variant of the general PGD method from (Madry et al., 2017).", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 317, + 667 + ], + "score": 1.0, + "content": "Table 2 compares error rates for the Step-LL and", + "type": "text" + }, + { + "bbox": [ + 318, + 655, + 336, + 666 + ], + "score": 0.5, + "content": "\\mathsf { R } { + } \\mathsf { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 655, + 430, + 667 + ], + "score": 1.0, + "content": "tep-LL methods (with", + "type": "text" + }, + { + "bbox": [ + 431, + 654, + 486, + 667 + ], + "score": 0.88, + "content": "\\epsilon \\ : = \\ : 1 6 / 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 146, + 678 + ], + "score": 0.89, + "content": "\\alpha = \\epsilon / 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "). The extra random step yields a stronger attack for all models, even those without ad-", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "versarial training. This suggests that a model’s loss function is generally less smooth near the data", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 104, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 242, + 700 + ], + "score": 1.0, + "content": "points. We further compared the", + "type": "text" + }, + { + "bbox": [ + 243, + 688, + 259, + 698 + ], + "score": 0.43, + "content": "\\mathsf { R } { + } \\iota", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Step-LL attack to a two-step Iter-LL attack, which computes", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 698, + 504, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 504, + 711 + ], + "score": 1.0, + "content": "two gradient steps. Surprisingly, we find that for the adversarially trained Inception v3 model, the", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 126, + 720 + ], + "score": 0.59, + "content": "\\mathbf { R } { + } \\mathbf { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "tep-LL attack is stronger than the two-step Iter-LL attack. That is, the local gradients learned by", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 721, + 503, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 503, + 733 + ], + "score": 1.0, + "content": "the adversarially trained model are worse than random directions for finding adversarial examples!", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 53 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 119, + 303, + 203 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 506, + 114 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 194, + 93 + ], + "score": 1.0, + "content": "Table 1: Error rates", + "type": "text" + }, + { + "bbox": [ + 194, + 80, + 219, + 91 + ], + "score": 0.82, + "content": "( \\mathbf { i n \\% } )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 79, + 505, + 93 + ], + "score": 1.0, + "content": "of adversarial examples transferred between models. We use Step-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 141, + 104 + ], + "score": 1.0, + "content": "LL with", + "type": "text" + }, + { + "bbox": [ + 141, + 91, + 184, + 103 + ], + "score": 0.92, + "content": "\\epsilon = { ^ { 1 6 } } / { 2 5 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 90, + 505, + 104 + ], + "score": 1.0, + "content": "for 10,000 random test inputs. Diagonal elements represent a white-box attack.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 101, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 505, + 115 + ], + "score": 1.0, + "content": "The best attack for each target appears in bold. Similar results for MNIST models appear in Table 7.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 106, + 119, + 303, + 203 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 119, + 303, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 303, + 203 + ], + "score": 0.68, + "html": "
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(2017b) found their adversarially trained model to be robust to various single-step", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "attacks. They conclude that this robustness should translate to attacks transferred from other models.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "As we have shown, the robustness to single-step attacks is actually misleading, as the model has", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "learned to degrade the information contained in the model’s gradient. As a consequence, we find that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 297, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 121, + 310 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 122, + 298, + 144, + 309 + ], + "score": 0.91, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 297, + 505, + 310 + ], + "score": 1.0, + "content": "model is substantially more vulnerable to single-step attacks than Kurakin et al. (2017b)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 308, + 477, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 414, + 321 + ], + "score": 1.0, + "content": "predicted, both in a white-box and black-box setting. The same holds for the", + "type": "text" + }, + { + "bbox": [ + 414, + 308, + 446, + 320 + ], + "score": 0.89, + "content": "\\mathrm { I R v } 2 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 308, + 477, + 321 + ], + "score": 1.0, + "content": "model.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 252, + 506, + 321 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 179, + 339 + ], + "score": 1.0, + "content": "In addition to the", + "type": "text" + }, + { + "bbox": [ + 180, + 326, + 201, + 337 + ], + "score": 0.9, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 324, + 221, + 339 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 221, + 326, + 253, + 337 + ], + "score": 0.9, + "content": "\\mathrm { I R v } 2 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 324, + 506, + 339 + ], + "score": 1.0, + "content": "models, we consider standard Inception v3, Inception v4 and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "Inception ResNet v2 models. These models are available in the TensorFlow-Slim library (Abadi", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "et al., 2015). We describe similar results for a variety of models trained on MNIST in Appendix C.2.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 324, + 506, + 360 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 365, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 504, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 504, + 377 + ], + "score": 1.0, + "content": "Black-box attacks. Table 1 shows error rates for single-step attacks transferred between models.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "score": 1.0, + "content": "We compute perturbations on one model (the source) and transfer them to all others (the targets).", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 386, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 400 + ], + "score": 1.0, + "content": "When the source and target are the same, the attack is white-box. Adversarial training greatly", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 399, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 410 + ], + "score": 1.0, + "content": "increases robustness to white-box single-step attacks, but incurs a higher error rate in a black-box", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 409, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 506, + 421 + ], + "score": 1.0, + "content": "setting. Thus, the robustness gain observed when evaluating defended models in isolation is mislead-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "ing. Given the ubiquity of this pitfall among proposed defenses against adversarial examples (Carlini", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 430, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 151, + 444 + ], + "score": 1.0, + "content": "& Wagner,", + "type": "text" + }, + { + "bbox": [ + 152, + 431, + 177, + 442 + ], + "score": 0.33, + "content": "2 0 1 7 \\mathrm { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 430, + 505, + 444 + ], + "score": 1.0, + "content": "; Brendel & Bethge, 2017; Papernot et al., 2016b), we advise researchers to always", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 441, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 455 + ], + "score": 1.0, + "content": "consider both white-box and black-box adversaries when evaluating defensive strategies. Notably,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 104, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "a similar discrepancy between white-box and black-box attacks was recently observed in Buckman", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 464, + 160, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 160, + 476 + ], + "score": 1.0, + "content": "et al. (2018).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 365, + 506, + 476 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 480, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "Attacks crafted on adversarial models are found to be weaker even against undefended models (i.e.,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 491, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 155, + 506 + ], + "score": 1.0, + "content": "when using", + "type": "text" + }, + { + "bbox": [ + 156, + 492, + 178, + 504 + ], + "score": 0.9, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 491, + 190, + 506 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 191, + 492, + 222, + 503 + ], + "score": 0.92, + "content": "\\mathrm { I R v } 2 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 491, + 506, + 506 + ], + "score": 1.0, + "content": "as source, the attack transfers with lower probability). This confirms", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "our intuition from Section 3.3: adversarial training does not just overfit to perturbations that affect", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 514, + 498, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 498, + 527 + ], + "score": 1.0, + "content": "standard models, but actively degrades the linear approximation underlying the single-step attack.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 480, + 506, + 527 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 531, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "A new randomized single-step attack. The loss function visualization in Figure 1 shows that", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 543, + 504, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 504, + 555 + ], + "score": 1.0, + "content": "sharp curvature artifacts localized near the data points can mask the true direction of steepest ascent.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "We thus suggest to prepend single-step attacks by a small random step, in order to “escape” the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "non-smooth vicinity of the data point before linearizing the model’s loss. Our new attack, called", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 576, + 501, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 147, + 586 + ], + "score": 0.69, + "content": "\\mathrm { R + F G S M }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 576, + 205, + 589 + ], + "score": 1.0, + "content": "(alternatively,", + "type": "text" + }, + { + "bbox": [ + 205, + 576, + 219, + 586 + ], + "score": 0.35, + "content": "\\mathrm { R } { + } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 576, + 406, + 589 + ], + "score": 1.0, + "content": "Step-LL), is defined as follows, for parameters", + "type": "text" + }, + { + "bbox": [ + 406, + 578, + 412, + 585 + ], + "score": 0.73, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 576, + 429, + 589 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 429, + 577, + 437, + 585 + ], + "score": 0.79, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 576, + 468, + 589 + ], + "score": 1.0, + "content": "(where", + "type": "text" + }, + { + "bbox": [ + 469, + 576, + 494, + 586 + ], + "score": 0.87, + "content": "\\alpha < \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 576, + 501, + 589 + ], + "score": 1.0, + "content": "):", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 531, + 505, + 589 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 594, + 480, + 610 + ], + "lines": [ + { + "bbox": [ + 118, + 594, + 480, + 610 + ], + "spans": [ + { + "bbox": [ + 118, + 594, + 480, + 610 + ], + "score": 0.9, + "content": "x ^ { \\mathrm { a d v } } = x ^ { \\prime } + ( \\varepsilon - \\alpha ) \\cdot \\mathsf { s i g n } \\left( \\nabla _ { x ^ { \\prime } } J ( x ^ { \\prime } , y _ { \\mathrm { t r u e } } ) \\right) , \\quad \\mathrm { w h e r e } \\quad x ^ { \\prime } = x + \\alpha \\cdot \\mathsf { s i g n } ( \\mathcal { N } ( \\mathbf { 0 } ^ { d } , \\mathbf { I } ^ { d } ) ) ~ .", + "type": "interline_equation", + "image_path": "a5a69dd287ac528c510c51f907431e520dfa67fc81620a099cda74fec32341a5.jpg" + } + ] + } + ], + "index": 46, + "virtual_lines": [ + { + "bbox": [ + 118, + 594, + 480, + 610 + ], + "spans": [], + "index": 46 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 615, + 506, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 376, + 629 + ], + "score": 1.0, + "content": "Note that the attack requires a single gradient computation. The", + "type": "text" + }, + { + "bbox": [ + 376, + 616, + 418, + 627 + ], + "score": 0.8, + "content": "\\mathrm { R + F G S M }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "is a computationally", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "efficient alternative to iterative methods that have high success rates in a white-box setting. Our", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 638, + 492, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 492, + 650 + ], + "score": 1.0, + "content": "attack can be seen as a single-step variant of the general PGD method from (Madry et al., 2017).", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 615, + 506, + 650 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 317, + 667 + ], + "score": 1.0, + "content": "Table 2 compares error rates for the Step-LL and", + "type": "text" + }, + { + "bbox": [ + 318, + 655, + 336, + 666 + ], + "score": 0.5, + "content": "\\mathsf { R } { + } \\mathsf { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 655, + 430, + 667 + ], + "score": 1.0, + "content": "tep-LL methods (with", + "type": "text" + }, + { + "bbox": [ + 431, + 654, + 486, + 667 + ], + "score": 0.88, + "content": "\\epsilon \\ : = \\ : 1 6 / 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 146, + 678 + ], + "score": 0.89, + "content": "\\alpha = \\epsilon / 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "). 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We further compared the", + "type": "text" + }, + { + "bbox": [ + 243, + 688, + 259, + 698 + ], + "score": 0.43, + "content": "\\mathsf { R } { + } \\iota", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Step-LL attack to a two-step Iter-LL attack, which computes", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 698, + 504, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 504, + 711 + ], + "score": 1.0, + "content": "two gradient steps. 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v4v3v3advIRv2IRv2adv
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Trained ModelPre-trained ModelsHoldout Models
Inception v3 (v3adv-ens3)Inception v3,ResNet v2 (50)Inception v4
Inception v3 (v3adv-ens4)Inception v3,ResNet v2 (5O), IncRes v2ResNet v1 (50)
IncRes v2 (IRv2adv-ens)Inception v3, IncRes v2ResNetv2(101))
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We also tried", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 212, + 321 + ], + "score": 1.0, + "content": "adversarial training using", + "type": "text" + }, + { + "bbox": [ + 212, + 309, + 253, + 320 + ], + "score": 0.77, + "content": "\\mathrm { R + F G S M }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "on MNIST, using a similar approach as (Madry et al., 2017).", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 319, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 426, + 333 + ], + "score": 1.0, + "content": "We adversarially train a CNN (model A in Table 5) for 100 epochs, and attain", + "type": "text" + }, + { + "bbox": [ + 426, + 320, + 466, + 331 + ], + "score": 0.91, + "content": "> 9 0 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 319, + 505, + 333 + ], + "score": 1.0, + "content": "accuracy", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 330, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 119, + 344 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 119, + 331, + 161, + 342 + ], + "score": 0.69, + "content": "\\mathrm { R + F G S M }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 330, + 291, + 344 + ], + "score": 1.0, + "content": "samples. However, training on", + "type": "text" + }, + { + "bbox": [ + 291, + 331, + 332, + 342 + ], + "score": 0.6, + "content": "\\mathrm { R + F G S M }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 330, + 506, + 344 + ], + "score": 1.0, + "content": "provides only little robustness to iterative", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 341, + 502, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 434, + 355 + ], + "score": 1.0, + "content": "attacks. For the PGD attack of (Madry et al., 2017) with 20 steps, the model attains", + "type": "text" + }, + { + "bbox": [ + 435, + 342, + 462, + 353 + ], + "score": 0.88, + "content": "1 8 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 341, + 502, + 355 + ], + "score": 1.0, + "content": "accuracy.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 108, + 373, + 290, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 372, + 291, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 291, + 385 + ], + "score": 1.0, + "content": "4.2 ENSEMBLE ADVERSARIAL TRAINING", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 395, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 107, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 107, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "We now evaluate our Ensemble Adversarial Training strategy described in Section 3.4. We recall", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "our intuition: by augmenting training data with adversarial examples crafted from static pre-trained", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "models, we decouple the generation of adversarial examples from the model being trained, so as to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "score": 1.0, + "content": "avoid the degenerate minimum described in Section 3.3. Moreover, our hope is that robustness to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 439, + 492, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 492, + 452 + ], + "score": 1.0, + "content": "attacks transferred from some fixed set of models will generalize to other black-box adversaries.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 579 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 470 + ], + "score": 1.0, + "content": "We train Inception v3 and Inception ResNet v2 models (Szegedy et al., 2016a) on ImageNet, using", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "the pre-trained models shown in Table 3. In each training batch, we rotate the source of adversarial", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 478, + 504, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 504, + 491 + ], + "score": 1.0, + "content": "examples between the currently trained model and one of the pre-trained models. We select the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 489, + 504, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 504, + 502 + ], + "score": 1.0, + "content": "source model at random in each batch, to diversify examples across epochs. The pre-trained models’", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "gradients can be precomputed for the full training set. The per-batch cost of Ensemble Adversarial", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 511, + 504, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 477, + 524 + ], + "score": 1.0, + "content": "Training is thus lower than that of standard adversarial training: using our method with", + "type": "text" + }, + { + "bbox": [ + 478, + 512, + 504, + 522 + ], + "score": 0.86, + "content": "n - 1", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 232, + 537 + ], + "score": 1.0, + "content": "pre-trained models, only every", + "type": "text" + }, + { + "bbox": [ + 233, + 523, + 246, + 534 + ], + "score": 0.87, + "content": "n ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "batch requires a forward-backward pass to compute adversarial", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 535, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 548 + ], + "score": 1.0, + "content": "gradients. We use synchronous distributed training on 50 machines, with minibatches of size 16 (we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "score": 1.0, + "content": "did not pre-compute gradients, and thus lower the batch size to fit all models in memory). Half of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 557, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 506, + 570 + ], + "score": 1.0, + "content": "the examples in a minibatch are replaced by Step-LL examples. As in Kurakin et al. 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For each model, we report the worst-case error rate over all black-box attacks", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 640, + 503, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 503, + 651 + ], + "score": 1.0, + "content": "transfered from each of the holdout models (20 attacks in total). Results for MNIST are in Table 8.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 678 + ], + "score": 1.0, + "content": "Convergence speed. 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v4v3v3advIRv2IRv2adv
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Trained ModelPre-trained ModelsHoldout Models
Inception v3 (v3adv-ens3)Inception v3,ResNet v2 (50)Inception v4
Inception v3 (v3adv-ens4)Inception v3,ResNet v2 (5O), IncRes v2ResNet v1 (50)
IncRes v2 (IRv2adv-ens)Inception v3, IncRes v2ResNetv2(101))
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We also tried", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 212, + 321 + ], + "score": 1.0, + "content": "adversarial training using", + "type": "text" + }, + { + "bbox": [ + 212, + 309, + 253, + 320 + ], + "score": 0.77, + "content": "\\mathrm { R + F G S M }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "on MNIST, using a similar approach as (Madry et al., 2017).", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 319, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 426, + 333 + ], + "score": 1.0, + "content": "We adversarially train a CNN (model A in Table 5) for 100 epochs, and attain", + "type": "text" + }, + { + "bbox": [ + 426, + 320, + 466, + 331 + ], + "score": 0.91, + "content": "> 9 0 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 319, + 505, + 333 + ], + "score": 1.0, + "content": "accuracy", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 330, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 119, + 344 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 119, + 331, + 161, + 342 + ], + "score": 0.69, + "content": "\\mathrm { R + F G S M }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 330, + 291, + 344 + ], + "score": 1.0, + "content": "samples. However, training on", + "type": "text" + }, + { + "bbox": [ + 291, + 331, + 332, + 342 + ], + "score": 0.6, + "content": "\\mathrm { R + F G S M }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 330, + 506, + 344 + ], + "score": 1.0, + "content": "provides only little robustness to iterative", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 341, + 502, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 434, + 355 + ], + "score": 1.0, + "content": "attacks. For the PGD attack of (Madry et al., 2017) with 20 steps, the model attains", + "type": "text" + }, + { + "bbox": [ + 435, + 342, + 462, + 353 + ], + "score": 0.88, + "content": "1 8 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 341, + 502, + 355 + ], + "score": 1.0, + "content": "accuracy.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 298, + 506, + 355 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 373, + 290, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 372, + 291, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 291, + 385 + ], + "score": 1.0, + "content": "4.2 ENSEMBLE ADVERSARIAL TRAINING", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 395, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 107, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 107, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "We now evaluate our Ensemble Adversarial Training strategy described in Section 3.4. We recall", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "our intuition: by augmenting training data with adversarial examples crafted from static pre-trained", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "models, we decouple the generation of adversarial examples from the model being trained, so as to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "score": 1.0, + "content": "avoid the degenerate minimum described in Section 3.3. Moreover, our hope is that robustness to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 439, + 492, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 492, + 452 + ], + "score": 1.0, + "content": "attacks transferred from some fixed set of models will generalize to other black-box adversaries.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 396, + 506, + 452 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 579 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 470 + ], + "score": 1.0, + "content": "We train Inception v3 and Inception ResNet v2 models (Szegedy et al., 2016a) on ImageNet, using", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "the pre-trained models shown in Table 3. In each training batch, we rotate the source of adversarial", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 478, + 504, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 504, + 491 + ], + "score": 1.0, + "content": "examples between the currently trained model and one of the pre-trained models. We select the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 489, + 504, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 504, + 502 + ], + "score": 1.0, + "content": "source model at random in each batch, to diversify examples across epochs. The pre-trained models’", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "gradients can be precomputed for the full training set. The per-batch cost of Ensemble Adversarial", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 511, + 504, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 477, + 524 + ], + "score": 1.0, + "content": "Training is thus lower than that of standard adversarial training: using our method with", + "type": "text" + }, + { + "bbox": [ + 478, + 512, + 504, + 522 + ], + "score": 0.86, + "content": "n - 1", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 232, + 537 + ], + "score": 1.0, + "content": "pre-trained models, only every", + "type": "text" + }, + { + "bbox": [ + 233, + 523, + 246, + 534 + ], + "score": 0.87, + "content": "n ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "batch requires a forward-backward pass to compute adversarial", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 535, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 548 + ], + "score": 1.0, + "content": "gradients. We use synchronous distributed training on 50 machines, with minibatches of size 16 (we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "score": 1.0, + "content": "did not pre-compute gradients, and thus lower the batch size to fit all models in memory). Half of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 557, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 506, + 570 + ], + "score": 1.0, + "content": "the examples in a minibatch are replaced by Step-LL examples. As in Kurakin et al. (2017b), we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 569, + 467, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 467, + 580 + ], + "score": 1.0, + "content": "use RMSProp with a learning rate of 0.045, decayed by a factor of 0.94 every two epochs.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 455, + 506, + 580 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 585, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "To evaluate how robustness to black-box attacks generalizes across models, we transfer various", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 596, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 607 + ], + "score": 1.0, + "content": "attacks crafted on three different holdout models (see Table 3), as well as on an ensemble of these", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 324, + 619 + ], + "score": 1.0, + "content": "models (as in Liu et al. (2017)). We use the Step-LL,", + "type": "text" + }, + { + "bbox": [ + 324, + 607, + 338, + 618 + ], + "score": 0.27, + "content": "\\mathbf { R } +", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "Step-LL, FGSM, I-FGSM and the PGD", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "attack from Madry et al. (2017) using the hinge-loss function from Carlini & Wagner (2017a). Our", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "results are in Table 4. For each model, we report the worst-case error rate over all black-box attacks", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 640, + 503, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 503, + 651 + ], + "score": 1.0, + "content": "transfered from each of the holdout models (20 attacks in total). Results for MNIST are in Table 8.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 585, + 505, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 678 + ], + "score": 1.0, + "content": "Convergence speed. Convergence of Ensemble Adversarial Training is slower than for standard", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "adversarial training, a result of training on “hard” adversarial examples and lowering the batch size.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 686, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 320, + 700 + ], + "score": 1.0, + "content": "Kurakin et al. 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The Inception", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "ResNet v2 model is trained for 175 epochs, where a baseline model converges at around 160 epochs.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 664, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 118, + 140, + 491, + 257 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 135 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 198, + 93 + ], + "score": 1.0, + "content": "Table 4: Error rates", + "type": "text" + }, + { + "bbox": [ + 198, + 80, + 223, + 91 + ], + "score": 0.77, + "content": "( \\mathbf { i n \\ \\% } )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 79, + 506, + 93 + ], + "score": 1.0, + "content": ") for Ensemble Adversarial Training on ImageNet. Error rates on", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 506, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 445, + 104 + ], + "score": 1.0, + "content": "clean data are computed over the full test set. For 10,000 random test set inputs, and", + "type": "text" + }, + { + "bbox": [ + 446, + 91, + 487, + 103 + ], + "score": 0.92, + "content": "\\epsilon = { ^ { 1 6 } } / { 2 5 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 90, + 506, + 104 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 104, + 102, + 505, + 115 + ], + "score": 1.0, + "content": "report error rates on white-box Step-LL and the worst-case error over a series of black-box attacks", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 113, + 506, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 146, + 126 + ], + "score": 1.0, + "content": "(Step-LL,", + "type": "text" + }, + { + "bbox": [ + 146, + 114, + 160, + 124 + ], + "score": 0.44, + "content": "R +", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 113, + 506, + 126 + ], + "score": 1.0, + "content": "Step-LL, FGSM, I-FGSM, PGD) transferred from the holdout models in Table 3. For", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 124, + 449, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 124, + 377, + 136 + ], + "score": 1.0, + "content": "both architectures, we mark methods tied for best in bold (based on", + "type": "text" + }, + { + "bbox": [ + 378, + 124, + 397, + 135 + ], + "score": 0.91, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 124, + 449, + 136 + ], + "score": 1.0, + "content": "confidence).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "table_body", + "bbox": [ + 118, + 140, + 491, + 257 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 118, + 140, + 491, + 257 + ], + "spans": [ + { + "bbox": [ + 118, + 140, + 491, + 257 + ], + "score": 0.976, + "html": "
ModelTop 1Top 5
CleanStep-LLMax.Black-BoxCleanStep-LLMax.Black-Box
v322.069.651.26.142.724.5
v3adv22.026.640.86.19.017.4
v3adv-ens323.630.034.07.610.111.2
v3adv-ens424.243.333.47.819.410.7
IRv219.650.744.44.824.017.8
IRv2adv19.821.434.54.95.811.7
IRv2adv-ens20.226.027.05.17.67.9
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For both architectures, the models trained with Ensemble Adversarial Train-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "ing are slightly less accurate on clean data, compared to standard adversarial training. Our models", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "are also more vulnerable to white-box single-step attacks, as they were only partially trained on", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 300, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 240, + 315 + ], + "score": 1.0, + "content": "such perturbations. Note that for", + "type": "text" + }, + { + "bbox": [ + 240, + 302, + 277, + 313 + ], + "score": 0.92, + "content": "{ \\mathbf { v } } 3 _ { \\mathrm { a d v - e n s } 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 300, + 505, + 315 + ], + "score": 1.0, + "content": ", the proportion of white-box Step-LL samples seen dur-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 311, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 167, + 326 + ], + "score": 1.0, + "content": "ing training is", + "type": "text" + }, + { + "bbox": [ + 167, + 312, + 180, + 324 + ], + "score": 0.78, + "content": "^ 1 / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 311, + 228, + 326 + ], + "score": 1.0, + "content": "(instead of", + "type": "text" + }, + { + "bbox": [ + 228, + 312, + 241, + 324 + ], + "score": 0.81, + "content": "^ 1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 311, + 286, + 326 + ], + "score": 1.0, + "content": "for model", + "type": "text" + }, + { + "bbox": [ + 286, + 313, + 324, + 324 + ], + "score": 0.89, + "content": "{ \\mathbf { v } } 3 _ { \\mathrm { a d v - e n s } 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 311, + 506, + 326 + ], + "score": 1.0, + "content": "). The negative impact on the robustness to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 321, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 338 + ], + "score": 1.0, + "content": "white-box attacks is large, for only a minor gain in robustness to transferred samples. Thus it ap-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 334, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 348 + ], + "score": 1.0, + "content": "pears that while increasing the diversity of adversarial examples seen during training can provide", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 343, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 104, + 343, + 505, + 359 + ], + "score": 1.0, + "content": "some marginal improvement, the main benefit of Ensemble Adversarial Training is in decoupling", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 357, + 452, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 452, + 369 + ], + "score": 1.0, + "content": "the attacks from the model being trained, which was the goal we stated in Section 3.4.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 373, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 372, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 383, + 386 + ], + "score": 1.0, + "content": "Ensemble Adversarial Training is not robust to white-box Iter-LL and", + "type": "text" + }, + { + "bbox": [ + 383, + 374, + 398, + 383 + ], + "score": 0.4, + "content": "\\mathrm { R } { + } \\mathrm { : }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 372, + 506, + 386 + ], + "score": 1.0, + "content": "Step-LL samples: the error", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 234, + 397 + ], + "score": 1.0, + "content": "rates are similar to those for the", + "type": "text" + }, + { + "bbox": [ + 235, + 384, + 257, + 395 + ], + "score": 0.9, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 383, + 505, + 397 + ], + "score": 1.0, + "content": "model, and omitted for brevity (see Kurakin et al. 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Yet, against black-box adversaries, these", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 417, + 408, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 408, + 429 + ], + "score": 1.0, + "content": "attacks are only a concern insofar as they reliably transfer between models.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 437, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "score": 1.0, + "content": "Black-box attacks. Ensemble Adversarial Training significantly boosts robustness to all attacks", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 447, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 297, + 462 + ], + "score": 1.0, + "content": "transferred from the holdout models. 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We find that the strongest attacks", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 445, + 483 + ], + "score": 1.0, + "content": "in our test suite (i.e., with highest transfer rates) are the FGSM attacks. Black-box", + "type": "text" + }, + { + "bbox": [ + 446, + 470, + 464, + 480 + ], + "score": 0.51, + "content": "\\mathbf { R } { + } \\mathbf { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "tep-LL or", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "iterative attacks are less effective, as they do not transfer with high probability (see Kurakin et al.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 492, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 504 + ], + "score": 1.0, + "content": "(2017b) and Table 9). Attacking an ensemble of all three holdout models, as in Liu et al. 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We also find that", + "type": "text" + }, + { + "bbox": [ + 357, + 542, + 394, + 554 + ], + "score": 0.91, + "content": "{ \\mathbf { v } } 3 _ { \\mathrm { a d v - e n s } 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 540, + 506, + 556 + ], + "score": 1.0, + "content": "is not vulnerable to pertur-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 553, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 207, + 566 + ], + "score": 1.0, + "content": "bations transferred from", + "type": "text" + }, + { + "bbox": [ + 207, + 554, + 244, + 564 + ], + "score": 0.92, + "content": "{ \\mathbf { v } } 3 _ { \\mathrm { a d v - e n s } 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 553, + 505, + 566 + ], + "score": 1.0, + "content": ". We obtain similar results on MNIST (see Appendix C.2), thus", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 564, + 480, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 480, + 576 + ], + "score": 1.0, + "content": "demonstrating the applicability of our approach to different datasets and model architectures.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 588, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "The NIPS 2017 competition on adversarial examples. 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ModelTop 1Top 5
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Thus it ap-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 334, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 348 + ], + "score": 1.0, + "content": "pears that while increasing the diversity of adversarial examples seen during training can provide", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 343, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 104, + 343, + 505, + 359 + ], + "score": 1.0, + "content": "some marginal improvement, the main benefit of Ensemble Adversarial Training is in decoupling", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 357, + 452, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 452, + 369 + ], + "score": 1.0, + "content": "the attacks from the model being trained, which was the goal we stated in Section 3.4.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12, + "bbox_fs": [ + 104, + 268, + 506, + 369 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 373, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 372, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 383, + 386 + ], + "score": 1.0, + "content": "Ensemble Adversarial Training is not robust to white-box Iter-LL and", + "type": "text" + }, + { + "bbox": [ + 383, + 374, + 398, + 383 + ], + "score": 0.4, + "content": "\\mathrm { R } { + } \\mathrm { : }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 372, + 506, + 386 + ], + "score": 1.0, + "content": "Step-LL samples: the error", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 383, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 234, + 397 + ], + "score": 1.0, + "content": "rates are similar to those for the", + "type": "text" + }, + { + "bbox": [ + 235, + 384, + 257, + 395 + ], + "score": 0.9, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 383, + 505, + 397 + ], + "score": 1.0, + "content": "model, and omitted for brevity (see Kurakin et al. 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Yet, against black-box adversaries, these", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 417, + 408, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 408, + 429 + ], + "score": 1.0, + "content": "attacks are only a concern insofar as they reliably transfer between models.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 372, + 506, + 429 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 437, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "score": 1.0, + "content": "Black-box attacks. Ensemble Adversarial Training significantly boosts robustness to all attacks", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 447, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 297, + 462 + ], + "score": 1.0, + "content": "transferred from the holdout models. For the", + "type": "text" + }, + { + "bbox": [ + 297, + 449, + 341, + 460 + ], + "score": 0.89, + "content": "\\mathrm { I R v } 2 _ { \\mathrm { a d v - e n s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 447, + 506, + 462 + ], + "score": 1.0, + "content": "model, the accuracy loss (compared to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 240, + 471 + ], + "score": 1.0, + "content": "IRv2’s accuracy on clean data) is", + "type": "text" + }, + { + "bbox": [ + 241, + 459, + 264, + 470 + ], + "score": 0.89, + "content": "7 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 459, + 312, + 471 + ], + "score": 1.0, + "content": "(top 1) and", + "type": "text" + }, + { + "bbox": [ + 313, + 460, + 336, + 470 + ], + "score": 0.87, + "content": "\\mathbf { 3 . 1 \\% }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 459, + 505, + 471 + ], + "score": 1.0, + "content": "(top 5). We find that the strongest attacks", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 445, + 483 + ], + "score": 1.0, + "content": "in our test suite (i.e., with highest transfer rates) are the FGSM attacks. Black-box", + "type": "text" + }, + { + "bbox": [ + 446, + 470, + 464, + 480 + ], + "score": 0.51, + "content": "\\mathbf { R } { + } \\mathbf { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "tep-LL or", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "iterative attacks are less effective, as they do not transfer with high probability (see Kurakin et al.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 492, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 504 + ], + "score": 1.0, + "content": "(2017b) and Table 9). Attacking an ensemble of all three holdout models, as in Liu et al. (2017), did", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 503, + 472, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 472, + 516 + ], + "score": 1.0, + "content": "not lead to stronger black-box attacks than when attacking the holdout models individually.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 436, + 506, + 516 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 520, + 505, + 575 + ], + "lines": [ + { + "bbox": [ + 106, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 433, + 533 + ], + "score": 1.0, + "content": "Our results have little variance with respect to the attack parameters (e.g., smaller", + "type": "text" + }, + { + "bbox": [ + 433, + 522, + 438, + 530 + ], + "score": 0.5, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 519, + 506, + 533 + ], + "score": 1.0, + "content": ") or to the use of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 527, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 104, + 527, + 506, + 546 + ], + "score": 1.0, + "content": "other holdout models for black-box attacks (e.g., we obtain similar results by attacking the v3adv-ens3", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 540, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 123, + 556 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 542, + 160, + 554 + ], + "score": 0.91, + "content": "{ \\mathbf { v } } 3 _ { \\mathrm { a d v - e n s } 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 540, + 357, + 556 + ], + "score": 1.0, + "content": "models with the IRv2 model). We also find that", + "type": "text" + }, + { + "bbox": [ + 357, + 542, + 394, + 554 + ], + "score": 0.91, + "content": "{ \\mathbf { v } } 3 _ { \\mathrm { a d v - e n s } 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 540, + 506, + 556 + ], + "score": 1.0, + "content": "is not vulnerable to pertur-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 553, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 207, + 566 + ], + "score": 1.0, + "content": "bations transferred from", + "type": "text" + }, + { + "bbox": [ + 207, + 554, + 244, + 564 + ], + "score": 0.92, + "content": "{ \\mathbf { v } } 3 _ { \\mathrm { a d v - e n s } 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 553, + 505, + 566 + ], + "score": 1.0, + "content": ". We obtain similar results on MNIST (see Appendix C.2), thus", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 564, + 480, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 480, + 576 + ], + "score": 1.0, + "content": "demonstrating the applicability of our approach to different datasets and model architectures.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 519, + 506, + 576 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 588, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "The NIPS 2017 competition on adversarial examples. 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The second placed defense (team “cihangxie” with a score of", + "type": "text" + }, + { + "bbox": [ + 435, + 264, + 463, + 275 + ], + "score": 0.86, + "content": "9 2 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 263, + 505, + 278 + ], + "score": 1.0, + "content": ") prepends", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 275, + 480, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 122, + 288 + ], + "score": 1.0, + "content": "our", + "type": "text" + }, + { + "bbox": [ + 123, + 276, + 165, + 288 + ], + "score": 0.83, + "content": "\\mathrm { I R v } 2 _ { \\mathrm { a d v - e n s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 275, + 480, + 288 + ], + "score": 1.0, + "content": "model with random padding and resizing of the input image (Xie et al., 2018).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 293, + 505, + 359 + ], + "lines": [ + { + "bbox": [ + 105, + 292, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 506, + 305 + ], + "score": 1.0, + "content": "It is noteworthy that the defenses that incorporated Ensemble Adversarial Training faired better", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "score": 1.0, + "content": "against the worst-case black-box adversary. Indeed, although very robust on average, the winning", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 220, + 327 + ], + "score": 1.0, + "content": "defense achieved as low as", + "type": "text" + }, + { + "bbox": [ + 220, + 315, + 248, + 326 + ], + "score": 0.87, + "content": "1 1 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 314, + 506, + 327 + ], + "score": 1.0, + "content": "accuracy on some attacks. The best defense under this metric", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 324, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 429, + 339 + ], + "score": 1.0, + "content": "(team “rafaelmm” which randomly perturbed images before feeding them to our", + "type": "text" + }, + { + "bbox": [ + 430, + 326, + 473, + 337 + ], + "score": 0.89, + "content": "\\mathrm { I R v } 2 _ { \\mathrm { a d v - e n s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 324, + 506, + 339 + ], + "score": 1.0, + "content": "model)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 175, + 349 + ], + "score": 1.0, + "content": "achieved at least", + "type": "text" + }, + { + "bbox": [ + 175, + 336, + 203, + 347 + ], + "score": 0.88, + "content": "5 3 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "accuracy against all submitted attacks, including the attacks that explicitly", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 347, + 311, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 311, + 360 + ], + "score": 1.0, + "content": "targeted our released model in a white-box setting.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "Decreasing gradient masking. Ensemble Adversarial Training decreases the magnitude of the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 395, + 507, + 411 + ], + "spans": [ + { + "bbox": [ + 104, + 395, + 325, + 411 + ], + "score": 1.0, + "content": "gradient masking effect described previously. For the", + "type": "text" + }, + { + "bbox": [ + 325, + 397, + 362, + 408 + ], + "score": 0.91, + "content": "{ \\mathbf { v } } 3 _ { \\mathrm { a d v - e n s } 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 395, + 381, + 411 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 381, + 397, + 419, + 407 + ], + "score": 0.89, + "content": "{ \\mathbf { v } } 3 _ { \\mathrm { a d v - e n s } 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 395, + 507, + 411 + ], + "score": 1.0, + "content": "models, we find that", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 360, + 420 + ], + "score": 1.0, + "content": "the loss incurred on a Step-LL attack gets within respectively", + "type": "text" + }, + { + "bbox": [ + 360, + 407, + 380, + 418 + ], + "score": 0.89, + "content": "1 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 407, + 399, + 420 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 399, + 408, + 419, + 418 + ], + "score": 0.88, + "content": "1 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "of the optimum loss", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 247, + 430 + ], + "score": 1.0, + "content": "(we recall that for models v3 and", + "type": "text" + }, + { + "bbox": [ + 248, + 419, + 269, + 430 + ], + "score": 0.9, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 417, + 443, + 430 + ], + "score": 1.0, + "content": ", the approximation ratio was respectively", + "type": "text" + }, + { + "bbox": [ + 444, + 418, + 464, + 429 + ], + "score": 0.89, + "content": "1 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 417, + 483, + 430 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 483, + 418, + 498, + 429 + ], + "score": 0.83, + "content": "7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 498, + 417, + 505, + 430 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 104, + 429, + 176, + 442 + ], + "score": 1.0, + "content": "Similarly, for the", + "type": "text" + }, + { + "bbox": [ + 177, + 429, + 220, + 441 + ], + "score": 0.87, + "content": "\\mathrm { I R v } 2 _ { \\mathrm { a d v - e n s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 429, + 347, + 442 + ], + "score": 1.0, + "content": "model, the ratio improves from", + "type": "text" + }, + { + "bbox": [ + 348, + 429, + 362, + 440 + ], + "score": 0.86, + "content": "8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 429, + 381, + 442 + ], + "score": 1.0, + "content": "(for", + "type": "text" + }, + { + "bbox": [ + 381, + 429, + 414, + 441 + ], + "score": 0.88, + "content": "\\mathrm { I R } \\mathrm { v } 2 _ { \\mathrm { a d v } } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 429, + 427, + 442 + ], + "score": 1.0, + "content": ") to", + "type": "text" + }, + { + "bbox": [ + 428, + 429, + 447, + 440 + ], + "score": 0.89, + "content": "1 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 429, + 506, + 442 + ], + "score": 1.0, + "content": ". As expected,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 452 + ], + "score": 1.0, + "content": "not solely training on a white-box single-step attack reduces gradient masking. We also verify that", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 451, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 430, + 463 + ], + "score": 1.0, + "content": "after Ensemble Adversarial Training, a two-step iterative attack outperforms the", + "type": "text" + }, + { + "bbox": [ + 430, + 451, + 444, + 461 + ], + "score": 0.44, + "content": "\\mathrm { R } +", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 451, + 506, + 463 + ], + "score": 1.0, + "content": "Step-LL attack", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 461, + 488, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 488, + 474 + ], + "score": 1.0, + "content": "from Section 4.1, thus providing further evidence that these models have meaningful gradients.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "Finally, we revisit the “Gradient-Aligned Adversarial Subspace” (GAAS) method of Tramer et al. `", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "(2017). Their method estimates the size of the space of adversarial examples in the vicinity of a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 104, + 500, + 347, + 514 + ], + "score": 1.0, + "content": "point, by finding a set of orthogonal perturbations of norm", + "type": "text" + }, + { + "bbox": [ + 348, + 503, + 353, + 511 + ], + "score": 0.66, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "that are all adversarial. We note that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "adversarial perturbations do not technically form a “subspace” (e.g., the 0 vector is not adversarial).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 521, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 104, + 521, + 432, + 537 + ], + "score": 1.0, + "content": "Rather, they may form a “cone”, the dimension of which varies as we increase", + "type": "text" + }, + { + "bbox": [ + 433, + 525, + 438, + 533 + ], + "score": 0.56, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 521, + 506, + 537 + ], + "score": 1.0, + "content": ". By linearizing", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 461, + 546 + ], + "score": 1.0, + "content": "the loss function, estimating the dimensionality of this cone reduces to finding vectors", + "type": "text" + }, + { + "bbox": [ + 461, + 535, + 470, + 545 + ], + "score": 0.85, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "that are", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 544, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 275, + 559 + ], + "score": 1.0, + "content": "strongly aligned with the model’s gradient", + "type": "text" + }, + { + "bbox": [ + 275, + 545, + 362, + 556 + ], + "score": 0.89, + "content": "g = \\nabla _ { x } L ( h ( x ) , y _ { \\mathrm { t r u e } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 544, + 506, + 559 + ], + "score": 1.0, + "content": ". Tramer et al. (2017) give a method `", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 553, + 507, + 573 + ], + "spans": [ + { + "bbox": [ + 104, + 553, + 145, + 573 + ], + "score": 1.0, + "content": "that finds", + "type": "text" + }, + { + "bbox": [ + 145, + 556, + 152, + 565 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 553, + 228, + 573 + ], + "score": 1.0, + "content": "orthogonal vectors", + "type": "text" + }, + { + "bbox": [ + 228, + 557, + 237, + 567 + ], + "score": 0.85, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 553, + 283, + 573 + ], + "score": 1.0, + "content": "that satisfy", + "type": "text" + }, + { + "bbox": [ + 284, + 556, + 361, + 570 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\boldsymbol { g } ^ { \\intercal } \\boldsymbol { r } _ { i } \\geq \\dot { \\epsilon } \\cdot \\| \\boldsymbol { g } \\| _ { 2 } \\cdot \\frac { 1 } { \\sqrt { k } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 553, + 507, + 573 + ], + "score": 1.0, + "content": "(this bound is tight). We extend this", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 155, + 581 + ], + "score": 1.0, + "content": "result to the", + "type": "text" + }, + { + "bbox": [ + 155, + 569, + 169, + 581 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "norm, an open question in Tramer et al. (2017). In Section E, we give a randomized `", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 579, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 329, + 593 + ], + "score": 1.0, + "content": "combinatorial construction (Colbourn, 2010), that finds", + "type": "text" + }, + { + "bbox": [ + 330, + 581, + 336, + 590 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 579, + 414, + 593 + ], + "score": 1.0, + "content": "orthogonal vectors", + "type": "text" + }, + { + "bbox": [ + 414, + 582, + 423, + 591 + ], + "score": 0.85, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 579, + 466, + 593 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 466, + 579, + 505, + 592 + ], + "score": 0.93, + "content": "\\| r _ { i } \\| _ { \\infty } =", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 590, + 397, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 112, + 602 + ], + "score": 0.67, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 590, + 130, + 609 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 130, + 591, + 231, + 606 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathbb { E } \\left[ \\boldsymbol { g } ^ { \\top } \\boldsymbol { r } _ { i } \\right] \\ge \\epsilon \\cdot \\| \\boldsymbol { g } \\| _ { 1 } \\cdot \\frac { 1 } { \\sqrt { k } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 590, + 397, + 609 + ], + "score": 1.0, + "content": ". We show that this result is tight as well.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 171, + 625 + ], + "score": 1.0, + "content": "For models v3,", + "type": "text" + }, + { + "bbox": [ + 171, + 611, + 193, + 623 + ], + "score": 0.89, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 610, + 212, + 625 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 212, + 612, + 249, + 623 + ], + "score": 0.9, + "content": "{ \\mathbf { v } } 3 _ { \\mathrm { a d v - e n s } 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 610, + 479, + 625 + ], + "score": 1.0, + "content": ", we select 500 correctly classified test points. For each", + "type": "text" + }, + { + "bbox": [ + 479, + 613, + 486, + 621 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 610, + 506, + 625 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 385, + 635 + ], + "score": 1.0, + "content": "search for a maximal number of orthogonal adversarial perturbations", + "type": "text" + }, + { + "bbox": [ + 386, + 623, + 395, + 633 + ], + "score": 0.85, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 622, + 417, + 635 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 417, + 622, + 463, + 633 + ], + "score": 0.91, + "content": "\\| r _ { i } \\| _ { \\infty } = \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 622, + 506, + 635 + ], + "score": 1.0, + "content": ". We limit", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 160, + 646 + ], + "score": 1.0, + "content": "our search to", + "type": "text" + }, + { + "bbox": [ + 161, + 633, + 196, + 644 + ], + "score": 0.91, + "content": "k \\leq 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 632, + 407, + 646 + ], + "score": 1.0, + "content": "directions per point. The results are in Figure 2. 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For the", + "type": "text" + }, + { + "bbox": [ + 325, + 397, + 362, + 408 + ], + "score": 0.91, + "content": "{ \\mathbf { v } } 3 _ { \\mathrm { a d v - e n s } 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 395, + 381, + 411 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 381, + 397, + 419, + 407 + ], + "score": 0.89, + "content": "{ \\mathbf { v } } 3 _ { \\mathrm { a d v - e n s } 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 395, + 507, + 411 + ], + "score": 1.0, + "content": "models, we find that", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 360, + 420 + ], + "score": 1.0, + "content": "the loss incurred on a Step-LL attack gets within respectively", + "type": "text" + }, + { + "bbox": [ + 360, + 407, + 380, + 418 + ], + "score": 0.89, + "content": "1 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 407, + 399, + 420 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 399, + 408, + 419, + 418 + ], + "score": 0.88, + "content": "1 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "of the optimum loss", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 247, + 430 + ], + "score": 1.0, + "content": "(we recall that for models v3 and", + "type": "text" + }, + { + "bbox": [ + 248, + 419, + 269, + 430 + ], + "score": 0.9, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 417, + 443, + 430 + ], + "score": 1.0, + "content": ", the approximation ratio was respectively", + "type": "text" + }, + { + "bbox": [ + 444, + 418, + 464, + 429 + ], + "score": 0.89, + "content": "1 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 417, + 483, + 430 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 483, + 418, + 498, + 429 + ], + "score": 0.83, + "content": "7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 498, + 417, + 505, + 430 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 104, + 429, + 176, + 442 + ], + "score": 1.0, + "content": "Similarly, for the", + "type": "text" + }, + { + "bbox": [ + 177, + 429, + 220, + 441 + ], + "score": 0.87, + "content": "\\mathrm { I R v } 2 _ { \\mathrm { a d v - e n s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 429, + 347, + 442 + ], + "score": 1.0, + "content": "model, the ratio improves from", + "type": "text" + }, + { + "bbox": [ + 348, + 429, + 362, + 440 + ], + "score": 0.86, + "content": "8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 429, + 381, + 442 + ], + "score": 1.0, + "content": "(for", + "type": "text" + }, + { + "bbox": [ + 381, + 429, + 414, + 441 + ], + "score": 0.88, + "content": "\\mathrm { I R } \\mathrm { v } 2 _ { \\mathrm { a d v } } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 429, + 427, + 442 + ], + "score": 1.0, + "content": ") to", + "type": "text" + }, + { + "bbox": [ + 428, + 429, + 447, + 440 + ], + "score": 0.89, + "content": "1 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 429, + 506, + 442 + ], + "score": 1.0, + "content": ". 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We also verify that", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 451, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 430, + 463 + ], + "score": 1.0, + "content": "after Ensemble Adversarial Training, a two-step iterative attack outperforms the", + "type": "text" + }, + { + "bbox": [ + 430, + 451, + 444, + 461 + ], + "score": 0.44, + "content": "\\mathrm { R } +", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 451, + 506, + 463 + ], + "score": 1.0, + "content": "Step-LL attack", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 461, + 488, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 488, + 474 + ], + "score": 1.0, + "content": "from Section 4.1, thus providing further evidence that these models have meaningful gradients.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 384, + 507, + 474 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "Finally, we revisit the “Gradient-Aligned Adversarial Subspace” (GAAS) method of Tramer et al. `", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "(2017). Their method estimates the size of the space of adversarial examples in the vicinity of a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 104, + 500, + 347, + 514 + ], + "score": 1.0, + "content": "point, by finding a set of orthogonal perturbations of norm", + "type": "text" + }, + { + "bbox": [ + 348, + 503, + 353, + 511 + ], + "score": 0.66, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "that are all adversarial. We note that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "adversarial perturbations do not technically form a “subspace” (e.g., the 0 vector is not adversarial).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 521, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 104, + 521, + 432, + 537 + ], + "score": 1.0, + "content": "Rather, they may form a “cone”, the dimension of which varies as we increase", + "type": "text" + }, + { + "bbox": [ + 433, + 525, + 438, + 533 + ], + "score": 0.56, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 521, + 506, + 537 + ], + "score": 1.0, + "content": ". By linearizing", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 461, + 546 + ], + "score": 1.0, + "content": "the loss function, estimating the dimensionality of this cone reduces to finding vectors", + "type": "text" + }, + { + "bbox": [ + 461, + 535, + 470, + 545 + ], + "score": 0.85, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "that are", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 544, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 275, + 559 + ], + "score": 1.0, + "content": "strongly aligned with the model’s gradient", + "type": "text" + }, + { + "bbox": [ + 275, + 545, + 362, + 556 + ], + "score": 0.89, + "content": "g = \\nabla _ { x } L ( h ( x ) , y _ { \\mathrm { t r u e } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 544, + 506, + 559 + ], + "score": 1.0, + "content": ". Tramer et al. (2017) give a method `", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 553, + 507, + 573 + ], + "spans": [ + { + "bbox": [ + 104, + 553, + 145, + 573 + ], + "score": 1.0, + "content": "that finds", + "type": "text" + }, + { + "bbox": [ + 145, + 556, + 152, + 565 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 553, + 228, + 573 + ], + "score": 1.0, + "content": "orthogonal vectors", + "type": "text" + }, + { + "bbox": [ + 228, + 557, + 237, + 567 + ], + "score": 0.85, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 553, + 283, + 573 + ], + "score": 1.0, + "content": "that satisfy", + "type": "text" + }, + { + "bbox": [ + 284, + 556, + 361, + 570 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\boldsymbol { g } ^ { \\intercal } \\boldsymbol { r } _ { i } \\geq \\dot { \\epsilon } \\cdot \\| \\boldsymbol { g } \\| _ { 2 } \\cdot \\frac { 1 } { \\sqrt { k } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 553, + 507, + 573 + ], + "score": 1.0, + "content": "(this bound is tight). We extend this", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 155, + 581 + ], + "score": 1.0, + "content": "result to the", + "type": "text" + }, + { + "bbox": [ + 155, + 569, + 169, + 581 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "norm, an open question in Tramer et al. (2017). In Section E, we give a randomized `", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 579, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 329, + 593 + ], + "score": 1.0, + "content": "combinatorial construction (Colbourn, 2010), that finds", + "type": "text" + }, + { + "bbox": [ + 330, + 581, + 336, + 590 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 579, + 414, + 593 + ], + "score": 1.0, + "content": "orthogonal vectors", + "type": "text" + }, + { + "bbox": [ + 414, + 582, + 423, + 591 + ], + "score": 0.85, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 579, + 466, + 593 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 466, + 579, + 505, + 592 + ], + "score": 0.93, + "content": "\\| r _ { i } \\| _ { \\infty } =", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 590, + 397, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 112, + 602 + ], + "score": 0.67, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 590, + 130, + 609 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 130, + 591, + 231, + 606 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathbb { E } \\left[ \\boldsymbol { g } ^ { \\top } \\boldsymbol { r } _ { i } \\right] \\ge \\epsilon \\cdot \\| \\boldsymbol { g } \\| _ { 1 } \\cdot \\frac { 1 } { \\sqrt { k } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 590, + 397, + 609 + ], + "score": 1.0, + "content": ". We show that this result is tight as well.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 479, + 507, + 609 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 171, + 625 + ], + "score": 1.0, + "content": "For models v3,", + "type": "text" + }, + { + "bbox": [ + 171, + 611, + 193, + 623 + ], + "score": 0.89, + "content": "{ \\mathrm { v } } 3 _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 610, + 212, + 625 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 212, + 612, + 249, + 623 + ], + "score": 0.9, + "content": "{ \\mathbf { v } } 3 _ { \\mathrm { a d v - e n s } 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 610, + 479, + 625 + ], + "score": 1.0, + "content": ", we select 500 correctly classified test points. For each", + "type": "text" + }, + { + "bbox": [ + 479, + 613, + 486, + 621 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 610, + 506, + 625 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 385, + 635 + ], + "score": 1.0, + "content": "search for a maximal number of orthogonal adversarial perturbations", + "type": "text" + }, + { + "bbox": [ + 386, + 623, + 395, + 633 + ], + "score": 0.85, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 622, + 417, + 635 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 417, + 622, + 463, + 633 + ], + "score": 0.91, + "content": "\\| r _ { i } \\| _ { \\infty } = \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 622, + 506, + 635 + ], + "score": 1.0, + "content": ". We limit", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 160, + 646 + ], + "score": 1.0, + "content": "our search to", + "type": "text" + }, + { + "bbox": [ + 161, + 633, + 196, + 644 + ], + "score": 0.91, + "content": "k \\leq 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 632, + 407, + 646 + ], + "score": 1.0, + "content": "directions per point. The results are in Figure 2. For", + "type": "text" + }, + { + "bbox": [ + 408, + 633, + 469, + 645 + ], + "score": 0.9, + "content": "\\epsilon \\in \\{ 4 , 1 0 , 1 6 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 632, + 506, + 646 + ], + "score": 1.0, + "content": ", we plot", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 275, + 657 + ], + "score": 1.0, + "content": "the proportion of points that have at least", + "type": "text" + }, + { + "bbox": [ + 275, + 644, + 282, + 654 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 644, + 480, + 657 + ], + "score": 1.0, + "content": "orthogonal adversarial perturbations. For a fixed", + "type": "text" + }, + { + "bbox": [ + 481, + 646, + 486, + 654 + ], + "score": 0.65, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 644, + 505, + 657 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 141, + 668 + ], + "score": 1.0, + "content": "value of", + "type": "text" + }, + { + "bbox": [ + 141, + 655, + 148, + 665 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "can be interpreted as the dimension of a “slice” of the cone of adversarial examples near", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "a data point. For the standard Inception v3 model, we find over 50 orthogonal adversarial directions", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 120, + 690 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 677, + 140, + 687 + ], + "score": 0.89, + "content": "3 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 676, + 215, + 690 + ], + "score": 1.0, + "content": "of the points. The", + "type": "text" + }, + { + "bbox": [ + 215, + 677, + 237, + 688 + ], + "score": 0.9, + "content": "{ \\bf V } { \\boldsymbol 3 } _ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 676, + 435, + 690 + ], + "score": 1.0, + "content": "model shows a curious bimodal phenomenon for", + "type": "text" + }, + { + "bbox": [ + 435, + 677, + 464, + 688 + ], + "score": 0.89, + "content": "\\epsilon \\geq 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 676, + 506, + 690 + ], + "score": 1.0, + "content": ": for most", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 134, + 700 + ], + "score": 1.0, + "content": "points", + "type": "text" + }, + { + "bbox": [ + 134, + 688, + 168, + 699 + ], + "score": 0.88, + "content": "( \\approx 8 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "), we find no adversarial direction aligned with the gradient, which is consistent with", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "the gradient masking effect. Yet, for most of the remaining points, the adversarial space is very", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 181, + 722 + ], + "score": 1.0, + "content": "high-dimensional", + "type": "text" + }, + { + "bbox": [ + 182, + 710, + 214, + 721 + ], + "score": 0.87, + "content": "k \\geq 9 0", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "). Ensemble Adversarial Training yields a more robust model, with only", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 330, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 330, + 733 + ], + "score": 1.0, + "content": "a small fraction of points near a large adversarial space.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 39, + "bbox_fs": [ + 104, + 610, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 303, + 94 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 304, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 304, + 96 + ], + "score": 1.0, + "content": "5 CONCLUSION AND FUTURE WORK", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 183 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "Previous work on adversarial training at scale has produced encouraging results, showing strong", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "robustness to (single-step) adversarial examples (Goodfellow et al., 2014b; Kurakin et al., 2017b).", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "Yet, these results are misleading, as the adversarially trained models remain vulnerable to simple", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 153 + ], + "score": 1.0, + "content": "black-box and white-box attacks. Our results, generic with respect to the application domain, sug-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 506, + 163 + ], + "score": 1.0, + "content": "gest that adversarial training can be improved by decoupling the generation of adversarial examples", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 505, + 173 + ], + "score": 1.0, + "content": "from the model being trained. Our experiments with Ensemble Adversarial Training show that the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 173, + 459, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 459, + 183 + ], + "score": 1.0, + "content": "robustness attained to attacks from some models transfers to attacks from other models.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 189, + 504, + 244 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 202 + ], + "score": 1.0, + "content": "We did not consider black-box adversaries that attack a model via other means than by transferring", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 199, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 506, + 212 + ], + "score": 1.0, + "content": "examples from a local model. For instance, generative techniques (Baluja & Fischer, 2017) might", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "provide an avenue for stronger attacks. Yet, a recent work by Xiao et al. (2018) found Ensemble", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 219, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 236 + ], + "score": 1.0, + "content": "Adversarial Training to be resilient to such attacks on MNIST and CIFAR10, and often attaining", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 233, + 432, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 432, + 245 + ], + "score": 1.0, + "content": "higher robustness than models that were adversarially trained on iterative attacks.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 249, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 262 + ], + "score": 1.0, + "content": "Moreover, interactive adversaries (see Appendix A) could try to exploit queries to the target model’s", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 260, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 506, + 274 + ], + "score": 1.0, + "content": "prediction function in their attack, as demonstrated in Papernot et al. (2017). If queries to the target", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "model yield prediction confidences, an adversary can estimate the target’s gradient at a given point", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 282, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 434, + 294 + ], + "score": 1.0, + "content": "(e.g., using finite-differences as in Chen et al. (2017)) and fool the target with our", + "type": "text" + }, + { + "bbox": [ + 434, + 282, + 475, + 293 + ], + "score": 0.62, + "content": "\\mathrm { R + F G S M }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 282, + 505, + 294 + ], + "score": 1.0, + "content": "attack.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 505, + 306 + ], + "score": 1.0, + "content": "Note that if queries only return the predicted label, the attack does not apply. Exploring the impact of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 303, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 318 + ], + "score": 1.0, + "content": "these classes of black-box attacks and evaluating their scalability to complex tasks is an interesting", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 316, + 202, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 202, + 326 + ], + "score": 1.0, + "content": "avenue for future work.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 108, + 344, + 218, + 355 + ], + "lines": [ + { + "bbox": [ + 106, + 343, + 220, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 220, + 358 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 367, + 505, + 456 + ], + "lines": [ + { + "bbox": [ + 106, + 368, + 504, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 504, + 379 + ], + "score": 1.0, + "content": "We thank Ben Poole and Jacob Steinhardt for feedback on early versions of this work. 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(2017). If queries to the target", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "model yield prediction confidences, an adversary can estimate the target’s gradient at a given point", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 282, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 434, + 294 + ], + "score": 1.0, + "content": "(e.g., using finite-differences as in Chen et al. (2017)) and fool the target with our", + "type": "text" + }, + { + "bbox": [ + 434, + 282, + 475, + 293 + ], + "score": 0.62, + "content": "\\mathrm { R + F G S M }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 282, + 505, + 294 + ], + "score": 1.0, + "content": "attack.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 505, + 306 + ], + "score": 1.0, + "content": "Note that if queries only return the predicted label, the attack does not apply. 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Nicolas", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "score": 1.0, + "content": "Papernot is supported by a Google PhD Fellowship in Security. Research was supported in part by", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "the Army Research Laboratory, under Cooperative Agreement Number W911NF-13-2-0045 (ARL", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "Cyber Security CRA), and the Army Research Office under grant W911NF-13-1-0421. 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We evaluate success of the attack as the error rate of the target model over", + "type": "text" + }, + { + "bbox": [ + 449, + 218, + 469, + 228 + ], + "score": 0.89, + "content": "X ^ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 216, + 475, + 232 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 231, + 374, + 263 + ], + "lines": [ + { + "bbox": [ + 236, + 231, + 374, + 263 + ], + "spans": [ + { + "bbox": [ + 236, + 231, + 374, + 263 + ], + "score": 0.94, + "content": "\\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\mathbb { 1 } ( \\arg \\operatorname* { m a x } h ( x _ { i } ^ { \\mathrm { a d v } } ) \\neq y _ { i } ) ~ .", + "type": "interline_equation", + "image_path": "7824d0154feca31f51f6d92c69765b2b67f54d8c254db27cbca380c77d387d2c.jpg" + } + ] + } + ], + "index": 11.5, + "virtual_lines": [ + { + "bbox": [ + 236, + 231, + 374, + 247.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 236, + 247.0, + 374, + 263.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 263, + 499, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 262, + 500, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 154, + 276 + ], + "score": 1.0, + "content": "We assume", + "type": "text" + }, + { + "bbox": [ + 154, + 264, + 163, + 273 + ], + "score": 0.8, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 262, + 372, + 276 + ], + "score": 1.0, + "content": "can sample inputs according to the data distribution", + "type": "text" + }, + { + "bbox": [ + 372, + 264, + 381, + 273 + ], + "score": 0.81, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 262, + 500, + 276 + ], + "score": 1.0, + "content": ". We define three adversaries.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 276, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 342, + 289 + ], + "score": 1.0, + "content": "Definition 2 (White-Box Adversary). For a target model", + "type": "text" + }, + { + "bbox": [ + 343, + 276, + 372, + 287 + ], + "score": 0.89, + "content": "h \\in \\mathcal H", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 276, + 505, + 289 + ], + "score": 1.0, + "content": ", a white-box adversary is given", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 475, + 300 + ], + "score": 1.0, + "content": "access to all elements of the training procedure, that is train (the training algorithm),", + "type": "text" + }, + { + "bbox": [ + 475, + 288, + 485, + 298 + ], + "score": 0.77, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 287, + 505, + 300 + ], + "score": 1.0, + "content": "(the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 298, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 263, + 312 + ], + "score": 1.0, + "content": "model architecture), the training data", + "type": "text" + }, + { + "bbox": [ + 264, + 298, + 312, + 310 + ], + "score": 0.75, + "content": "X _ { t r a i n } , Y _ { t r a i n } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 298, + 383, + 312 + ], + "score": 1.0, + "content": ", the randomness", + "type": "text" + }, + { + "bbox": [ + 383, + 301, + 389, + 308 + ], + "score": 0.5, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 298, + 473, + 312 + ], + "score": 1.0, + "content": "and the parameters", + "type": "text" + }, + { + "bbox": [ + 474, + 299, + 480, + 308 + ], + "score": 0.54, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 298, + 505, + 312 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 310, + 436, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 436, + 322 + ], + "score": 1.0, + "content": "adversary can use any attack (e.g., those in Section 3.2) to find adversarial inputs.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 328, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 340 + ], + "score": 1.0, + "content": "White-box access to the internal model weights corresponds to a very strong adversarial model. We", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 338, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 353 + ], + "score": 1.0, + "content": "thus also consider the following relaxed and arguably more realistic notion of a black-box adversary.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 402, + 365 + ], + "score": 1.0, + "content": "Definition 3 (Non-Interactive Black-Box Adversary). For a target model", + "type": "text" + }, + { + "bbox": [ + 402, + 352, + 430, + 363 + ], + "score": 0.87, + "content": "h \\in \\mathcal H", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 352, + 505, + 365 + ], + "score": 1.0, + "content": ", a non-interactive", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 476, + 376 + ], + "score": 1.0, + "content": "black-box adversary only gets access to train (the target model’s training procedure) and", + "type": "text" + }, + { + "bbox": [ + 476, + 364, + 486, + 374 + ], + "score": 0.72, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "(the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 372, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 419, + 388 + ], + "score": 1.0, + "content": "model architecture). The adversary can sample from the data distribution", + "type": "text" + }, + { + "bbox": [ + 419, + 375, + 428, + 384 + ], + "score": 0.73, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 372, + 506, + 388 + ], + "score": 1.0, + "content": ", and uses a local", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 103, + 383, + 292, + 401 + ], + "spans": [ + { + "bbox": [ + 103, + 383, + 292, + 401 + ], + "score": 1.0, + "content": "algorithm to craft adversarial examples Xadv.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 404, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 419 + ], + "score": 1.0, + "content": "Attacks based on transferability (Szegedy et al., 2013) fall in this category, wherein the adversary", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 310, + 429 + ], + "score": 1.0, + "content": "selects a procedure train0 and model architecture", + "type": "text" + }, + { + "bbox": [ + 311, + 416, + 323, + 427 + ], + "score": 0.88, + "content": "\\mathcal { H } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 417, + 405, + 429 + ], + "score": 1.0, + "content": ", trains a local model", + "type": "text" + }, + { + "bbox": [ + 406, + 416, + 415, + 427 + ], + "score": 0.87, + "content": "\\mathit { \\Pi } _ { h ^ { ' } }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 417, + 435, + 429 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 435, + 417, + 444, + 427 + ], + "score": 0.79, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 417, + 505, + 429 + ], + "score": 1.0, + "content": ", and computes", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 427, + 412, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 266, + 440 + ], + "score": 1.0, + "content": "adversarial examples on its local model", + "type": "text" + }, + { + "bbox": [ + 266, + 427, + 276, + 438 + ], + "score": 0.87, + "content": "h ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 428, + 412, + 440 + ], + "score": 1.0, + "content": "using white-box attack strategies.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 106, + 443, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 400, + 458 + ], + "score": 1.0, + "content": "Most importantly, a black-box adversary does not learn the randomness", + "type": "text" + }, + { + "bbox": [ + 401, + 447, + 407, + 455 + ], + "score": 0.65, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 443, + 505, + 458 + ], + "score": 1.0, + "content": "used to train the target,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 215, + 468 + ], + "score": 1.0, + "content": "nor the target’s parameters", + "type": "text" + }, + { + "bbox": [ + 216, + 456, + 223, + 465 + ], + "score": 0.75, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 455, + 506, + 468 + ], + "score": 1.0, + "content": ". The black-box adversaries in our paper are actually slightly stronger", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 465, + 504, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 375, + 482 + ], + "score": 1.0, + "content": "than the ones defined above, in that they use the same training data", + "type": "text" + }, + { + "bbox": [ + 375, + 467, + 421, + 479 + ], + "score": 0.75, + "content": "X _ { \\mathrm { t r a i n } } , Y _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 465, + 504, + 482 + ], + "score": 1.0, + "content": "as the target model.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 154, + 495 + ], + "score": 1.0, + "content": "We provide", + "type": "text" + }, + { + "bbox": [ + 155, + 484, + 164, + 494 + ], + "score": 0.64, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "with the target’s training procedure train to capture knowledge of defensive strate-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 494, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 104, + 494, + 506, + 507 + ], + "score": 1.0, + "content": "gies applied at training time, e.g., adversarial training (Szegedy et al., 2013; Goodfellow et al.,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 476, + 518 + ], + "score": 1.0, + "content": "2014b) or ensemble adversarial training (see Section 4.2). For ensemble adversarial training,", + "type": "text" + }, + { + "bbox": [ + 477, + 506, + 486, + 515 + ], + "score": 0.72, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 505, + 505, + 518 + ], + "score": 1.0, + "content": "also", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 528 + ], + "score": 1.0, + "content": "knows the architectures of all pre-trained models. In this work, we always mount black-box attacks", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "that train a local model with a different architecture than the target model. We actually find that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 537, + 459, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 459, + 551 + ], + "score": 1.0, + "content": "black-box attacks on adversarially trained models are stronger in this case (see Table 1).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "The main focus of our paper is on non-interactive black-box adversaries as defined above. For com-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "pleteness, we also formalize a stronger notion of interactive black-box adversaries that additionally", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "issue prediction queries to the target model (Papernot et al., 2017). We note that in cases where ML", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "models are deployed as part of a larger system (e.g., a self driving car), an adversary may not have", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 599, + 283, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 283, + 612 + ], + "score": 1.0, + "content": "direct access to the model’s query interface.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 612, + 505, + 670 + ], + "lines": [ + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 385, + 625 + ], + "score": 1.0, + "content": "Definition 4 (Interactive Black-Box Adversary). 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The adversary issues (adaptive) oracle queries to the target model. That is, for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 645, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 172, + 660 + ], + "score": 1.0, + "content": "arbitrary inputs", + "type": "text" + }, + { + "bbox": [ + 172, + 645, + 216, + 659 + ], + "score": 0.93, + "content": "x \\in [ 0 , 1 ] ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 645, + 309, + 660 + ], + "score": 1.0, + "content": ", the adversary obtains", + "type": "text" + }, + { + "bbox": [ + 309, + 646, + 384, + 658 + ], + "score": 0.85, + "content": "y = \\arg \\operatorname* { m a x } h ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 645, + 505, + 660 + ], + "score": 1.0, + "content": "and uses a local algorithm to", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 657, + 421, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 298, + 672 + ], + "score": 1.0, + "content": "craft adversarial examples (given knowledge of", + "type": "text" + }, + { + "bbox": [ + 298, + 658, + 307, + 668 + ], + "score": 0.7, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 657, + 390, + 672 + ], + "score": 1.0, + "content": ", train, and tuples", + "type": "text" + }, + { + "bbox": [ + 391, + 658, + 415, + 670 + ], + "score": 0.9, + "content": "( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 657, + 421, + 672 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "Papernot et al. 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Note that if the target model’s prediction interface additionally", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 186, + 712 + ], + "score": 1.0, + "content": "returns class scores", + "type": "text" + }, + { + "bbox": [ + 186, + 699, + 207, + 711 + ], + "score": 0.92, + "content": "h ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 698, + 505, + 712 + ], + "score": 1.0, + "content": ", interactive black-box adversaries could use queries to the target model to", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "estimate the model’s gradient (e.g., using finite differences) (Chen et al., 2017), and then apply the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 479, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 479, + 732 + ], + "score": 1.0, + "content": "attacks in Section 3.2. We further discuss interactive black-box attack strategies in Section 5.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48 + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "13", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 335, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 337, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 337, + 96 + ], + "score": 1.0, + "content": "A THREAT MODEL: FORMAL DEFINITIONS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 104, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "We provide formal definitions for the threat model introduced in Section 3.1. In the following,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 285, + 128 + ], + "score": 1.0, + "content": "we explicitly identify the hypothesis space", + "type": "text" + }, + { + "bbox": [ + 285, + 117, + 295, + 127 + ], + "score": 0.76, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 117, + 505, + 128 + ], + "score": 1.0, + "content": "that a model belongs to as describing the model’s", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 274, + 140 + ], + "score": 1.0, + "content": "architecture. 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More precisely, we write", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 104, + 505, + 150 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 150, + 371, + 164 + ], + "lines": [ + { + "bbox": [ + 238, + 150, + 371, + 164 + ], + "spans": [ + { + "bbox": [ + 238, + 150, + 371, + 164 + ], + "score": 0.9, + "content": "h \\gets \\mathsf { t r a i n } ( \\mathcal { H } , X _ { \\mathrm { t r a i n } } , Y _ { \\mathrm { t r a i n } } , r ) \\ ,", + "type": "interline_equation", + "image_path": "574c912208ce5ad6f00dbc7752843f6142bc28b6ce68dfb6f2a014ec1485fa18.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 238, + 150, + 371, + 164 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 502, + 187 + ], + "lines": [ + { + "bbox": [ + 106, + 164, + 504, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 504, + 177 + ], + "score": 1.0, + "content": "where train is a randomized training procedure that takes in a description of the model architec-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 175, + 387, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 124, + 189 + ], + "score": 1.0, + "content": "ture", + "type": "text" + }, + { + "bbox": [ + 124, + 176, + 134, + 186 + ], + "score": 0.79, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 175, + 192, + 189 + ], + "score": 1.0, + "content": ", a training set", + "type": "text" + }, + { + "bbox": [ + 192, + 176, + 238, + 187 + ], + "score": 0.9, + "content": "X _ { \\mathrm { t r a i n } } , Y _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 175, + 297, + 189 + ], + "score": 1.0, + "content": "sampled from", + "type": "text" + }, + { + "bbox": [ + 297, + 176, + 306, + 185 + ], + "score": 0.83, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 175, + 377, + 189 + ], + "score": 1.0, + "content": ", and randomness", + "type": "text" + }, + { + "bbox": [ + 377, + 178, + 383, + 185 + ], + "score": 0.73, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 175, + 387, + 189 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 164, + 504, + 189 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 191, + 505, + 230 + ], + "lines": [ + { + "bbox": [ + 116, + 190, + 509, + 224 + ], + "spans": [ + { + "bbox": [ + 116, + 190, + 205, + 224 + ], + "score": 1.0, + "content": "iven a set of test inputs produces adversarial", + "type": "text" + }, + { + "bbox": [ + 205, + 192, + 341, + 204 + ], + "score": 0.9, + "content": "X , Y = \\{ ( x _ { 1 } , y _ { 1 } ) , \\dots , ( x _ { m } , y _ { m } ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 190, + 363, + 224 + ], + "score": 1.0, + "content": "om, su", + "type": "text" + }, + { + "bbox": [ + 364, + 193, + 373, + 202 + ], + "score": 0.82, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 190, + 395, + 224 + ], + "score": 1.0, + "content": "and a that", + "type": "text" + }, + { + "bbox": [ + 426, + 193, + 449, + 203 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 190, + 509, + 224 + ], + "score": 1.0, + "content": "versaryfor all", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 204, + 475, + 218 + ], + "spans": [ + { + "bbox": [ + 107, + 205, + 116, + 215 + ], + "score": 0.76, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 205, + 349, + 218 + ], + "score": 0.91, + "content": "X ^ { \\mathrm { a d v } } = \\{ x _ { 1 } ^ { \\mathrm { a d v } } , \\ldots , x _ { m } ^ { \\mathrm { a d v } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 204, + 475, + 218 + ], + "score": 0.91, + "content": "\\| x _ { i } - x _ { i } ^ { \\mathrm { a d v } } \\| _ { \\infty } ~ \\leq ~ \\epsilon", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 216, + 475, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 147, + 231 + ], + "score": 0.92, + "content": "i \\in [ 1 , m ]", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 216, + 449, + 232 + ], + "score": 1.0, + "content": ". 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We define three adversaries.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 106, + 262, + 500, + 276 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 276, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 342, + 289 + ], + "score": 1.0, + "content": "Definition 2 (White-Box Adversary). 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The", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 310, + 436, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 436, + 322 + ], + "score": 1.0, + "content": "adversary can use any attack (e.g., those in Section 3.2) to find adversarial inputs.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 276, + 505, + 322 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 328, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 340 + ], + "score": 1.0, + "content": "White-box access to the internal model weights corresponds to a very strong adversarial model. We", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 338, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 353 + ], + "score": 1.0, + "content": "thus also consider the following relaxed and arguably more realistic notion of a black-box adversary.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 402, + 365 + ], + "score": 1.0, + "content": "Definition 3 (Non-Interactive Black-Box Adversary). For a target model", + "type": "text" + }, + { + "bbox": [ + 402, + 352, + 430, + 363 + ], + "score": 0.87, + "content": "h \\in \\mathcal H", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 352, + 505, + 365 + ], + "score": 1.0, + "content": ", a non-interactive", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 476, + 376 + ], + "score": 1.0, + "content": "black-box adversary only gets access to train (the target model’s training procedure) and", + "type": "text" + }, + { + "bbox": [ + 476, + 364, + 486, + 374 + ], + "score": 0.72, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "(the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 372, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 419, + 388 + ], + "score": 1.0, + "content": "model architecture). The adversary can sample from the data distribution", + "type": "text" + }, + { + "bbox": [ + 419, + 375, + 428, + 384 + ], + "score": 0.73, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 372, + 506, + 388 + ], + "score": 1.0, + "content": ", and uses a local", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 103, + 383, + 292, + 401 + ], + "spans": [ + { + "bbox": [ + 103, + 383, + 292, + 401 + ], + "score": 1.0, + "content": "algorithm to craft adversarial examples Xadv.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 103, + 329, + 506, + 401 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 404, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 419 + ], + "score": 1.0, + "content": "Attacks based on transferability (Szegedy et al., 2013) fall in this category, wherein the adversary", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 310, + 429 + ], + "score": 1.0, + "content": "selects a procedure train0 and model architecture", + "type": "text" + }, + { + "bbox": [ + 311, + 416, + 323, + 427 + ], + "score": 0.88, + "content": "\\mathcal { H } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 417, + 405, + 429 + ], + "score": 1.0, + "content": ", trains a local model", + "type": "text" + }, + { + "bbox": [ + 406, + 416, + 415, + 427 + ], + "score": 0.87, + "content": "\\mathit { \\Pi } _ { h ^ { ' } }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 417, + 435, + 429 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 435, + 417, + 444, + 427 + ], + "score": 0.79, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 417, + 505, + 429 + ], + "score": 1.0, + "content": ", and computes", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 427, + 412, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 266, + 440 + ], + "score": 1.0, + "content": "adversarial examples on its local model", + "type": "text" + }, + { + "bbox": [ + 266, + 427, + 276, + 438 + ], + "score": 0.87, + "content": "h ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 428, + 412, + 440 + ], + "score": 1.0, + "content": "using white-box attack strategies.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 404, + 505, + 440 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 106, + 443, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 400, + 458 + ], + "score": 1.0, + "content": "Most importantly, a black-box adversary does not learn the randomness", + "type": "text" + }, + { + "bbox": [ + 401, + 447, + 407, + 455 + ], + "score": 0.65, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 443, + 505, + 458 + ], + "score": 1.0, + "content": "used to train the target,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 215, + 468 + ], + "score": 1.0, + "content": "nor the target’s parameters", + "type": "text" + }, + { + "bbox": [ + 216, + 456, + 223, + 465 + ], + "score": 0.75, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 455, + 506, + 468 + ], + "score": 1.0, + "content": ". The black-box adversaries in our paper are actually slightly stronger", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 465, + 504, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 375, + 482 + ], + "score": 1.0, + "content": "than the ones defined above, in that they use the same training data", + "type": "text" + }, + { + "bbox": [ + 375, + 467, + 421, + 479 + ], + "score": 0.75, + "content": "X _ { \\mathrm { t r a i n } } , Y _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 465, + 504, + 482 + ], + "score": 1.0, + "content": "as the target model.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 443, + 506, + 482 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 154, + 495 + ], + "score": 1.0, + "content": "We provide", + "type": "text" + }, + { + "bbox": [ + 155, + 484, + 164, + 494 + ], + "score": 0.64, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "with the target’s training procedure train to capture knowledge of defensive strate-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 494, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 104, + 494, + 506, + 507 + ], + "score": 1.0, + "content": "gies applied at training time, e.g., adversarial training (Szegedy et al., 2013; Goodfellow et al.,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 476, + 518 + ], + "score": 1.0, + "content": "2014b) or ensemble adversarial training (see Section 4.2). For ensemble adversarial training,", + "type": "text" + }, + { + "bbox": [ + 477, + 506, + 486, + 515 + ], + "score": 0.72, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 505, + 505, + 518 + ], + "score": 1.0, + "content": "also", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 516, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 528 + ], + "score": 1.0, + "content": "knows the architectures of all pre-trained models. In this work, we always mount black-box attacks", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "that train a local model with a different architecture than the target model. We actually find that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 537, + 459, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 459, + 551 + ], + "score": 1.0, + "content": "black-box attacks on adversarially trained models are stronger in this case (see Table 1).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 483, + 506, + 551 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 555, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "The main focus of our paper is on non-interactive black-box adversaries as defined above. For com-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "pleteness, we also formalize a stronger notion of interactive black-box adversaries that additionally", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "issue prediction queries to the target model (Papernot et al., 2017). We note that in cases where ML", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "models are deployed as part of a larger system (e.g., a self driving car), an adversary may not have", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 599, + 283, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 283, + 612 + ], + "score": 1.0, + "content": "direct access to the model’s query interface.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 555, + 505, + 612 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 612, + 505, + 670 + ], + "lines": [ + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 385, + 625 + ], + "score": 1.0, + "content": "Definition 4 (Interactive Black-Box Adversary). For a target model", + "type": "text" + }, + { + "bbox": [ + 386, + 612, + 415, + 623 + ], + "score": 0.89, + "content": "h \\in \\mathcal H", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 612, + 505, + 625 + ], + "score": 1.0, + "content": ", an interactive black-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 449, + 636 + ], + "score": 1.0, + "content": "box adversary only gets access to train (the target model’s training procedure) and", + "type": "text" + }, + { + "bbox": [ + 449, + 624, + 459, + 633 + ], + "score": 0.68, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "(the model", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 633, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 647 + ], + "score": 1.0, + "content": "architecture). The adversary issues (adaptive) oracle queries to the target model. That is, for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 645, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 172, + 660 + ], + "score": 1.0, + "content": "arbitrary inputs", + "type": "text" + }, + { + "bbox": [ + 172, + 645, + 216, + 659 + ], + "score": 0.93, + "content": "x \\in [ 0 , 1 ] ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 645, + 309, + 660 + ], + "score": 1.0, + "content": ", the adversary obtains", + "type": "text" + }, + { + "bbox": [ + 309, + 646, + 384, + 658 + ], + "score": 0.85, + "content": "y = \\arg \\operatorname* { m a x } h ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 645, + 505, + 660 + ], + "score": 1.0, + "content": "and uses a local algorithm to", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 657, + 421, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 298, + 672 + ], + "score": 1.0, + "content": "craft adversarial examples (given knowledge of", + "type": "text" + }, + { + "bbox": [ + 298, + 658, + 307, + 668 + ], + "score": 0.7, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 657, + 390, + 672 + ], + "score": 1.0, + "content": ", train, and tuples", + "type": "text" + }, + { + "bbox": [ + 391, + 658, + 415, + 670 + ], + "score": 0.9, + "content": "( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 657, + 421, + 672 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 612, + 506, + 672 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "Papernot et al. (2017) show that such attacks are possible even if the adversary only gets access to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 686, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 104, + 686, + 238, + 701 + ], + "score": 1.0, + "content": "a small number of samples from", + "type": "text" + }, + { + "bbox": [ + 239, + 688, + 247, + 698 + ], + "score": 0.77, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 686, + 505, + 701 + ], + "score": 1.0, + "content": ". Note that if the target model’s prediction interface additionally", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 186, + 712 + ], + "score": 1.0, + "content": "returns class scores", + "type": "text" + }, + { + "bbox": [ + 186, + 699, + 207, + 711 + ], + "score": 0.92, + "content": "h ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 698, + 505, + 712 + ], + "score": 1.0, + "content": ", interactive black-box adversaries could use queries to the target model to", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "estimate the model’s gradient (e.g., using finite differences) (Chen et al., 2017), and then apply the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 479, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 479, + 732 + ], + "score": 1.0, + "content": "attacks in Section 3.2. We further discuss interactive black-box attack strategies in Section 5.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48, + "bbox_fs": [ + 104, + 676, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 104, + 81, + 478, + 94 + ], + "lines": [ + { + "bbox": [ + 104, + 80, + 479, + 95 + ], + "spans": [ + { + "bbox": [ + 104, + 80, + 479, + 95 + ], + "score": 1.0, + "content": "B GENERALIZATION BOUND FOR ENSEMBLE ADVERSARIAL TRAINING", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 106, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "We provide a formal statement of Theorem 1 in Section 3.4, regarding the generalization guarantees", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 116, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 130 + ], + "score": 1.0, + "content": "of Ensemble Adversarial Training. For simplicity, we assume that the model is trained solely on", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 128, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 505, + 140 + ], + "score": 1.0, + "content": "adversarial examples computed on the pre-trained models (i.e., we ignore the clean training data and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 138, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 152 + ], + "score": 1.0, + "content": "the adversarial examples computed on the model being trained). Our results are easily extended to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 150, + 232, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 232, + 162 + ], + "score": 1.0, + "content": "also consider these data points.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 166, + 505, + 227 + ], + "lines": [ + { + "bbox": [ + 105, + 165, + 504, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 122, + 180 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 167, + 131, + 177 + ], + "score": 0.76, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 165, + 241, + 180 + ], + "score": 1.0, + "content": "be the data distribution and", + "type": "text" + }, + { + "bbox": [ + 241, + 166, + 306, + 178 + ], + "score": 0.89, + "content": "\\mathcal { A } _ { 1 } , \\ldots , \\mathcal { A } _ { k } , \\mathcal { A } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 165, + 480, + 180 + ], + "score": 1.0, + "content": "be adversarial distributions where a sample", + "type": "text" + }, + { + "bbox": [ + 480, + 167, + 504, + 179 + ], + "score": 0.91, + "content": "( x , y )", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 103, + 175, + 507, + 195 + ], + "spans": [ + { + "bbox": [ + 103, + 175, + 210, + 195 + ], + "score": 1.0, + "content": "is obtained by sampling", + "type": "text" + }, + { + "bbox": [ + 210, + 180, + 245, + 192 + ], + "score": 0.91, + "content": "( x , y _ { \\mathrm { t r u e } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 175, + 271, + 195 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 271, + 180, + 280, + 190 + ], + "score": 0.79, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 175, + 345, + 195 + ], + "score": 1.0, + "content": ", computing an", + "type": "text" + }, + { + "bbox": [ + 345, + 178, + 363, + 190 + ], + "score": 0.87, + "content": "x ^ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 175, + 407, + 195 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 408, + 178, + 486, + 192 + ], + "score": 0.92, + "content": "\\| x ^ { \\mathrm { a d v } } - x \\| _ { \\infty } \\leq \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 175, + 507, + 195 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 103, + 189, + 508, + 209 + ], + "spans": [ + { + "bbox": [ + 103, + 189, + 146, + 209 + ], + "score": 1.0, + "content": "returning", + "type": "text" + }, + { + "bbox": [ + 146, + 192, + 191, + 205 + ], + "score": 0.9, + "content": "( x ^ { \\mathrm { a d v } } , y _ { \\mathrm { t r u e } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 189, + 337, + 209 + ], + "score": 1.0, + "content": ". We assume the model is trained on", + "type": "text" + }, + { + "bbox": [ + 338, + 193, + 348, + 203 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 189, + 395, + 209 + ], + "score": 1.0, + "content": "data points", + "type": "text" + }, + { + "bbox": [ + 395, + 193, + 416, + 204 + ], + "score": 0.89, + "content": "Z _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 189, + 447, + 209 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 448, + 192, + 457, + 205 + ], + "score": 0.86, + "content": "\\textstyle { \\frac { N } { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 189, + 508, + 209 + ], + "score": 1.0, + "content": "data points", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 203, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 249, + 218 + ], + "score": 1.0, + "content": "are sampled from each distribution", + "type": "text" + }, + { + "bbox": [ + 250, + 204, + 262, + 216 + ], + "score": 0.89, + "content": "A _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 203, + 281, + 218 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 281, + 204, + 325, + 215 + ], + "score": 0.92, + "content": "1 \\leq i \\leq k", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 203, + 376, + 218 + ], + "score": 1.0, + "content": ". We denote", + "type": "text" + }, + { + "bbox": [ + 376, + 204, + 470, + 216 + ], + "score": 0.92, + "content": "\\mathcal { A } _ { \\mathrm { t r a i n } } = \\{ \\mathcal { A } _ { 1 } , . . . , \\overset { \\because } { \\mathcal { A } } _ { k } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 203, + 506, + 218 + ], + "score": 1.0, + "content": ". At test", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 214, + 358, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 341, + 227 + ], + "score": 1.0, + "content": "time, the model is evaluated on adversarial examples from", + "type": "text" + }, + { + "bbox": [ + 342, + 214, + 354, + 225 + ], + "score": 0.88, + "content": "\\mathcal { A } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 215, + 358, + 227 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 231, + 299, + 243 + ], + "lines": [ + { + "bbox": [ + 106, + 231, + 299, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 157, + 244 + ], + "score": 1.0, + "content": "For a model", + "type": "text" + }, + { + "bbox": [ + 157, + 232, + 185, + 242 + ], + "score": 0.9, + "content": "h \\in \\mathcal H", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 231, + 299, + 244 + ], + "score": 1.0, + "content": "we define the empirical risk", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 246, + 407, + 281 + ], + "lines": [ + { + "bbox": [ + 203, + 246, + 407, + 281 + ], + "spans": [ + { + "bbox": [ + 203, + 246, + 407, + 281 + ], + "score": 0.93, + "content": "\\hat { R } ( h , { \\mathcal A } _ { \\mathrm { t r a i n } } ) : = \\frac { 1 } { N } \\sum _ { ( x ^ { \\mathrm { a d v } } , y _ { \\mathrm { t r u e } } ) \\in Z _ { \\mathrm { t r a i n } } } L ( h ( x ^ { \\mathrm { a d v } } ) , y _ { \\mathrm { t r u e } } ) \\ ,", + "type": "interline_equation", + "image_path": "f95f5a93a7a03ef3af8fd50b6bad89e0003e2ec105c3044243de34c44707df50.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 203, + 246, + 407, + 263.5 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 203, + 263.5, + 407, + 281.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 284, + 348, + 296 + ], + "lines": [ + { + "bbox": [ + 105, + 282, + 348, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 348, + 298 + ], + "score": 1.0, + "content": "and the risk over the target distribution (or future adversary)", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 300, + 396, + 324 + ], + "lines": [ + { + "bbox": [ + 214, + 300, + 396, + 324 + ], + "spans": [ + { + "bbox": [ + 214, + 300, + 396, + 324 + ], + "score": 0.93, + "content": "R ( h , \\mathcal { A } ^ { \\ast } ) : = \\underset { ( x ^ { \\mathrm { a d v } } , y _ { \\mathrm { t r u e } } ) \\sim \\mathcal { A } ^ { \\ast } } { \\mathbb { E } } [ L ( h ( x ^ { \\mathrm { a d v } } ) , y _ { \\mathrm { t r u e } } ) ] .", + "type": "interline_equation", + "image_path": "af0c33838d357dd04f064c1748c3e63cc727f68787a70ecccec78b9da227ca24.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 214, + 300, + 396, + 324 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 328, + 504, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 504, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 492, + 342 + ], + "score": 1.0, + "content": "We further define the average discrepancy distance (Mansour et al., 2009) between distributions", + "type": "text" + }, + { + "bbox": [ + 492, + 330, + 504, + 340 + ], + "score": 0.87, + "content": "A _ { i }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 339, + 299, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 123, + 352 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 339, + 137, + 349 + ], + "score": 0.88, + "content": "\\mathcal { A } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 339, + 276, + 352 + ], + "score": 1.0, + "content": "with respect to a hypothesis space", + "type": "text" + }, + { + "bbox": [ + 276, + 340, + 286, + 349 + ], + "score": 0.83, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 339, + 299, + 352 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 355, + 474, + 389 + ], + "lines": [ + { + "bbox": [ + 120, + 355, + 474, + 389 + ], + "spans": [ + { + "bbox": [ + 120, + 355, + 474, + 389 + ], + "score": 0.94, + "content": "\\operatorname { d i s c } _ { { \\mathcal { H } } } ( { \\mathcal { A } } _ { \\mathrm { t r a i n } } , { \\mathcal { A } } ^ { * } ) : = \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } \\operatorname* { s u p } _ { h _ { 1 } , h _ { 2 } \\in { \\mathcal { H } } } \\left| \\frac { \\mathbb { E } } { A _ { i } } \\big [ \\mathbb { 1 } _ { \\{ h _ { 1 } ( x ^ { \\mathrm { a b } } ) = h _ { 2 } ( x ^ { \\mathrm { a b } } ) \\} } \\big ] - \\underline { { \\mathbb { E } } } _ { * } \\big [ \\mathbb { 1 } _ { \\{ h _ { 1 } ( x ^ { \\mathrm { a b } } ) = h _ { 2 } ( x ^ { \\mathrm { a b } } ) \\} } \\big ] \\right| .", + "type": "interline_equation", + "image_path": "6a12ecf33155f68f7b97ca554c11b0753afc76a027e28f5ae3ddbebb40ad2192.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 120, + 355, + 474, + 366.3333333333333 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 120, + 366.3333333333333, + 474, + 377.66666666666663 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 120, + 377.66666666666663, + 474, + 388.99999999999994 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "This quantity characterizes how “different” the future adversary is from the train-time adversaries.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 219, + 417 + ], + "score": 1.0, + "content": "Intuitively, the distance disc", + "type": "text" + }, + { + "bbox": [ + 219, + 403, + 265, + 416 + ], + "score": 0.85, + "content": "( \\mathcal { A } _ { \\mathrm { t r a i n } } , \\mathcal { A } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "is small if the difference in robustness between two models", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 414, + 504, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 182, + 426 + ], + "score": 1.0, + "content": "to the target attack", + "type": "text" + }, + { + "bbox": [ + 182, + 414, + 195, + 425 + ], + "score": 0.91, + "content": "\\mathcal { A } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 415, + 504, + 426 + ], + "score": 1.0, + "content": "is somewhat similar to the difference in robustness between these two models", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 374, + 438 + ], + "score": 1.0, + "content": "to the attacks used for training (e.g., if the static black-box attacks", + "type": "text" + }, + { + "bbox": [ + 374, + 426, + 387, + 437 + ], + "score": 0.89, + "content": "\\mathbf { \\mathcal { A } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "induce much higher error on", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 436, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 158, + 448 + ], + "score": 1.0, + "content": "some model", + "type": "text" + }, + { + "bbox": [ + 158, + 437, + 169, + 448 + ], + "score": 0.88, + "content": "h _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 436, + 264, + 448 + ], + "score": 1.0, + "content": "than on another model", + "type": "text" + }, + { + "bbox": [ + 265, + 437, + 276, + 448 + ], + "score": 0.88, + "content": "h _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 436, + 471, + 448 + ], + "score": 1.0, + "content": ", then the same should hold for the target attack", + "type": "text" + }, + { + "bbox": [ + 472, + 436, + 485, + 447 + ], + "score": 0.82, + "content": "\\mathcal { A } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 436, + 505, + 448 + ], + "score": 1.0, + "content": "). In", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 447, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 326, + 459 + ], + "score": 1.0, + "content": "other words, the ranking of the robustness of models", + "type": "text" + }, + { + "bbox": [ + 326, + 447, + 357, + 457 + ], + "score": 0.91, + "content": "h \\in \\mathcal H", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 448, + 505, + 459 + ], + "score": 1.0, + "content": "should be similar for the attacks in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 458, + 173, + 471 + ], + "spans": [ + { + "bbox": [ + 107, + 459, + 129, + 470 + ], + "score": 0.9, + "content": "\\mathcal { A } _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 458, + 155, + 471 + ], + "score": 1.0, + "content": "as for", + "type": "text" + }, + { + "bbox": [ + 155, + 458, + 168, + 469 + ], + "score": 0.88, + "content": "\\mathcal { A } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 458, + 173, + 471 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 475, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 152, + 489 + ], + "score": 1.0, + "content": "Finally, let", + "type": "text" + }, + { + "bbox": [ + 152, + 475, + 185, + 487 + ], + "score": 0.93, + "content": "R _ { N } ( \\mathscr { H } )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 474, + 424, + 489 + ], + "score": 1.0, + "content": "be the average Rademacher complexity of the distributions", + "type": "text" + }, + { + "bbox": [ + 424, + 475, + 472, + 487 + ], + "score": 0.91, + "content": "\\mathcal { A } _ { 1 } , \\ldots , \\mathcal { A } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 474, + 505, + 489 + ], + "score": 1.0, + "content": "(Zhang", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 485, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 202, + 500 + ], + "score": 1.0, + "content": "et al., 2012). Note that", + "type": "text" + }, + { + "bbox": [ + 203, + 486, + 258, + 498 + ], + "score": 0.93, + "content": "R _ { N } ( \\mathscr { H } ) \\dot { \\mathbf { \\Omega } } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 485, + 270, + 500 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 271, + 486, + 308, + 496 + ], + "score": 0.9, + "content": "N \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 485, + 505, + 500 + ], + "score": 1.0, + "content": ". The following theorem is a corollary of Zhang", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 496, + 216, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 216, + 510 + ], + "score": 1.0, + "content": "et al. (2012, Theorem 5.2):", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 504, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 504, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 211, + 524 + ], + "score": 1.0, + "content": "Theorem 5. Assume that", + "type": "text" + }, + { + "bbox": [ + 212, + 512, + 221, + 522 + ], + "score": 0.76, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 510, + 504, + 524 + ], + "score": 1.0, + "content": "is a function class consisting of bounded functions. Then, with proba-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 520, + 188, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 161, + 536 + ], + "score": 1.0, + "content": "bility at least", + "type": "text" + }, + { + "bbox": [ + 161, + 523, + 184, + 533 + ], + "score": 0.72, + "content": "1 - \\epsilon ,", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 520, + 188, + 536 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "interline_equation", + "bbox": [ + 124, + 537, + 469, + 571 + ], + "lines": [ + { + "bbox": [ + 124, + 537, + 469, + 571 + ], + "spans": [ + { + "bbox": [ + 124, + 537, + 469, + 571 + ], + "score": 0.91, + "content": "\\operatorname* { s u p } _ { h \\in \\mathcal { H } } \\vert \\hat { R } ( h , { \\mathcal A } _ { t r a i n } ) - R ( h , { \\mathcal A } ^ { * } ) \\vert \\leq d i s c _ { \\mathcal { H } } ( { \\mathcal A } _ { t r a i n } , { \\mathcal A } ^ { * } ) + 2 R _ { N } ( \\mathcal { H } ) + O \\left( \\sqrt { \\frac { \\ln ( 1 / \\epsilon ) } { N } } \\right) .", + "type": "interline_equation", + "image_path": "968b4a6a65dca5201584f84e64ee8b3494df09efba8c9219df7cf5ceb85db708.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 124, + 537, + 469, + 548.3333333333334 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 124, + 548.3333333333334, + 469, + 559.6666666666667 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 124, + 559.6666666666667, + 469, + 571.0000000000001 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 580, + 505, + 648 + ], + "lines": [ + { + "bbox": [ + 106, + 581, + 504, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 504, + 592 + ], + "score": 1.0, + "content": "Compared to the standard generalization bound for supervised learning, the generalization bound for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 591, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 296, + 605 + ], + "score": 1.0, + "content": "Domain Adaptation incorporates the extra term", + "type": "text" + }, + { + "bbox": [ + 297, + 591, + 366, + 604 + ], + "score": 0.92, + "content": "\\mathrm { d i s c } _ { \\mathcal { H } } ( \\mathcal { A } _ { \\mathrm { t r a i n } } , \\mathcal { A } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "to capture the divergence between", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 407, + 615 + ], + "score": 1.0, + "content": "the target and source distributions. 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For this simpler task, Madry et al. 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In", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 447, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 326, + 459 + ], + "score": 1.0, + "content": "other words, the ranking of the robustness of models", + "type": "text" + }, + { + "bbox": [ + 326, + 447, + 357, + 457 + ], + "score": 0.91, + "content": "h \\in \\mathcal H", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 448, + 505, + 459 + ], + "score": 1.0, + "content": "should be similar for the attacks in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 458, + 173, + 471 + ], + "spans": [ + { + "bbox": [ + 107, + 459, + 129, + 470 + ], + "score": 0.9, + "content": "\\mathcal { A } _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 458, + 155, + 471 + ], + "score": 1.0, + "content": "as for", + "type": "text" + }, + { + "bbox": [ + 155, + 458, + 168, + 469 + ], + "score": 0.88, + "content": "\\mathcal { A } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 458, + 173, + 471 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 392, + 506, + 471 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 475, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 152, + 489 + ], + "score": 1.0, + "content": "Finally, let", + "type": "text" + }, + { + "bbox": [ + 152, + 475, + 185, + 487 + ], + "score": 0.93, + "content": "R _ { N } ( \\mathscr { H } )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 474, + 424, + 489 + ], + "score": 1.0, + "content": "be the average Rademacher complexity of the distributions", + "type": "text" + }, + { + "bbox": [ + 424, + 475, + 472, + 487 + ], + "score": 0.91, + "content": "\\mathcal { A } _ { 1 } , \\ldots , \\mathcal { A } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 474, + 505, + 489 + ], + "score": 1.0, + "content": "(Zhang", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 485, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 202, + 500 + ], + "score": 1.0, + "content": "et al., 2012). Note that", + "type": "text" + }, + { + "bbox": [ + 203, + 486, + 258, + 498 + ], + "score": 0.93, + "content": "R _ { N } ( \\mathscr { H } ) \\dot { \\mathbf { \\Omega } } \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 485, + 270, + 500 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 271, + 486, + 308, + 496 + ], + "score": 0.9, + "content": "N \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 485, + 505, + 500 + ], + "score": 1.0, + "content": ". The following theorem is a corollary of Zhang", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 496, + 216, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 216, + 510 + ], + "score": 1.0, + "content": "et al. (2012, Theorem 5.2):", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 474, + 505, + 510 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 504, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 504, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 211, + 524 + ], + "score": 1.0, + "content": "Theorem 5. Assume that", + "type": "text" + }, + { + "bbox": [ + 212, + 512, + 221, + 522 + ], + "score": 0.76, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 510, + 504, + 524 + ], + "score": 1.0, + "content": "is a function class consisting of bounded functions. Then, with proba-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 520, + 188, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 161, + 536 + ], + "score": 1.0, + "content": "bility at least", + "type": "text" + }, + { + "bbox": [ + 161, + 523, + 184, + 533 + ], + "score": 0.72, + "content": "1 - \\epsilon ,", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 520, + 188, + 536 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 510, + 504, + 536 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 124, + 537, + 469, + 571 + ], + "lines": [ + { + "bbox": [ + 124, + 537, + 469, + 571 + ], + "spans": [ + { + "bbox": [ + 124, + 537, + 469, + 571 + ], + "score": 0.91, + "content": "\\operatorname* { s u p } _ { h \\in \\mathcal { H } } \\vert \\hat { R } ( h , { \\mathcal A } _ { t r a i n } ) - R ( h , { \\mathcal A } ^ { * } ) \\vert \\leq d i s c _ { \\mathcal { H } } ( { \\mathcal A } _ { t r a i n } , { \\mathcal A } ^ { * } ) + 2 R _ { N } ( \\mathcal { H } ) + O \\left( \\sqrt { \\frac { \\ln ( 1 / \\epsilon ) } { N } } \\right) .", + "type": "interline_equation", + "image_path": "968b4a6a65dca5201584f84e64ee8b3494df09efba8c9219df7cf5ceb85db708.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 124, + 537, + 469, + 548.3333333333334 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 124, + 548.3333333333334, + 469, + 559.6666666666667 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 124, + 559.6666666666667, + 469, + 571.0000000000001 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 580, + 505, + 648 + ], + "lines": [ + { + "bbox": [ + 106, + 581, + 504, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 504, + 592 + ], + "score": 1.0, + "content": "Compared to the standard generalization bound for supervised learning, the generalization bound for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 591, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 296, + 605 + ], + "score": 1.0, + "content": "Domain Adaptation incorporates the extra term", + "type": "text" + }, + { + "bbox": [ + 297, + 591, + 366, + 604 + ], + "score": 0.92, + "content": "\\mathrm { d i s c } _ { \\mathcal { H } } ( \\mathcal { A } _ { \\mathrm { t r a i n } } , \\mathcal { A } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "to capture the divergence between", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 407, + 615 + ], + "score": 1.0, + "content": "the target and source distributions. 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Note that", + "type": "text" + }, + { + "bbox": [ + 388, + 624, + 401, + 635 + ], + "score": 0.88, + "content": "\\mathcal { A } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 624, + 506, + 637 + ], + "score": 1.0, + "content": "need not restrict itself to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 635, + 342, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 214, + 648 + ], + "score": 1.0, + "content": "perturbation with bounded", + "type": "text" + }, + { + "bbox": [ + 215, + 636, + 228, + 647 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 635, + 342, + 648 + ], + "score": 1.0, + "content": "norm for this result to hold.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 581, + 506, + 648 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 662, + 263, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 662, + 264, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 264, + 677 + ], + "score": 1.0, + "content": "C EXPERIMENTS ON MNIST", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "We re-iterate our ImageNet experiments on MNIST. For this simpler task, Madry et al. (2017) show", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 462, + 711 + ], + "score": 1.0, + "content": "that training on iterative attacks conveys robustness to white-box attacks with bounded", + "type": "text" + }, + { + "bbox": [ + 463, + 699, + 477, + 710 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "norm.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "Our goal is not to attain similarly strong white-box robustness on MNIST, but to show that our ob-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "servations on limitations of single-step adversarial training, extend to other datasets than ImageNet.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 688, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 227, + 82, + 384, + 129 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 227, + 82, + 384, + 129 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 227, + 82, + 384, + 129 + ], + "spans": [ + { + "bbox": [ + 227, + 82, + 384, + 129 + ], + "score": 0.936, + "type": "image", + "image_path": "2b6af3b5b8bdd08fb1a44e32ac65797040e19643a1e40965efbd54da313b7abf.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 227, + 82, + 384, + 97.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 227, + 97.66666666666667, + 384, + 113.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 227, + 113.33333333333334, + 384, + 129.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 133, + 504, + 155 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 131, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 146 + ], + "score": 1.0, + "content": "Figure 3: Adversarial Examples on MNIST. (top) clean examples. (middle) inputs are rotated by", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 144, + 442, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 122, + 154 + ], + "score": 0.85, + "content": "2 0 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 144, + 366, + 155 + ], + "score": 1.0, + "content": "and 5 random pixels are flipped. (bottom) The I-FGSM with", + "type": "text" + }, + { + "bbox": [ + 366, + 144, + 398, + 154 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 144, + 442, + 155 + ], + "score": 1.0, + "content": "is applied.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 167, + 308, + 180 + ], + "lines": [ + { + "bbox": [ + 106, + 167, + 309, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 184, + 181 + ], + "score": 1.0, + "content": "C.1 A NOTE ON", + "type": "text" + }, + { + "bbox": [ + 184, + 169, + 198, + 180 + ], + "score": 0.86, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 167, + 309, + 181 + ], + "score": 1.0, + "content": "ROBUSTNESS ON MNIST", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 189, + 505, + 279 + ], + "lines": [ + { + "bbox": [ + 105, + 188, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 506, + 202 + ], + "score": 1.0, + "content": "The MNIST dataset is a simple baseline for assessing the potential of a defense, but the obtained", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 200, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 505, + 213 + ], + "score": 1.0, + "content": "results do not always generalize to harder tasks. We suggest that this is because achieving robustness", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 211, + 504, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 116, + 223 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 211, + 130, + 223 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 211, + 504, + 223 + ], + "score": 1.0, + "content": "perturbations admits a simple “closed-form” solution, given the near-binary nature of the data.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 222, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 283, + 235 + ], + "score": 1.0, + "content": "Indeed, for an average MNIST image, over", + "type": "text" + }, + { + "bbox": [ + 284, + 222, + 303, + 233 + ], + "score": 0.88, + "content": "8 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 222, + 383, + 235 + ], + "score": 1.0, + "content": "of the pixels are in", + "type": "text" + }, + { + "bbox": [ + 384, + 222, + 409, + 234 + ], + "score": 0.93, + "content": "\\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 222, + 448, + 235 + ], + "score": 1.0, + "content": "and only", + "type": "text" + }, + { + "bbox": [ + 448, + 222, + 463, + 233 + ], + "score": 0.88, + "content": "6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 222, + 506, + 235 + ], + "score": 1.0, + "content": "are in the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 103, + 234, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 103, + 234, + 294, + 248 + ], + "score": 1.0, + "content": "range [0.2, 0.8]. Thus, for a perturbation with", + "type": "text" + }, + { + "bbox": [ + 294, + 235, + 327, + 246 + ], + "score": 0.89, + "content": "\\epsilon \\leq 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 234, + 418, + 248 + ], + "score": 1.0, + "content": ", binarized versions of", + "type": "text" + }, + { + "bbox": [ + 419, + 237, + 425, + 245 + ], + "score": 0.78, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 234, + 444, + 248 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 444, + 234, + 462, + 245 + ], + "score": 0.89, + "content": "x ^ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 234, + 505, + 248 + ], + "score": 1.0, + "content": "can differ", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 149, + 258 + ], + "score": 1.0, + "content": "in at most", + "type": "text" + }, + { + "bbox": [ + 149, + 246, + 164, + 257 + ], + "score": 0.87, + "content": "6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "of the input dimensions. By binarizing the inputs of a standard CNN trained without", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 256, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 505, + 270 + ], + "score": 1.0, + "content": "adversarial training, we obtain a model that enjoys robustness similar to the model trained by Madry", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 268, + 450, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 398, + 280 + ], + "score": 1.0, + "content": "et al. (2017). Concretely, for a white-box I-FGSM attack, we get at most", + "type": "text" + }, + { + "bbox": [ + 398, + 268, + 425, + 279 + ], + "score": 0.87, + "content": "1 1 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 268, + 450, + 280 + ], + "score": 1.0, + "content": "error.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 285, + 505, + 329 + ], + "lines": [ + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "score": 1.0, + "content": "The existence of such a simple robust representation begs the question of why learning a robust", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "model with adversarial training takes so much effort. Finding techniques to improve the performance", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "of adversarial training, even on simple tasks, could provide useful insights for more complex tasks", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 318, + 455, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 455, + 330 + ], + "score": 1.0, + "content": "such as ImageNet, where we do not know of a similarly simple “denoising” procedure.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 334, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 268, + 348 + ], + "score": 1.0, + "content": "These positive results on MNIST for the", + "type": "text" + }, + { + "bbox": [ + 268, + 335, + 282, + 346 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 334, + 505, + 348 + ], + "score": 1.0, + "content": "norm also leave open the question of defining a general", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "norm for adversarial examples. Let us motivate the need for such a definition: we find that if we", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 232, + 369 + ], + "score": 1.0, + "content": "first rotate an MNIST digit by", + "type": "text" + }, + { + "bbox": [ + 233, + 357, + 248, + 367 + ], + "score": 0.86, + "content": "2 0 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 356, + 505, + 369 + ], + "score": 1.0, + "content": ", and then use the I-FGSM, our rounding model and the model", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 261, + 380 + ], + "score": 1.0, + "content": "from Madry et al. (2017) achieve only", + "type": "text" + }, + { + "bbox": [ + 261, + 367, + 281, + 379 + ], + "score": 0.88, + "content": "6 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 367, + 473, + 380 + ], + "score": 1.0, + "content": "accuracy (on “clean” rotated inputs, the error is", + "type": "text" + }, + { + "bbox": [ + 473, + 367, + 498, + 378 + ], + "score": 0.9, + "content": "< 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 367, + 505, + 380 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 482, + 392 + ], + "score": 1.0, + "content": "If we further randomly “flip” 5 pixels per image, the accuracy of both models drops to under", + "type": "text" + }, + { + "bbox": [ + 482, + 379, + 501, + 389 + ], + "score": 0.87, + "content": "5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 378, + 505, + 392 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 389, + 478, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 478, + 402 + ], + "score": 1.0, + "content": "Thus, we successfully evade the model by slightly extending the threat model (see Figure 3).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 406, + 505, + 472 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "Of course, we could augment the training set with such perturbations (see Engstrom et al. (2017)).", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "An open question is whether we can enumerate all types of “adversarial” perturbations. In this", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 195, + 441 + ], + "score": 1.0, + "content": "work, we focus on the", + "type": "text" + }, + { + "bbox": [ + 196, + 429, + 210, + 440 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "norm to illustrate our findings on the limitations of single-step adversarial", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "training on ImageNet and MNIST, and to showcase the benefits of our Ensemble Adversarial Train-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 451, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 463 + ], + "score": 1.0, + "content": "ing variant. Our approach can easily be extended to consider multiple perturbation metrics. We", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 461, + 267, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 267, + 473 + ], + "score": 1.0, + "content": "leave such an evaluation to future work.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 107, + 487, + 172, + 499 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 173, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 173, + 501 + ], + "score": 1.0, + "content": "C.2 RESULTS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 108, + 508, + 504, + 531 + ], + "lines": [ + { + "bbox": [ + 107, + 508, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 107, + 508, + 506, + 521 + ], + "score": 1.0, + "content": "We repeat experiments from Section 4 on MNIST. We use the architectures in Table 5. We train a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 520, + 489, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 394, + 533 + ], + "score": 1.0, + "content": "standard model for 6 epochs, and an adversarial model with the FGSM", + "type": "text" + }, + { + "bbox": [ + 394, + 520, + 427, + 531 + ], + "score": 0.83, + "content": "\\epsilon = 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 520, + 489, + 533 + ], + "score": 1.0, + "content": ") for 12 epochs.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 536, + 502, + 559 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 550 + ], + "score": 1.0, + "content": "During adversarial training, we avoid the label leaking effect described by Kurakin et al. (2017b) by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 547, + 470, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 276, + 560 + ], + "score": 1.0, + "content": "using the model’s predicted class arg max", + "type": "text" + }, + { + "bbox": [ + 277, + 547, + 297, + 560 + ], + "score": 0.9, + "content": "h ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 547, + 394, + 560 + ], + "score": 1.0, + "content": "instead of the true label", + "type": "text" + }, + { + "bbox": [ + 394, + 549, + 412, + 559 + ], + "score": 0.9, + "content": "y _ { \\mathrm { t r u e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 547, + 470, + 560 + ], + "score": 1.0, + "content": "in the FGSM,", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 564, + 505, + 598 + ], + "lines": [ + { + "bbox": [ + 106, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "We first analyze the “degenerate” minimum of adversarial training, described in Section 3.3. For", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 576, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 587 + ], + "score": 1.0, + "content": "each trained model, we compute the approximation-ratio of the FGSM for the inner maximization", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 587, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 599 + ], + "score": 1.0, + "content": "problem in equation (1). That is, we compare the loss produced by the FGSM with the loss of a", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "table", + "bbox": [ + 106, + 637, + 503, + 728 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 610, + 502, + 633 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 608, + 503, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 503, + 623 + ], + "score": 1.0, + "content": "Table 5: Neural network architectures used in this work for the MNIST dataset. 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(top) clean examples. (middle) inputs are rotated by", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 144, + 442, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 122, + 154 + ], + "score": 0.85, + "content": "2 0 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 144, + 366, + 155 + ], + "score": 1.0, + "content": "and 5 random pixels are flipped. (bottom) The I-FGSM with", + "type": "text" + }, + { + "bbox": [ + 366, + 144, + 398, + 154 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 144, + 442, + 155 + ], + "score": 1.0, + "content": "is applied.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 167, + 308, + 180 + ], + "lines": [ + { + "bbox": [ + 106, + 167, + 309, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 184, + 181 + ], + "score": 1.0, + "content": "C.1 A NOTE ON", + "type": "text" + }, + { + "bbox": [ + 184, + 169, + 198, + 180 + ], + "score": 0.86, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 167, + 309, + 181 + ], + "score": 1.0, + "content": "ROBUSTNESS ON MNIST", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 106, + 167, + 309, + 181 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 189, + 505, + 279 + ], + "lines": [ + { + "bbox": [ + 105, + 188, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 506, + 202 + ], + "score": 1.0, + "content": "The MNIST dataset is a simple baseline for assessing the potential of a defense, but the obtained", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 200, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 505, + 213 + ], + "score": 1.0, + "content": "results do not always generalize to harder tasks. We suggest that this is because achieving robustness", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 211, + 504, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 116, + 223 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 211, + 130, + 223 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 211, + 504, + 223 + ], + "score": 1.0, + "content": "perturbations admits a simple “closed-form” solution, given the near-binary nature of the data.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 222, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 283, + 235 + ], + "score": 1.0, + "content": "Indeed, for an average MNIST image, over", + "type": "text" + }, + { + "bbox": [ + 284, + 222, + 303, + 233 + ], + "score": 0.88, + "content": "8 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 222, + 383, + 235 + ], + "score": 1.0, + "content": "of the pixels are in", + "type": "text" + }, + { + "bbox": [ + 384, + 222, + 409, + 234 + ], + "score": 0.93, + "content": "\\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 222, + 448, + 235 + ], + "score": 1.0, + "content": "and only", + "type": "text" + }, + { + "bbox": [ + 448, + 222, + 463, + 233 + ], + "score": 0.88, + "content": "6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 222, + 506, + 235 + ], + "score": 1.0, + "content": "are in the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 103, + 234, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 103, + 234, + 294, + 248 + ], + "score": 1.0, + "content": "range [0.2, 0.8]. Thus, for a perturbation with", + "type": "text" + }, + { + "bbox": [ + 294, + 235, + 327, + 246 + ], + "score": 0.89, + "content": "\\epsilon \\leq 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 234, + 418, + 248 + ], + "score": 1.0, + "content": ", binarized versions of", + "type": "text" + }, + { + "bbox": [ + 419, + 237, + 425, + 245 + ], + "score": 0.78, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 234, + 444, + 248 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 444, + 234, + 462, + 245 + ], + "score": 0.89, + "content": "x ^ { \\mathrm { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 234, + 505, + 248 + ], + "score": 1.0, + "content": "can differ", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 149, + 258 + ], + "score": 1.0, + "content": "in at most", + "type": "text" + }, + { + "bbox": [ + 149, + 246, + 164, + 257 + ], + "score": 0.87, + "content": "6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "of the input dimensions. By binarizing the inputs of a standard CNN trained without", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 256, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 505, + 270 + ], + "score": 1.0, + "content": "adversarial training, we obtain a model that enjoys robustness similar to the model trained by Madry", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 268, + 450, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 398, + 280 + ], + "score": 1.0, + "content": "et al. (2017). Concretely, for a white-box I-FGSM attack, we get at most", + "type": "text" + }, + { + "bbox": [ + 398, + 268, + 425, + 279 + ], + "score": 0.87, + "content": "1 1 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 268, + 450, + 280 + ], + "score": 1.0, + "content": "error.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5, + "bbox_fs": [ + 103, + 188, + 506, + 280 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 285, + 505, + 329 + ], + "lines": [ + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 298 + ], + "score": 1.0, + "content": "The existence of such a simple robust representation begs the question of why learning a robust", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "model with adversarial training takes so much effort. Finding techniques to improve the performance", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "of adversarial training, even on simple tasks, could provide useful insights for more complex tasks", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 318, + 455, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 455, + 330 + ], + "score": 1.0, + "content": "such as ImageNet, where we do not know of a similarly simple “denoising” procedure.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 284, + 506, + 330 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 334, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 268, + 348 + ], + "score": 1.0, + "content": "These positive results on MNIST for the", + "type": "text" + }, + { + "bbox": [ + 268, + 335, + 282, + 346 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 334, + 505, + 348 + ], + "score": 1.0, + "content": "norm also leave open the question of defining a general", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "norm for adversarial examples. Let us motivate the need for such a definition: we find that if we", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 232, + 369 + ], + "score": 1.0, + "content": "first rotate an MNIST digit by", + "type": "text" + }, + { + "bbox": [ + 233, + 357, + 248, + 367 + ], + "score": 0.86, + "content": "2 0 ^ { \\circ }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 356, + 505, + 369 + ], + "score": 1.0, + "content": ", and then use the I-FGSM, our rounding model and the model", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 261, + 380 + ], + "score": 1.0, + "content": "from Madry et al. (2017) achieve only", + "type": "text" + }, + { + "bbox": [ + 261, + 367, + 281, + 379 + ], + "score": 0.88, + "content": "6 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 367, + 473, + 380 + ], + "score": 1.0, + "content": "accuracy (on “clean” rotated inputs, the error is", + "type": "text" + }, + { + "bbox": [ + 473, + 367, + 498, + 378 + ], + "score": 0.9, + "content": "< 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 367, + 505, + 380 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 482, + 392 + ], + "score": 1.0, + "content": "If we further randomly “flip” 5 pixels per image, the accuracy of both models drops to under", + "type": "text" + }, + { + "bbox": [ + 482, + 379, + 501, + 389 + ], + "score": 0.87, + "content": "5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 378, + 505, + 392 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 389, + 478, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 478, + 402 + ], + "score": 1.0, + "content": "Thus, we successfully evade the model by slightly extending the threat model (see Figure 3).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 334, + 505, + 402 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 406, + 505, + 472 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "Of course, we could augment the training set with such perturbations (see Engstrom et al. (2017)).", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "An open question is whether we can enumerate all types of “adversarial” perturbations. In this", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 195, + 441 + ], + "score": 1.0, + "content": "work, we focus on the", + "type": "text" + }, + { + "bbox": [ + 196, + 429, + 210, + 440 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "norm to illustrate our findings on the limitations of single-step adversarial", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "training on ImageNet and MNIST, and to showcase the benefits of our Ensemble Adversarial Train-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 451, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 463 + ], + "score": 1.0, + "content": "ing variant. Our approach can easily be extended to consider multiple perturbation metrics. We", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 461, + 267, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 267, + 473 + ], + "score": 1.0, + "content": "leave such an evaluation to future work.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 406, + 506, + 473 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 487, + 172, + 499 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 173, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 173, + 501 + ], + "score": 1.0, + "content": "C.2 RESULTS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 108, + 508, + 504, + 531 + ], + "lines": [ + { + "bbox": [ + 107, + 508, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 107, + 508, + 506, + 521 + ], + "score": 1.0, + "content": "We repeat experiments from Section 4 on MNIST. We use the architectures in Table 5. We train a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 520, + 489, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 394, + 533 + ], + "score": 1.0, + "content": "standard model for 6 epochs, and an adversarial model with the FGSM", + "type": "text" + }, + { + "bbox": [ + 394, + 520, + 427, + 531 + ], + "score": 0.83, + "content": "\\epsilon = 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 520, + 489, + 533 + ], + "score": 1.0, + "content": ") for 12 epochs.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 106, + 508, + 506, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 536, + 502, + 559 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 550 + ], + "score": 1.0, + "content": "During adversarial training, we avoid the label leaking effect described by Kurakin et al. (2017b) by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 547, + 470, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 276, + 560 + ], + "score": 1.0, + "content": "using the model’s predicted class arg max", + "type": "text" + }, + { + "bbox": [ + 277, + 547, + 297, + 560 + ], + "score": 0.9, + "content": "h ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 547, + 394, + 560 + ], + "score": 1.0, + "content": "instead of the true label", + "type": "text" + }, + { + "bbox": [ + 394, + 549, + 412, + 559 + ], + "score": 0.9, + "content": "y _ { \\mathrm { t r u e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 547, + 470, + 560 + ], + "score": 1.0, + "content": "in the FGSM,", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 535, + 505, + 560 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 564, + 505, + 598 + ], + "lines": [ + { + "bbox": [ + 106, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "We first analyze the “degenerate” minimum of adversarial training, described in Section 3.3. For", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 576, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 587 + ], + "score": 1.0, + "content": "each trained model, we compute the approximation-ratio of the FGSM for the inner maximization", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 587, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 599 + ], + "score": 1.0, + "content": "problem in equation (1). That is, we compare the loss produced by the FGSM with the loss of a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "strong iterative attack. The results appear in Table 6. As we can see, for all model architectures,", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 161, + 452, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 452, + 173 + ], + "score": 1.0, + "content": "adversarial training degraded the quality of a linear approximation to the model’s loss.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 564, + 506, + 599 + ] + }, + { + "type": "table", + "bbox": [ + 106, + 637, + 503, + 728 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 610, + 502, + 633 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 608, + 503, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 503, + 623 + ], + "score": 1.0, + "content": "Table 5: Neural network architectures used in this work for the MNIST dataset. Conv: convo-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 620, + 270, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 270, + 634 + ], + "score": 1.0, + "content": "lutional layer, FC: fully connected layer.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 637, + 503, + 728 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 637, + 503, + 728 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 503, + 728 + ], + "score": 0.984, + "html": "
ABCD
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Conv(64,5,5) + ReluConv(64,8,8) + ReluMaxPool(2,2)Dropout(0.5）
Dropout(0.25)Conv(128,6,6) +ReluConv(64,3,3) + TanhFC +Softmax
FC(128) + ReluConv(128,5,5) +ReluMaxPool(2,2)
Dropout(0.5)Dropout(0.5)FC(128) + Relu
FC + SoftmaxFC + SoftmaxFC + Softmax
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AAadvBBadvB*BdvCCadvDDady
17%0%25%8%23%1%25%0%49%16%
", + "type": "table", + "image_path": "f794efb8aaa8dd491508cb1da781c576c3834f69ba9a421c42ba723c5e898e0d.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 138, + 106, + 469, + 116.66666666666667 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 138, + 116.66666666666667, + 469, + 127.33333333333334 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 138, + 127.33333333333334, + 469, + 138.0 + ], + "spans": [], + "index": 4 + } + ] + } + ], + "index": 1.75 + }, + { + "type": "text", + "bbox": [ + 107, + 149, + 504, + 172 + ], + "lines": [ + { + "bbox": [ + 106, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "strong iterative attack. The results appear in Table 6. As we can see, for all model architectures,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 161, + 452, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 452, + 173 + ], + "score": 1.0, + "content": "adversarial training degraded the quality of a linear approximation to the model’s loss.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 177, + 505, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 177, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 190 + ], + "score": 1.0, + "content": "We find that input dropout (Srivastava et al., 2014) (i.e., randomly dropping a fraction of input fea-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 102, + 188, + 507, + 205 + ], + "spans": [ + { + "bbox": [ + 102, + 188, + 507, + 205 + ], + "score": 1.0, + "content": "tures during training) as used in architecture B limits this unwarranted effect of adversarial training.2", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 329, + 213 + ], + "score": 1.0, + "content": "If we omit the input dropout (we call this architecture", + "type": "text" + }, + { + "bbox": [ + 329, + 201, + 342, + 212 + ], + "score": 0.82, + "content": "\\boldsymbol { \\mathrm { B } } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 201, + 505, + 213 + ], + "score": 1.0, + "content": ") the single-step attack degrades signifi-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 505, + 225 + ], + "score": 1.0, + "content": "cantly. We discuss this effect in more detail below. For the fully connected architecture D, we find", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 223, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 236 + ], + "score": 1.0, + "content": "that the learned model is very close to linear and thus also less prone to the degenerate solution to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 234, + 327, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 327, + 246 + ], + "score": 1.0, + "content": "the min-max problem, as we postulated in Section 3.3.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 254, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "Attacks. Table 7 compares error rates of undefended and adversarially trained models on white-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 265, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 279 + ], + "score": 1.0, + "content": "box and black-box attacks, as in Section 4.1. Again, model B presents an anomaly. For all other", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "models, we corroborate our findings on ImageNet for adversarial training: (1) black-box attacks", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 288, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 506, + 300 + ], + "score": 1.0, + "content": "trump white-box single-step attacks; (2) white-box single-step attacks are significantly stronger if", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 273, + 312 + ], + "score": 1.0, + "content": "prepended by a random step. For model", + "type": "text" + }, + { + "bbox": [ + 273, + 299, + 291, + 311 + ], + "score": 0.92, + "content": "\\mathrm { \\Delta B _ { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 299, + 505, + 312 + ], + "score": 1.0, + "content": ", the opposite holds true. We believe this is because", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 309, + 488, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 488, + 324 + ], + "score": 1.0, + "content": "input dropout increases diversity of attack samples similarly to Ensemble Adversarial Training.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "table", + "bbox": [ + 137, + 360, + 472, + 525 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 332, + 504, + 355 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 333, + 504, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 504, + 344 + ], + "score": 1.0, + "content": "Table 7: White-box and black-box attacks against standard and adversarially trained models.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 343, + 483, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 483, + 355 + ], + "score": 1.0, + "content": "For each model, the strongest single-step white-box and black box attacks are marked in bold.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "table_body", + "bbox": [ + 137, + 360, + 472, + 525 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 137, + 360, + 472, + 525 + ], + "spans": [ + { + "bbox": [ + 137, + 360, + 472, + 525 + ], + "score": 0.982, + "html": "
white-boxblack-box
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A64.769.7161.553.246.841.5
Aadv2.214.86.610.78.86.58.3
B85.086.045.7169.959.985.9
Badv11.611.16.48.98.54.96.1
B*75.774.144.372.8-46.062.6
Badv4.340.616.114.715.017.99.1
C81.881.840.255.849.5159.4
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Moreover, Mishkin et al. (2017) find that input dropout signifi-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "cantly degrades accuracy on ImageNet. We thus did not incorporate it into our models on ImageNet.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "Ensemble Adversarial Training. To evaluate Ensemble Adversarial Training 3.4, we train two", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 648, + 507, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 278, + 664 + ], + "score": 1.0, + "content": "models per architecture. The first, denoted", + "type": "text" + }, + { + "bbox": [ + 279, + 650, + 326, + 661 + ], + "score": 0.84, + "content": "[ \\mathrm { A } \\mathrm { - } \\mathrm { D } ] _ { \\mathrm { a d v - e n s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 648, + 507, + 664 + ], + "score": 1.0, + "content": ", uses a single pre-trained model of the same", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 661, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 145, + 672 + ], + "score": 1.0, + "content": "type (i.e.,", + "type": "text" + }, + { + "bbox": [ + 146, + 661, + 177, + 672 + ], + "score": 0.9, + "content": "\\mathbf { A } _ { \\mathrm { a d v - e n s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 661, + 505, + 672 + ], + "score": 1.0, + "content": "is trained on perturbations from another model A). The second model, denoted [A-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 671, + 503, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 144, + 683 + ], + "score": 0.65, + "content": "\\mathrm { D l a d v } { \\mathrm { - e n s } } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 671, + 254, + 685 + ], + "score": 1.0, + "content": ", uses 3 pre-trained models", + "type": "text" + }, + { + "bbox": [ + 254, + 672, + 299, + 684 + ], + "score": 0.92, + "content": "( \\{ A , C , D \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 671, + 311, + 685 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 312, + 671, + 357, + 684 + ], + "score": 0.92, + "content": "\\{ B , C , D \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 671, + 503, + 685 + ], + "score": 1.0, + "content": ". We train all models for 12 epochs.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 109, + 688, + 504, + 711 + ], + "lines": [ + { + "bbox": [ + 107, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 107, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "We evaluate our models on black-box attacks crafted on models A,B,C,D (for a fair comparison, we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 107, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "do not use the same pre-trained models for evaluation, but retrain them with different random seeds).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + } + ], + "page_idx": 15, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 118, + 721, + 370, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 719, + 371, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 719, + 371, + 734 + ], + "score": 1.0, + "content": "2We thank Arjun Bhagoji, Bo Li and Dawn Song for this observation.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "16", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 138, + 106, + 469, + 138 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 503, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 79, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 106, + 79, + 505, + 93 + ], + "score": 1.0, + "content": "Table 6: Approximation ratio between optimal loss and loss induced by single-step attack on", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 90, + 403, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 90, + 195, + 104 + ], + "score": 1.0, + "content": "MNIST. 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We discuss this effect in more detail below. For the fully connected architecture D, we find", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 223, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 236 + ], + "score": 1.0, + "content": "that the learned model is very close to linear and thus also less prone to the degenerate solution to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 234, + 327, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 327, + 246 + ], + "score": 1.0, + "content": "the min-max problem, as we postulated in Section 3.3.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5, + "bbox_fs": [ + 102, + 177, + 507, + 246 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 254, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "Attacks. Table 7 compares error rates of undefended and adversarially trained models on white-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 265, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 279 + ], + "score": 1.0, + "content": "box and black-box attacks, as in Section 4.1. Again, model B presents an anomaly. For all other", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "models, we corroborate our findings on ImageNet for adversarial training: (1) black-box attacks", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 288, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 506, + 300 + ], + "score": 1.0, + "content": "trump white-box single-step attacks; (2) white-box single-step attacks are significantly stronger if", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 273, + 312 + ], + "score": 1.0, + "content": "prepended by a random step. For model", + "type": "text" + }, + { + "bbox": [ + 273, + 299, + 291, + 311 + ], + "score": 0.92, + "content": "\\mathrm { \\Delta B _ { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 299, + 505, + 312 + ], + "score": 1.0, + "content": ", the opposite holds true. We believe this is because", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 309, + 488, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 488, + 324 + ], + "score": 1.0, + "content": "input dropout increases diversity of attack samples similarly to Ensemble Adversarial Training.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 254, + 506, + 324 + ] + }, + { + "type": "table", + "bbox": [ + 137, + 360, + 472, + 525 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 332, + 504, + 355 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 333, + 504, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 504, + 344 + ], + "score": 1.0, + "content": "Table 7: White-box and black-box attacks against standard and adversarially trained models.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 343, + 483, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 483, + 355 + ], + "score": 1.0, + "content": "For each model, the strongest single-step white-box and black box attacks are marked in bold.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "table_body", + "bbox": [ + 137, + 360, + 472, + 525 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 137, + 360, + 472, + 525 + ], + "spans": [ + { + "bbox": [ + 137, + 360, + 472, + 525 + ], + "score": 0.982, + "html": "
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Moreover, Mishkin et al. (2017) find that input dropout signifi-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "cantly degrades accuracy on ImageNet. We thus did not incorporate it into our models on ImageNet.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 584, + 506, + 631 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "Ensemble Adversarial Training. To evaluate Ensemble Adversarial Training 3.4, we train two", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 648, + 507, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 278, + 664 + ], + "score": 1.0, + "content": "models per architecture. 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We train all models for 12 epochs.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 638, + 507, + 685 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 688, + 504, + 711 + ], + "lines": [ + { + "bbox": [ + 107, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 107, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "We evaluate our models on black-box attacks crafted on models A,B,C,D (for a fair comparison, we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 107, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "do not use the same pre-trained models for evaluation, but retrain them with different random seeds).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 107, + 687, + 506, + 711 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 166, + 129, + 443, + 295 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 80, + 505, + 124 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 80, + 505, + 92 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 505, + 92 + ], + "score": 1.0, + "content": "Table 8: Ensemble Adversarial Training on MNIST. For black-box robustness, we report the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 102 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 102 + ], + "score": 1.0, + "content": "maximum and average error rate over a suite of 12 attacks, comprised of the FGSM, I-FGSM and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 406, + 115 + ], + "score": 1.0, + "content": "PGD (Madry et al., 2017) attacks applied to models A,B,C and D. We use", + "type": "text" + }, + { + "bbox": [ + 407, + 102, + 437, + 113 + ], + "score": 0.89, + "content": "\\epsilon = 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 102, + 506, + 115 + ], + "score": 1.0, + "content": "in all cases. For", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 113, + 483, + 125 + ], + "spans": [ + { + "bbox": [ + 106, + 113, + 359, + 125 + ], + "score": 1.0, + "content": "each model architecture, we mark the models tied for best (at a", + "type": "text" + }, + { + "bbox": [ + 360, + 113, + 379, + 124 + ], + "score": 0.88, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 113, + 483, + 125 + ], + "score": 1.0, + "content": "confidence level) in bold.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 166, + 129, + 443, + 295 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 166, + 129, + 443, + 295 + ], + "spans": [ + { + "bbox": [ + 166, + 129, + 443, + 295 + ], + "score": 0.979, + "html": "
CleanFGSMMax.Black BoxAvg. Black Box
Aadv0.82.210.87.7
Aadv-ens0.87.06.65.2
Aadv-ens30.75.46.54.3
Badv0.811.68.95.5
Badv-ens0.710.56.85.3
Badv-ens30.814.08.85.1
Cadv1.03.729.318.7
Cadv-ens1.31.917.210.7
Cadv-ens31.43.614.58.4
Dadv2.625.532.523.5
Dadv-ens2.621.538.628.0
Dadv-ens32.629.429.815.6
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(2017) with", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 345, + 339 + ], + "score": 1.0, + "content": "the loss function from Carlini & Wagner (2017a)), all with", + "type": "text" + }, + { + "bbox": [ + 345, + 327, + 378, + 338 + ], + "score": 0.9, + "content": "\\epsilon = 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 326, + 505, + 339 + ], + "score": 1.0, + "content": ". The results appear in Table 8.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 338, + 490, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 490, + 350 + ], + "score": 1.0, + "content": "For each model, we report the worst-case and average-case error rate over all black-box attacks.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 355, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 505, + 366 + ], + "score": 1.0, + "content": "Ensemble Adversarial Training significantly increases robustness to black-box attacks, except for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "architecture B, which we previously found to not suffer from the same overfitting phenomenon", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 375, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 104, + 375, + 403, + 390 + ], + "score": 1.0, + "content": "that affects the other adversarially trained networks. Nevertheless, model", + "type": "text" + }, + { + "bbox": [ + 404, + 377, + 434, + 388 + ], + "score": 0.89, + "content": "\\mathrm { B _ { a d v - e n s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 375, + 505, + 390 + ], + "score": 1.0, + "content": "achieves slightly", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 387, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 344, + 402 + ], + "score": 1.0, + "content": "better robustness to white-box and black-box attacks than", + "type": "text" + }, + { + "bbox": [ + 344, + 388, + 362, + 399 + ], + "score": 0.9, + "content": "\\mathrm { \\Delta B _ { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 387, + 506, + 402 + ], + "score": 1.0, + "content": ". In the majority of cases, we find", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 397, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 412 + ], + "score": 1.0, + "content": "that using a single pre-trained model produces good results, but that the extra diversity of including", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 409, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 506, + 422 + ], + "score": 1.0, + "content": "three pre-trained models can sometimes increase robustness even further. Our experiments confirm", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "score": 1.0, + "content": "our conjecture that robustness to black-box attacks generalizes across models. Indeed, we find that", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "when training with three external models, we attain very good robustness against attacks initiated", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "from models with the same architecture (as evidenced by the average error on our attack suite), but", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 453, + 409, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 409, + 464 + ], + "score": 1.0, + "content": "also increased robustness to attacks initiated from the fourth holdout model", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 105, + 492, + 481, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 483, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 483, + 506 + ], + "score": 1.0, + "content": "D TRANSFERABILITY OF RANDOMIZED SINGLE-STEP PERTURBATIONS.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 523, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 250, + 536 + ], + "score": 1.0, + "content": "In Section 4.1, we introduced the", + "type": "text" + }, + { + "bbox": [ + 251, + 524, + 268, + 534 + ], + "score": 0.4, + "content": "\\mathbb { R } { + } \\mathbb { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 524, + 505, + 536 + ], + "score": 1.0, + "content": "tep-LL attack, an extension of the Step-LL method that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "prepends the attack with a small random perturbation. 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Source
Targetv4v3v3advIRv2IRv2ady
v470.537.223.234.024.6
v342.680.026.738.527.6
v3ady31.430.764.830.434.0
IRv236.235.723.056.324.6
IRv2adv26.826.325.226.937.5
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Source
Targetv4v3v3advIRv2IRv2adv
v442.814.36.311.96.9
v318.057.18.015.68.6
v3ady10.710.437.110.112.9
IRv212.813.66.129.37.0
IRv2adv8.08.07.78.315.0
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CleanFGSMMax.Black BoxAvg. Black Box
Aadv0.82.210.87.7
Aadv-ens0.87.06.65.2
Aadv-ens30.75.46.54.3
Badv0.811.68.95.5
Badv-ens0.710.56.85.3
Badv-ens30.814.08.85.1
Cadv1.03.729.318.7
Cadv-ens1.31.917.210.7
Cadv-ens31.43.614.58.4
Dadv2.625.532.523.5
Dadv-ens2.621.538.628.0
Dadv-ens32.629.429.815.6
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(2017) with", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 345, + 339 + ], + "score": 1.0, + "content": "the loss function from Carlini & Wagner (2017a)), all with", + "type": "text" + }, + { + "bbox": [ + 345, + 327, + 378, + 338 + ], + "score": 0.9, + "content": "\\epsilon = 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 326, + 505, + 339 + ], + "score": 1.0, + "content": ". The results appear in Table 8.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 338, + 490, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 490, + 350 + ], + "score": 1.0, + "content": "For each model, we report the worst-case and average-case error rate over all black-box attacks.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 315, + 505, + 350 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 355, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 505, + 366 + ], + "score": 1.0, + "content": "Ensemble Adversarial Training significantly increases robustness to black-box attacks, except for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "architecture B, which we previously found to not suffer from the same overfitting phenomenon", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 375, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 104, + 375, + 403, + 390 + ], + "score": 1.0, + "content": "that affects the other adversarially trained networks. Nevertheless, model", + "type": "text" + }, + { + "bbox": [ + 404, + 377, + 434, + 388 + ], + "score": 0.89, + "content": "\\mathrm { B _ { a d v - e n s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 375, + 505, + 390 + ], + "score": 1.0, + "content": "achieves slightly", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 387, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 344, + 402 + ], + "score": 1.0, + "content": "better robustness to white-box and black-box attacks than", + "type": "text" + }, + { + "bbox": [ + 344, + 388, + 362, + 399 + ], + "score": 0.9, + "content": "\\mathrm { \\Delta B _ { a d v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 387, + 506, + 402 + ], + "score": 1.0, + "content": ". In the majority of cases, we find", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 397, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 412 + ], + "score": 1.0, + "content": "that using a single pre-trained model produces good results, but that the extra diversity of including", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 409, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 506, + 422 + ], + "score": 1.0, + "content": "three pre-trained models can sometimes increase robustness even further. Our experiments confirm", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "score": 1.0, + "content": "our conjecture that robustness to black-box attacks generalizes across models. Indeed, we find that", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "when training with three external models, we attain very good robustness against attacks initiated", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "from models with the same architecture (as evidenced by the average error on our attack suite), but", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 453, + 409, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 409, + 464 + ], + "score": 1.0, + "content": "also increased robustness to attacks initiated from the fourth holdout model", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 355, + 506, + 464 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 492, + 481, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 483, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 483, + 506 + ], + "score": 1.0, + "content": "D TRANSFERABILITY OF RANDOMIZED SINGLE-STEP PERTURBATIONS.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 523, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 250, + 536 + ], + "score": 1.0, + "content": "In Section 4.1, we introduced the", + "type": "text" + }, + { + "bbox": [ + 251, + 524, + 268, + 534 + ], + "score": 0.4, + "content": "\\mathbb { R } { + } \\mathbb { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 524, + 505, + 536 + ], + "score": 1.0, + "content": "tep-LL attack, an extension of the Step-LL method that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "prepends the attack with a small random perturbation. In Table 9, we evaluate the transferability", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 118, + 558 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 546, + 132, + 556 + ], + "score": 0.48, + "content": "\\mathrm { R } +", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "Step-LL adversarial examples on ImageNet. We find that the randomized variant produces", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 557, + 465, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 465, + 569 + ], + "score": 1.0, + "content": "perturbations that transfer at a much lower rate (see Table 1 for the deterministic variant).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 524, + 505, + 569 + ] + }, + { + "type": "table", + "bbox": [ + 106, + 627, + 303, + 711 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 588, + 504, + 622 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 195, + 601 + ], + "score": 1.0, + "content": "Table 9: Error rates", + "type": "text" + }, + { + "bbox": [ + 196, + 588, + 220, + 599 + ], + "score": 0.76, + "content": "( \\mathbf { i n \\ } \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "of randomized single-step attacks transferred between models on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 185, + 612 + ], + "score": 1.0, + "content": "ImageNet. 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v470.537.223.234.024.6
v342.680.026.738.527.6
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Given the", + "type": "text" + }, + { + "bbox": [ + 209, + 177, + 223, + 189 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 177, + 382, + 189 + ], + "score": 1.0, + "content": "constraint, we find orthogonal vectors", + "type": "text" + }, + { + "bbox": [ + 383, + 179, + 392, + 189 + ], + "score": 0.86, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "that are maximally aligned", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 189, + 490, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 208, + 201 + ], + "score": 1.0, + "content": "with the signed gradient,", + "type": "text" + }, + { + "bbox": [ + 208, + 189, + 245, + 200 + ], + "score": 0.79, + "content": "\\mathtt { s i g n } ( g )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 189, + 490, + 201 + ], + "score": 1.0, + "content": ". We first prove an analog of (Tramer et al., 2017, Lemma 1). `", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 203, + 504, + 230 + ], + "lines": [ + { + "bbox": [ + 105, + 201, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 170, + 218 + ], + "score": 1.0, + "content": "Lemma 6. Let", + "type": "text" + }, + { + "bbox": [ + 171, + 202, + 227, + 216 + ], + "score": 0.92, + "content": "v \\in \\{ - 1 , 1 \\} ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 201, + 246, + 218 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 246, + 203, + 289, + 216 + ], + "score": 0.91, + "content": "\\alpha \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 201, + 369, + 218 + ], + "score": 1.0, + "content": ". 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Then", + "type": "text" + }, + { + "bbox": [ + 266, + 216, + 304, + 229 + ], + "score": 0.91, + "content": "\\alpha \\leq { k ^ { - \\frac { 1 } { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 215, + 309, + 231 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 240, + 287, + 257 + ], + "lines": [ + { + "bbox": [ + 101, + 235, + 292, + 261 + ], + "spans": [ + { + "bbox": [ + 101, + 235, + 152, + 261 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 152, + 241, + 222, + 257 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\hat { r _ { i } } = \\frac { r _ { i } } { \\Vert r _ { i } \\Vert _ { 2 } } = \\frac { r _ { i } } { \\sqrt { d } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 235, + 292, + 261 + ], + "score": 1.0, + "content": ". Then, we have", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 257, + 461, + 293 + ], + "lines": [ + { + "bbox": [ + 149, + 257, + 461, + 293 + ], + "spans": [ + { + "bbox": [ + 149, + 257, + 461, + 293 + ], + "score": 0.92, + "content": "d = \\| \\boldsymbol { v } \\| _ { 2 } ^ { 2 } \\geq \\sum _ { i = 1 } ^ { k } | \\boldsymbol { v } ^ { \\top } \\hat { \\boldsymbol { r _ { i } } } | ^ { 2 } = d ^ { - 1 } \\sum _ { i = 1 } ^ { k } | \\boldsymbol { v } ^ { \\top } \\boldsymbol { r _ { i } } | ^ { 2 } \\geq d ^ { - 1 } \\cdot k \\cdot \\left( { \\boldsymbol { \\alpha } } \\cdot d \\right) ^ { 2 } = k \\cdot \\alpha ^ { 2 } \\cdot d \\ ,", + "type": "interline_equation", + "image_path": "f2e38f4be0edea35abc342b4470994b5d5019d7e07b8ec4f5d373271c5e314df.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 149, + 257, + 461, + 269.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 149, + 269.0, + 461, + 281.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 149, + 281.0, + 461, + 293.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 297, + 239, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 294, + 240, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 196, + 309 + ], + "score": 1.0, + "content": "from which we obtain", + "type": "text" + }, + { + "bbox": [ + 197, + 294, + 235, + 308 + ], + "score": 0.94, + "content": "\\alpha \\leq k ^ { - { \\frac { 1 } { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 296, + 240, + 309 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 320, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "score": 1.0, + "content": "This result bounds the number of orthogonal perturbations we can expect to find, for a given align-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 332, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 482, + 345 + ], + "score": 1.0, + "content": "ment with the signed gradient. As a warm-up consider the following trivial construction of", + "type": "text" + }, + { + "bbox": [ + 482, + 333, + 489, + 342 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 332, + 505, + 345 + ], + "score": 1.0, + "content": "or-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 343, + 504, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 186, + 358 + ], + "score": 1.0, + "content": "thogonal vectors in", + "type": "text" + }, + { + "bbox": [ + 186, + 343, + 223, + 357 + ], + "score": 0.93, + "content": "\\{ - 1 , 1 \\} ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 343, + 361, + 358 + ], + "score": 1.0, + "content": "that are “somewhat” aligned with", + "type": "text" + }, + { + "bbox": [ + 362, + 345, + 399, + 357 + ], + "score": 0.77, + "content": "\\mathtt { s i g n } ( g )", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 343, + 439, + 358 + ], + "score": 1.0, + "content": ". We split", + "type": "text" + }, + { + "bbox": [ + 440, + 344, + 477, + 357 + ], + "score": 0.3, + "content": "\\mathtt { s i g n } ( g )", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 343, + 497, + 358 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 497, + 345, + 504, + 355 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 356, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 175, + 370 + ], + "score": 1.0, + "content": "“chunks” of size", + "type": "text" + }, + { + "bbox": [ + 176, + 357, + 183, + 371 + ], + "score": 0.87, + "content": "\\frac { d } { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 357, + 228, + 370 + ], + "score": 1.0, + "content": "and define", + "type": "text" + }, + { + "bbox": [ + 228, + 360, + 237, + 370 + ], + "score": 0.84, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 357, + 365, + 370 + ], + "score": 1.0, + "content": "to be the vector that is equal to", + "type": "text" + }, + { + "bbox": [ + 365, + 358, + 403, + 370 + ], + "score": 0.83, + "content": "\\mathtt { s i g n } ( g )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 357, + 429, + 370 + ], + "score": 1.0, + "content": "in the", + "type": "text" + }, + { + "bbox": [ + 429, + 356, + 441, + 368 + ], + "score": 0.85, + "content": "i ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "chunk and zero", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 369, + 476, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 194, + 386 + ], + "score": 1.0, + "content": "otherwise. We obtain", + "type": "text" + }, + { + "bbox": [ + 194, + 370, + 266, + 385 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathrm { s i g n } ( g ) ^ { \\top } r _ { i } = \\frac { d } { k } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 369, + 303, + 386 + ], + "score": 1.0, + "content": ", a factor", + "type": "text" + }, + { + "bbox": [ + 303, + 370, + 318, + 383 + ], + "score": 0.92, + "content": "\\sqrt { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 369, + 476, + 386 + ], + "score": 1.0, + "content": "worse than the the bound in Lemma 6.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "We now provide a construction that meets this upper bound. We make use of Regular Hadamard", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 399, + 504, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 178, + 413 + ], + "score": 1.0, + "content": "Matrices of order", + "type": "text" + }, + { + "bbox": [ + 178, + 401, + 185, + 410 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 399, + 366, + 413 + ], + "score": 1.0, + "content": "(Colbourn, 2010). These are square matrices", + "type": "text" + }, + { + "bbox": [ + 366, + 400, + 380, + 411 + ], + "score": 0.91, + "content": "H _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 399, + 489, + 413 + ], + "score": 1.0, + "content": "such that: (1) all entries of √", + "type": "text" + }, + { + "bbox": [ + 489, + 400, + 504, + 411 + ], + "score": 0.88, + "content": "H _ { k }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 410, + 491, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 132, + 427 + ], + "score": 1.0, + "content": "are in", + "type": "text" + }, + { + "bbox": [ + 132, + 411, + 169, + 425 + ], + "score": 0.93, + "content": "\\{ - 1 , 1 \\} ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 410, + 235, + 427 + ], + "score": 1.0, + "content": "; (2) the rows of", + "type": "text" + }, + { + "bbox": [ + 235, + 413, + 249, + 424 + ], + "score": 0.9, + "content": "H _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 410, + 470, + 427 + ], + "score": 1.0, + "content": "are mutually orthogonal; (3) All row sums are equal to", + "type": "text" + }, + { + "bbox": [ + 471, + 411, + 486, + 424 + ], + "score": 0.91, + "content": "\\sqrt { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 410, + 491, + 427 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 108, + 430, + 502, + 455 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 345, + 445 + ], + "score": 1.0, + "content": "The order of a Regular Hadamard Matrix is of the form", + "type": "text" + }, + { + "bbox": [ + 346, + 430, + 362, + 442 + ], + "score": 0.89, + "content": "4 u ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 429, + 425, + 445 + ], + "score": 1.0, + "content": "for an integer", + "type": "text" + }, + { + "bbox": [ + 426, + 434, + 433, + 442 + ], + "score": 0.68, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 429, + 505, + 445 + ], + "score": 1.0, + "content": ". We use known", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 442, + 277, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 176, + 455 + ], + "score": 1.0, + "content": "constructions for", + "type": "text" + }, + { + "bbox": [ + 177, + 442, + 272, + 455 + ], + "score": 0.91, + "content": "k \\overset { \\cdot } { \\in } \\{ 4 , 1 6 , 3 6 , 6 4 , 1 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 442, + 277, + 455 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 457, + 505, + 497 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 169, + 471 + ], + "score": 1.0, + "content": "Lemma 7. Let", + "type": "text" + }, + { + "bbox": [ + 169, + 456, + 199, + 470 + ], + "score": 0.91, + "content": "\\boldsymbol { g } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 456, + 217, + 471 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 218, + 459, + 225, + 469 + ], + "score": 0.64, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 456, + 470, + 471 + ], + "score": 1.0, + "content": "be an integer for which a Regular Hadamard Matrix of order", + "type": "text" + }, + { + "bbox": [ + 470, + 459, + 477, + 468 + ], + "score": 0.61, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 456, + 505, + 471 + ], + "score": 1.0, + "content": "exists.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 469, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 286, + 484 + ], + "score": 1.0, + "content": "Then, there is a randomized construction of", + "type": "text" + }, + { + "bbox": [ + 286, + 471, + 293, + 481 + ], + "score": 0.69, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 470, + 372, + 484 + ], + "score": 1.0, + "content": "orthogonal vectors", + "type": "text" + }, + { + "bbox": [ + 373, + 469, + 461, + 483 + ], + "score": 0.92, + "content": "r _ { 1 } , . . . r _ { n } \\in \\{ - 1 , 1 \\} ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 470, + 506, + 484 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 479, + 353, + 500 + ], + "spans": [ + { + "bbox": [ + 107, + 482, + 206, + 497 + ], + "score": 0.91, + "content": "s i g n ( g ) ^ { \\top } r _ { i } = d \\cdot k ^ { - 1 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 479, + 253, + 500 + ], + "score": 1.0, + "content": ". Moreover,", + "type": "text" + }, + { + "bbox": [ + 254, + 482, + 348, + 497 + ], + "score": 0.92, + "content": "\\mathbb { E } [ g ^ { \\top } r _ { i } ] = k ^ { - 1 / 2 } \\cdot \\lVert g \\rVert _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 479, + 353, + 500 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 509, + 505, + 570 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 190, + 523 + ], + "score": 1.0, + "content": "Proof. We construct", + "type": "text" + }, + { + "bbox": [ + 191, + 511, + 198, + 520 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 509, + 275, + 523 + ], + "score": 1.0, + "content": "orthogonal vectors", + "type": "text" + }, + { + "bbox": [ + 276, + 509, + 366, + 522 + ], + "score": 0.93, + "content": "r _ { 1 } , \\ldots , r _ { k } \\in \\{ - 1 , 1 \\} ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 509, + 396, + 523 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 397, + 512, + 406, + 522 + ], + "score": 0.84, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 509, + 506, + 523 + ], + "score": 1.0, + "content": "is obtained by repeating", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 519, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 104, + 519, + 121, + 538 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 121, + 522, + 132, + 533 + ], + "score": 0.83, + "content": "\\mathrm { i } ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 519, + 163, + 538 + ], + "score": 1.0, + "content": "row of", + "type": "text" + }, + { + "bbox": [ + 163, + 523, + 192, + 535 + ], + "score": 0.92, + "content": "H _ { k } \\ d / k", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 519, + 346, + 538 + ], + "score": 1.0, + "content": "times (for simplicity, we assume that", + "type": "text" + }, + { + "bbox": [ + 346, + 524, + 353, + 533 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 519, + 385, + 538 + ], + "score": 1.0, + "content": "divides", + "type": "text" + }, + { + "bbox": [ + 386, + 524, + 392, + 533 + ], + "score": 0.79, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 519, + 473, + 538 + ], + "score": 1.0, + "content": ". 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By construction, the", + "type": "text" + }, + { + "bbox": [ + 465, + 535, + 472, + 544 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 534, + 506, + 547 + ], + "score": 1.0, + "content": "vectors", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 545, + 507, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 165, + 560 + ], + "score": 0.93, + "content": "r _ { i } \\in \\{ - 1 , 1 \\} ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 545, + 322, + 561 + ], + "score": 1.0, + "content": "are mutually orthogonal, and we have", + "type": "text" + }, + { + "bbox": [ + 322, + 545, + 464, + 561 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathrm { s i g n } ( g ) ^ { \\top } r _ { i } = \\frac { d } { k } \\cdot \\sqrt { k } = d \\cdot k ^ { - 1 / 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 545, + 507, + 561 + ], + "score": 1.0, + "content": ", which is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 558, + 222, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 222, + 570 + ], + "score": 1.0, + "content": "tight according to Lemma 6.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 575, + 503, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 224, + 590 + ], + "score": 1.0, + "content": "As the weight of the gradient", + "type": "text" + }, + { + "bbox": [ + 224, + 578, + 231, + 587 + ], + "score": 0.78, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 574, + 406, + 590 + ], + "score": 1.0, + "content": "may not be uniformly distributed among its", + "type": "text" + }, + { + "bbox": [ + 406, + 577, + 413, + 586 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 574, + 505, + 590 + ], + "score": 1.0, + "content": "components, we apply", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 587, + 430, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 430, + 599 + ], + "score": 1.0, + "content": "our construction to a random permutation of the signed gradient. We then obtain", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 600, + 429, + 673 + ], + "lines": [ + { + "bbox": [ + 181, + 600, + 429, + 673 + ], + "spans": [ + { + "bbox": [ + 181, + 600, + 429, + 673 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathbb { E } [ g ^ { \\top } \\boldsymbol { r } _ { i } ] = \\mathbb { E } \\left[ \\displaystyle \\sum _ { j = 1 } ^ { d } \\vert g ^ { ( j ) } \\vert \\cdot \\mathsf { s i g n } ( g ^ { ( j ) } ) \\cdot r _ { i } ^ { ( j ) } \\right] } \\\\ & { \\quad \\quad = \\displaystyle \\sum _ { j = 1 } ^ { d } \\vert g ^ { ( j ) } \\vert \\cdot \\mathbb { E } \\left[ \\mathsf { s i g n } ( g ^ { ( j ) } ) \\cdot r _ { i } ^ { ( j ) } \\right] = k ^ { - 1 / 2 } \\cdot \\| g \\| _ { 1 } . } \\end{array}", + "type": "interline_equation", + "image_path": "af4f888667eda85a9f15c4424ad4a3d59e967346e9de7fe74800593f72ac6847.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 181, + 600, + 429, + 624.3333333333334 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 181, + 624.3333333333334, + 429, + 648.6666666666667 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 181, + 648.6666666666667, + 429, + 673.0000000000001 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 696, + 505, + 730 + ], + "lines": [ + { + "bbox": [ + 105, + 696, + 504, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 410, + 709 + ], + "score": 1.0, + "content": "It can be shown that the bound in Lemma 7 can be attained if and only if the", + "type": "text" + }, + { + "bbox": [ + 410, + 698, + 419, + 708 + ], + "score": 0.84, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 696, + 504, + 709 + ], + "score": 1.0, + "content": "are constructed from", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 707, + 505, + 720 + ], + "spans": [ + { + "bbox": [ + 106, + 707, + 423, + 720 + ], + "score": 1.0, + "content": "the rows of a Regular Hadamard Matrix (Colbourn, 2010). 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(2017) consider the following task for a given model `", + "type": "text" + }, + { + "bbox": [ + 383, + 106, + 390, + 115 + ], + "score": 0.73, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 105, + 504, + 117 + ], + "score": 1.0, + "content": ": for a (correctly classified)", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 129, + 129 + ], + "score": 1.0, + "content": "point", + "type": "text" + }, + { + "bbox": [ + 129, + 118, + 136, + 126 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 115, + 158, + 129 + ], + "score": 1.0, + "content": ", find", + "type": "text" + }, + { + "bbox": [ + 158, + 117, + 164, + 126 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 115, + 242, + 129 + ], + "score": 1.0, + "content": "orthogonal vectors", + "type": "text" + }, + { + "bbox": [ + 243, + 117, + 294, + 128 + ], + "score": 0.94, + "content": "\\{ r _ { 1 } , \\ldots , r _ { k } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 115, + 333, + 129 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 333, + 117, + 374, + 128 + ], + "score": 0.93, + "content": "\\lVert \\boldsymbol { r } _ { i } \\rVert _ { 2 } \\leq \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 115, + 417, + 129 + ], + "score": 1.0, + "content": "and all the", + "type": "text" + }, + { + "bbox": [ + 418, + 117, + 443, + 128 + ], + "score": 0.91, + "content": "\\boldsymbol { x } + \\boldsymbol { r } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 115, + 505, + 129 + ], + "score": 1.0, + "content": "are adversarial", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 504, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 163, + 142 + ], + "score": 1.0, + "content": "(i.e., arg max", + "type": "text" + }, + { + "bbox": [ + 164, + 127, + 236, + 140 + ], + "score": 0.91, + "content": "h ( x + r _ { i } ) \\neq y _ { \\mathrm { t r u e } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 127, + 497, + 142 + ], + "score": 1.0, + "content": ". By linearizing the model’s loss function, this reduces to finding", + "type": "text" + }, + { + "bbox": [ + 497, + 128, + 504, + 137 + ], + "score": 0.76, + "content": "k", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 136, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 104, + 136, + 184, + 153 + ], + "score": 1.0, + "content": "orthogonal vectors", + "type": "text" + }, + { + "bbox": [ + 184, + 141, + 193, + 150 + ], + "score": 0.86, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 136, + 412, + 153 + ], + "score": 1.0, + "content": "that are maximally aligned with the model’s gradient", + "type": "text" + }, + { + "bbox": [ + 412, + 138, + 501, + 150 + ], + "score": 0.94, + "content": "g = \\nabla _ { x } L ( h ( x ) , \\hat { y _ { \\mathrm { t r u e } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 136, + 506, + 153 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 149, + 413, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 291, + 163 + ], + "score": 1.0, + "content": "Tramer et al. (2017) left a construction for the`", + "type": "text" + }, + { + "bbox": [ + 292, + 150, + 306, + 161 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 149, + 413, + 163 + ], + "score": 1.0, + "content": "norm as an open problem.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3, + "bbox_fs": [ + 104, + 105, + 506, + 163 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 166, + 505, + 200 + ], + "lines": [ + { + "bbox": [ + 105, + 165, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 282, + 180 + ], + "score": 1.0, + "content": "We provide an optimal construction for the", + "type": "text" + }, + { + "bbox": [ + 282, + 167, + 296, + 178 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 165, + 505, + 180 + ], + "score": 1.0, + "content": "norm, based on Regular Hadamard Matrices (Col-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 209, + 189 + ], + "score": 1.0, + "content": "bourn, 2010). Given the", + "type": "text" + }, + { + "bbox": [ + 209, + 177, + 223, + 189 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 177, + 382, + 189 + ], + "score": 1.0, + "content": "constraint, we find orthogonal vectors", + "type": "text" + }, + { + "bbox": [ + 383, + 179, + 392, + 189 + ], + "score": 0.86, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "that are maximally aligned", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 189, + 490, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 208, + 201 + ], + "score": 1.0, + "content": "with the signed gradient,", + "type": "text" + }, + { + "bbox": [ + 208, + 189, + 245, + 200 + ], + "score": 0.79, + "content": "\\mathtt { s i g n } ( g )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 189, + 490, + 201 + ], + "score": 1.0, + "content": ". We first prove an analog of (Tramer et al., 2017, Lemma 1). `", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 165, + 505, + 201 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 203, + 504, + 230 + ], + "lines": [ + { + "bbox": [ + 105, + 201, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 170, + 218 + ], + "score": 1.0, + "content": "Lemma 6. Let", + "type": "text" + }, + { + "bbox": [ + 171, + 202, + 227, + 216 + ], + "score": 0.92, + "content": "v \\in \\{ - 1 , 1 \\} ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 201, + 246, + 218 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 246, + 203, + 289, + 216 + ], + "score": 0.91, + "content": "\\alpha \\in ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 201, + 369, + 218 + ], + "score": 1.0, + "content": ". Suppose there are", + "type": "text" + }, + { + "bbox": [ + 370, + 204, + 376, + 214 + ], + "score": 0.59, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 201, + 456, + 218 + ], + "score": 1.0, + "content": "orthogonal vectors", + "type": "text" + }, + { + "bbox": [ + 456, + 205, + 505, + 216 + ], + "score": 0.86, + "content": "r _ { 1 } , \\ldots r _ { n } \\in", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 215, + 309, + 231 + ], + "spans": [ + { + "bbox": [ + 107, + 216, + 144, + 230 + ], + "score": 0.92, + "content": "\\{ - 1 , 1 \\} ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 215, + 186, + 231 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 186, + 216, + 240, + 230 + ], + "score": 0.9, + "content": "v ^ { \\top } r _ { i } \\geq \\alpha \\cdot d .", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 215, + 266, + 231 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 266, + 216, + 304, + 229 + ], + "score": 0.91, + "content": "\\alpha \\leq { k ^ { - \\frac { 1 } { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 215, + 309, + 231 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 201, + 505, + 231 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 240, + 287, + 257 + ], + "lines": [ + { + "bbox": [ + 101, + 235, + 292, + 261 + ], + "spans": [ + { + "bbox": [ + 101, + 235, + 152, + 261 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 152, + 241, + 222, + 257 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\hat { r _ { i } } = \\frac { r _ { i } } { \\Vert r _ { i } \\Vert _ { 2 } } = \\frac { r _ { i } } { \\sqrt { d } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 235, + 292, + 261 + ], + "score": 1.0, + "content": ". Then, we have", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 101, + 235, + 292, + 261 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 257, + 461, + 293 + ], + "lines": [ + { + "bbox": [ + 149, + 257, + 461, + 293 + ], + "spans": [ + { + "bbox": [ + 149, + 257, + 461, + 293 + ], + "score": 0.92, + "content": "d = \\| \\boldsymbol { v } \\| _ { 2 } ^ { 2 } \\geq \\sum _ { i = 1 } ^ { k } | \\boldsymbol { v } ^ { \\top } \\hat { \\boldsymbol { r _ { i } } } | ^ { 2 } = d ^ { - 1 } \\sum _ { i = 1 } ^ { k } | \\boldsymbol { v } ^ { \\top } \\boldsymbol { r _ { i } } | ^ { 2 } \\geq d ^ { - 1 } \\cdot k \\cdot \\left( { \\boldsymbol { \\alpha } } \\cdot d \\right) ^ { 2 } = k \\cdot \\alpha ^ { 2 } \\cdot d \\ ,", + "type": "interline_equation", + "image_path": "f2e38f4be0edea35abc342b4470994b5d5019d7e07b8ec4f5d373271c5e314df.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 149, + 257, + 461, + 269.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 149, + 269.0, + 461, + 281.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 149, + 281.0, + 461, + 293.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 297, + 239, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 294, + 240, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 196, + 309 + ], + "score": 1.0, + "content": "from which we obtain", + "type": "text" + }, + { + "bbox": [ + 197, + 294, + 235, + 308 + ], + "score": 0.94, + "content": "\\alpha \\leq k ^ { - { \\frac { 1 } { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 296, + 240, + 309 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 294, + 240, + 309 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 320, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "score": 1.0, + "content": "This result bounds the number of orthogonal perturbations we can expect to find, for a given align-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 332, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 482, + 345 + ], + "score": 1.0, + "content": "ment with the signed gradient. As a warm-up consider the following trivial construction of", + "type": "text" + }, + { + "bbox": [ + 482, + 333, + 489, + 342 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 332, + 505, + 345 + ], + "score": 1.0, + "content": "or-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 343, + 504, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 186, + 358 + ], + "score": 1.0, + "content": "thogonal vectors in", + "type": "text" + }, + { + "bbox": [ + 186, + 343, + 223, + 357 + ], + "score": 0.93, + "content": "\\{ - 1 , 1 \\} ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 343, + 361, + 358 + ], + "score": 1.0, + "content": "that are “somewhat” aligned with", + "type": "text" + }, + { + "bbox": [ + 362, + 345, + 399, + 357 + ], + "score": 0.77, + "content": "\\mathtt { s i g n } ( g )", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 343, + 439, + 358 + ], + "score": 1.0, + "content": ". We split", + "type": "text" + }, + { + "bbox": [ + 440, + 344, + 477, + 357 + ], + "score": 0.3, + "content": "\\mathtt { s i g n } ( g )", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 343, + 497, + 358 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 497, + 345, + 504, + 355 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 356, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 175, + 370 + ], + "score": 1.0, + "content": "“chunks” of size", + "type": "text" + }, + { + "bbox": [ + 176, + 357, + 183, + 371 + ], + "score": 0.87, + "content": "\\frac { d } { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 357, + 228, + 370 + ], + "score": 1.0, + "content": "and define", + "type": "text" + }, + { + "bbox": [ + 228, + 360, + 237, + 370 + ], + "score": 0.84, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 357, + 365, + 370 + ], + "score": 1.0, + "content": "to be the vector that is equal to", + "type": "text" + }, + { + "bbox": [ + 365, + 358, + 403, + 370 + ], + "score": 0.83, + "content": "\\mathtt { s i g n } ( g )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 357, + 429, + 370 + ], + "score": 1.0, + "content": "in the", + "type": "text" + }, + { + "bbox": [ + 429, + 356, + 441, + 368 + ], + "score": 0.85, + "content": "i ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "chunk and zero", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 369, + 476, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 194, + 386 + ], + "score": 1.0, + "content": "otherwise. We obtain", + "type": "text" + }, + { + "bbox": [ + 194, + 370, + 266, + 385 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathrm { s i g n } ( g ) ^ { \\top } r _ { i } = \\frac { d } { k } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 369, + 303, + 386 + ], + "score": 1.0, + "content": ", a factor", + "type": "text" + }, + { + "bbox": [ + 303, + 370, + 318, + 383 + ], + "score": 0.92, + "content": "\\sqrt { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 369, + 476, + 386 + ], + "score": 1.0, + "content": "worse than the the bound in Lemma 6.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 320, + 505, + 386 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "We now provide a construction that meets this upper bound. We make use of Regular Hadamard", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 399, + 504, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 178, + 413 + ], + "score": 1.0, + "content": "Matrices of order", + "type": "text" + }, + { + "bbox": [ + 178, + 401, + 185, + 410 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 399, + 366, + 413 + ], + "score": 1.0, + "content": "(Colbourn, 2010). These are square matrices", + "type": "text" + }, + { + "bbox": [ + 366, + 400, + 380, + 411 + ], + "score": 0.91, + "content": "H _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 399, + 489, + 413 + ], + "score": 1.0, + "content": "such that: (1) all entries of √", + "type": "text" + }, + { + "bbox": [ + 489, + 400, + 504, + 411 + ], + "score": 0.88, + "content": "H _ { k }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 410, + 491, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 132, + 427 + ], + "score": 1.0, + "content": "are in", + "type": "text" + }, + { + "bbox": [ + 132, + 411, + 169, + 425 + ], + "score": 0.93, + "content": "\\{ - 1 , 1 \\} ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 410, + 235, + 427 + ], + "score": 1.0, + "content": "; (2) the rows of", + "type": "text" + }, + { + "bbox": [ + 235, + 413, + 249, + 424 + ], + "score": 0.9, + "content": "H _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 410, + 470, + 427 + ], + "score": 1.0, + "content": "are mutually orthogonal; (3) All row sums are equal to", + "type": "text" + }, + { + "bbox": [ + 471, + 411, + 486, + 424 + ], + "score": 0.91, + "content": "\\sqrt { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 410, + 491, + 427 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 388, + 505, + 427 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 430, + 502, + 455 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 345, + 445 + ], + "score": 1.0, + "content": "The order of a Regular Hadamard Matrix is of the form", + "type": "text" + }, + { + "bbox": [ + 346, + 430, + 362, + 442 + ], + "score": 0.89, + "content": "4 u ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 429, + 425, + 445 + ], + "score": 1.0, + "content": "for an integer", + "type": "text" + }, + { + "bbox": [ + 426, + 434, + 433, + 442 + ], + "score": 0.68, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 429, + 505, + 445 + ], + "score": 1.0, + "content": ". We use known", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 442, + 277, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 176, + 455 + ], + "score": 1.0, + "content": "constructions for", + "type": "text" + }, + { + "bbox": [ + 177, + 442, + 272, + 455 + ], + "score": 0.91, + "content": "k \\overset { \\cdot } { \\in } \\{ 4 , 1 6 , 3 6 , 6 4 , 1 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 442, + 277, + 455 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 429, + 505, + 455 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 457, + 505, + 497 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 169, + 471 + ], + "score": 1.0, + "content": "Lemma 7. Let", + "type": "text" + }, + { + "bbox": [ + 169, + 456, + 199, + 470 + ], + "score": 0.91, + "content": "\\boldsymbol { g } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 456, + 217, + 471 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 218, + 459, + 225, + 469 + ], + "score": 0.64, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 456, + 470, + 471 + ], + "score": 1.0, + "content": "be an integer for which a Regular Hadamard Matrix of order", + "type": "text" + }, + { + "bbox": [ + 470, + 459, + 477, + 468 + ], + "score": 0.61, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 456, + 505, + 471 + ], + "score": 1.0, + "content": "exists.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 469, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 286, + 484 + ], + "score": 1.0, + "content": "Then, there is a randomized construction of", + "type": "text" + }, + { + "bbox": [ + 286, + 471, + 293, + 481 + ], + "score": 0.69, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 470, + 372, + 484 + ], + "score": 1.0, + "content": "orthogonal vectors", + "type": "text" + }, + { + "bbox": [ + 373, + 469, + 461, + 483 + ], + "score": 0.92, + "content": "r _ { 1 } , . . . r _ { n } \\in \\{ - 1 , 1 \\} ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 470, + 506, + 484 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 479, + 353, + 500 + ], + "spans": [ + { + "bbox": [ + 107, + 482, + 206, + 497 + ], + "score": 0.91, + "content": "s i g n ( g ) ^ { \\top } r _ { i } = d \\cdot k ^ { - 1 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 479, + 253, + 500 + ], + "score": 1.0, + "content": ". Moreover,", + "type": "text" + }, + { + "bbox": [ + 254, + 482, + 348, + 497 + ], + "score": 0.92, + "content": "\\mathbb { E } [ g ^ { \\top } r _ { i } ] = k ^ { - 1 / 2 } \\cdot \\lVert g \\rVert _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 479, + 353, + 500 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 456, + 506, + 500 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 509, + 505, + 570 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 190, + 523 + ], + "score": 1.0, + "content": "Proof. We construct", + "type": "text" + }, + { + "bbox": [ + 191, + 511, + 198, + 520 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 509, + 275, + 523 + ], + "score": 1.0, + "content": "orthogonal vectors", + "type": "text" + }, + { + "bbox": [ + 276, + 509, + 366, + 522 + ], + "score": 0.93, + "content": "r _ { 1 } , \\ldots , r _ { k } \\in \\{ - 1 , 1 \\} ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 509, + 396, + 523 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 397, + 512, + 406, + 522 + ], + "score": 0.84, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 509, + 506, + 523 + ], + "score": 1.0, + "content": "is obtained by repeating", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 519, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 104, + 519, + 121, + 538 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 121, + 522, + 132, + 533 + ], + "score": 0.83, + "content": "\\mathrm { i } ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 519, + 163, + 538 + ], + "score": 1.0, + "content": "row of", + "type": "text" + }, + { + "bbox": [ + 163, + 523, + 192, + 535 + ], + "score": 0.92, + "content": "H _ { k } \\ d / k", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 519, + 346, + 538 + ], + "score": 1.0, + "content": "times (for simplicity, we assume that", + "type": "text" + }, + { + "bbox": [ + 346, + 524, + 353, + 533 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 519, + 385, + 538 + ], + "score": 1.0, + "content": "divides", + "type": "text" + }, + { + "bbox": [ + 386, + 524, + 392, + 533 + ], + "score": 0.79, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 519, + 473, + 538 + ], + "score": 1.0, + "content": ". Otherwise we pad", + "type": "text" + }, + { + "bbox": [ + 474, + 525, + 483, + 535 + ], + "score": 0.83, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 519, + 506, + 538 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 233, + 547 + ], + "score": 1.0, + "content": "zeros). We then multiply each", + "type": "text" + }, + { + "bbox": [ + 234, + 536, + 243, + 546 + ], + "score": 0.85, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 534, + 335, + 547 + ], + "score": 1.0, + "content": "component-wise with", + "type": "text" + }, + { + "bbox": [ + 335, + 534, + 372, + 546 + ], + "score": 0.6, + "content": "\\mathtt { s i g n } ( g )", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 534, + 465, + 547 + ], + "score": 1.0, + "content": ". 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We then obtain", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 574, + 505, + 599 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 600, + 429, + 673 + ], + "lines": [ + { + "bbox": [ + 181, + 600, + 429, + 673 + ], + "spans": [ + { + "bbox": [ + 181, + 600, + 429, + 673 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathbb { E } [ g ^ { \\top } \\boldsymbol { r } _ { i } ] = \\mathbb { E } \\left[ \\displaystyle \\sum _ { j = 1 } ^ { d } \\vert g ^ { ( j ) } \\vert \\cdot \\mathsf { s i g n } ( g ^ { ( j ) } ) \\cdot r _ { i } ^ { ( j ) } \\right] } \\\\ & { \\quad \\quad = \\displaystyle \\sum _ { j = 1 } ^ { d } \\vert g ^ { ( j ) } \\vert \\cdot \\mathbb { E } \\left[ \\mathsf { s i g n } ( g ^ { ( j ) } ) \\cdot r _ { i } ^ { ( j ) } \\right] = k ^ { - 1 / 2 } \\cdot \\| g \\| _ { 1 } . } \\end{array}", + "type": "interline_equation", + "image_path": "af4f888667eda85a9f15c4424ad4a3d59e967346e9de7fe74800593f72ac6847.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 181, + 600, + 429, + 624.3333333333334 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 181, + 624.3333333333334, + 429, + 648.6666666666667 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 181, + 648.6666666666667, + 429, + 673.0000000000001 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 696, + 505, + 730 + ], + "lines": [ + { + "bbox": [ + 105, + 696, + 504, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 410, + 709 + ], + "score": 1.0, + "content": "It can be shown that the bound in Lemma 7 can be attained if and only if the", + "type": "text" + }, + { + "bbox": [ + 410, + 698, + 419, + 708 + ], + "score": 0.84, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 696, + 504, + 709 + ], + "score": 1.0, + "content": "are constructed from", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 707, + 505, + 720 + ], + "spans": [ + { + "bbox": [ + 106, + 707, + 423, + 720 + ], + "score": 1.0, + "content": "the rows of a Regular Hadamard Matrix (Colbourn, 2010). 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