diff --git a/parse/train/o966_Is_nPA/o966_Is_nPA_content_list.json b/parse/train/o966_Is_nPA/o966_Is_nPA_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..f1c4820c5ee966e551b40dff5da6018712de1836 --- /dev/null +++ b/parse/train/o966_Is_nPA/o966_Is_nPA_content_list.json @@ -0,0 +1,1928 @@ +[ + { + "type": "text", + "text": "NEURAL PRUNING VIA GROWING REGULARIZATION ", + "text_level": 1, + "bbox": [ + 173, + 98, + 802, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Huan Wang, Can Qin, Yulun Zhang∗, Yun Fu Northeastern University, Boston, MA, USA {wang.huan, qin.ca}@northeastern.edu, yulun100@gmail.com, yunfu@ece.neu.edu ", + "bbox": [ + 183, + 143, + 547, + 202 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 238, + 544, + 252 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Regularization has long been utilized to learn sparsity in deep neural network pruning. However, its role is mainly explored in the small penalty strength regime. In this work, we extend its application to a new scenario where the regularization grows large gradually to tackle two central problems of pruning: pruning schedule and weight importance scoring. (1) The former topic is newly brought up in this work, which we find critical to the pruning performance while receives little research attention. Specifically, we propose an $L _ { 2 }$ regularization variant with rising penalty factors and show it can bring significant accuracy gains compared with its one-shot counterpart, even when the same weights are removed. (2) The growing penalty scheme also brings us an approach to exploit the Hessian information for more accurate pruning without knowing their specific values, thus not bothered by the common Hessian approximation problems. Empirically, the proposed algorithms are easy to implement and scalable to large datasets and networks in both structured and unstructured pruning. Their effectiveness is demonstrated with modern deep neural networks on the CIFAR and ImageNet datasets, achieving competitive results compared to many state-of-the-art algorithms. Our code and trained models are publicly available at https://github.com/mingsuntse/regularization-pruning. ", + "bbox": [ + 233, + 267, + 764, + 518 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 178, + 544, + 336, + 560 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "As deep neural networks advance in recent years LeCun et al. (2015); Schmidhuber (2015), their remarkable effectiveness comes at a cost of rising storage, memory footprint, computing resources and energy consumption Cheng et al. (2017); Deng et al. (2020). Neural network pruning Han et al. (2015; 2016); Li et al. (2017); Wen et al. (2016); He et al. (2017); Gale et al. (2019) is deemed as a promising force to alleviate this problem. Since its early debut Mozer & Smolensky (1989); Reed (1993), the central problem of neural network pruning has been (arguably) how to choose weights to discard, i.e., the weight importance scoring problem LeCun et al. (1990); Hassibi & Stork (1993); Molchanov et al. (2017b; 2019); Wang et al. (2019a); He et al. (2020). ", + "bbox": [ + 174, + 575, + 825, + 686 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The approaches to the scoring problem generally fall into two groups: importance-based and regularization-based Reed (1993). The former focuses on directly proposing certain theoretically sound importance criterion so that we can prune the unimportant weights once for all. Thus, the pruning process is typically one-shot. In contrast, regularization-based approaches typically select unimportant weights through training with a penalty term Han et al. (2015); Wen et al. (2016); Liu et al. (2017). However, the penalty strength is usually maintained in a small regime to avoid damaging the model expressivity. Whereas, a large penalty strength can be helpful, specifically in two aspects. (1) A large penalty can push unimportant weights rather close to zero, then the pruning later barely hurts the performance even if the simple weight magnitude is adopted as criterion. (2) It is well-known that different weights of a neural network lie on the regions with different local quadratic structures, i.e., Hessian information. Many methods try to tap into this to build a more accurate scoring LeCun et al. (1990); Hassibi & Stork (1993); Wang et al. (2019a); Singh & Alistarh (2020). However, for deep networks, it is especially hard to estimate Hessian. Sometimes, even the computing itself can be intractable without resorting to proper approximation Wang et al. (2019a). On this problem, we ask: Is it possible to exploit the Hessian information without knowing their specific values? This is the second scenario where a growing regularization can help. We will show under a growing regularization, the weight magnitude will naturally separate because of their different underlying local quadratic structure, therein we can pick the unimportant weights more faithfully even using the simple magnitude-based criterion. Corresponding to these two aspects, we will present two algorithms based on a growing $L _ { 2 }$ regularization paradigm, in which the first highlights a better pruning schedule1 and the second explores a better pruning criterion. ", + "bbox": [ + 174, + 694, + 825, + 901 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our contributions. (1) We propose a simple yet effective growing regularization scheme, which can help transfer the model expressivity to the remaining part during pruning. The encouraging performance inspires us that the pruning schedule may be as critical as the weight importance criterion and deserve more research attention. (2) We further adopt growing regularization to exploit Hessian implicitly, without knowing their specific values. The method can help choose the unimportant weights more faithfully with a theoretically sound basis. In this regard, our paper is the first to show the connection between magnitude-based pruning and Hessian-based pruning, pointing out that the latter can be turned into the first one through our proposed growing regularization scheme. (3) The proposed two algorithms are easy to implement and scalable to large-scale datasets and networks. We show their effectiveness compared with many state-of-the-arts. Especially, the methods can work seamlessly for both filter pruning and unstructured pruning. ", + "bbox": [ + 173, + 194, + 825, + 347 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 369, + 344, + 386 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Regularization-based pruning. The first group of relevant works is those applying regularization to learn sparsity. The most famous probably is to use $L _ { 0 }$ or $L _ { 1 }$ regularization Louizos et al. (2018); Liu et al. (2017); Ye et al. (2018) due to their sparsity-inducing nature. In addition, the common $L _ { 2 }$ regularization is also explored for approximated sparsity Han et al. (2015; 2016). The early papers focus more on unstructured pruning, which is beneficial to model compression yet not to acceleration. For structured pruning in favor of acceleration, Group-wise Brain Damage Lebedev & Lempitsky (2016) and SSL Wen et al. (2016) propose to use Group LASSO Yuan & Lin (2006) to learn regular sparsity, where the penalty strength is still kept in small scale because the penalty is uniformly applied to all the weights. To resolve this, Ding et al. (2018) and Wang et al. (2019c) propose to employ different penalty factors for different weights, enabling large regularization. ", + "bbox": [ + 173, + 402, + 825, + 541 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Importance-based pruning. Importance-based pruning tries to establish certain advanced importance criteria that can reflect the true relative importance among weights as faithfully as possible. The pruned weights are usually decided immediately by some proposed formula instead of by training (although the whole pruning process can involve training, e.g., iterative pruning). The most widely used criterion is the magnitude-based: weight absolute value for unstructured pruningHan et al. (2015; 2016) or $L _ { 1 } / L _ { 2 }$ -norm for structured pruning Li et al. (2017). This heuristic criterion was proposed a long time ago Reed (1993) and has been argued to be inaccurate. In this respect, improvement mainly comes from using Hessian information to obtain a more accurate approximation of the increased loss when a weight is removed LeCun et al. (1990); Hassibi & Stork (1993). Hessian is intractable to compute for large networks, so some methods (e.g., EigenDamage Wang et al. (2019a), WoodFisher Singh & Alistarh (2020)) employ cheap approximation (such as K-FAC Fisher Martens & Grosse (2015)) to make the 2nd-order criteria tractable on deep networks. ", + "bbox": [ + 173, + 549, + 825, + 715 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Note that, there is no a hard boundary between the importance-based and regularization-based. Many papers present their schemes in the combination of the two Ding et al. (2018); Wang et al. (2019c). The difference mainly lies in their emphasis: Regularization-based method focuses more on an advanced penalty scheme so that the subsequent pruning criterion can be simple; while the importance-based one focus more on an advanced importance criterion itself. Meanwhile, regularization paradigm always involves iterative training, while the importance-based can be one-shot LeCun et al. (1990); Hassibi & Stork (1993); Wang et al. (2019a) (no training for picking weights to prune) or involve iterative training Molchanov et al. (2017b; 2019); Ding et al. (2019a;b). ", + "bbox": [ + 173, + 722, + 825, + 833 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Other model compression methods. Apart from pruning, there are also many other model compression approaches, e.g., quantization Courbariaux & Bengio (2016); Courbariaux et al. (2016); Rastegari et al. (2016), knowledge distillation Bucilua et al. ˇ (2006); Hinton et al. (2014), lowrank decomposition Denton et al. (2014); Jaderberg et al. (2014); Lebedev et al. (2014); Zhang et al. (2015), and efficient architecture design or search Howard et al. (2017); Sandler et al. (2018); Howard et al. (2019); Zhang et al. (2018); Tan & Le (2019); Zoph & Le (2017); Elsken et al. (2019). They are orthogonal to network pruning and can work with the proposed methods to compress more. ", + "bbox": [ + 176, + 840, + 823, + 882 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 PROPOSED METHOD ", + "text_level": 1, + "bbox": [ + 176, + 179, + 377, + 195 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 PROBLEM FORMULATION ", + "text_level": 1, + "bbox": [ + 176, + 209, + 392, + 223 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Pruning can be formulated as a transformation $T ( * )$ that takes a pretrained big model w as input and output a small model $\\mathbf { w } _ { 1 }$ , typically followed by a fine-tuning process $F ( * )$ , which gives us the final output $\\mathbf { w } _ { 2 } = F ( \\mathbf { w } _ { 1 } )$ . We do not focus on $F ( * )$ since it is simply a standard neural network training process, but focus on the process of $\\mathbf { w } _ { 1 } = T ( \\mathbf { w } )$ . The effect of pruning can be further specified into two sub-transformations: (1) $M = T _ { 1 } ( \\mathbf { w } )$ , which obtains a binary mask vector $M$ that decides which weights will be removed; (2) $T _ { 2 } ( \\mathbf { w } )$ , which adjusts the values of remaining weights. That is, ", + "bbox": [ + 173, + 234, + 825, + 319 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/2d1ceb0dc13e488fbfecab35f8e6c3c27d51269645ded8779c1531fcf956c376.jpg", + "text": "$$\n\\mathbf { w } _ { 1 } = T ( \\mathbf { w } ) = T _ { 1 } ( \\mathbf { w } ) \\odot T _ { 2 } ( \\mathbf { w } ) = M \\odot T _ { 2 } ( \\mathbf { w } ) .\n$$", + "text_format": "latex", + "bbox": [ + 339, + 320, + 658, + 338 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "For one-shot pruning, there is no iterative training at $T _ { 1 }$ . It depends on a specific algorithm to decide whether to adjust the remaining weights. For example, OBD LeCun et al. (1990) and $L _ { 1 }$ - norm pruning Li et al. (2017) do not adjust the kept weights (i.e., $T _ { 2 }$ is the identity function) while OBS Hassibi & Stork (1993) does. For learning-based pruning, both $T _ { 1 }$ and $T _ { 2 }$ involve iterative training and the kept weights will always be adjusted. ", + "bbox": [ + 173, + 345, + 825, + 417 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In the following, we will present our algorithms in the filter pruning scenario since we mainly focus on model acceleration instead of compression in this work. Nevertheless, the methodology can seamlessly translate to the unstructured pruning case. The difference lies in how we define the weight group: For filter pruning, a 4-d tensor convolutional filter (or 2-d tensor for fully-connected layers) is regarded as a weight group, while for unstructured pruning, a single weight makes a group. ", + "bbox": [ + 173, + 422, + 825, + 493 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 PRUNING SCHEDULE: GREG-1 ", + "text_level": 1, + "bbox": [ + 176, + 508, + 426, + 523 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our first method (GReg-1) is a variant of $L _ { 1 }$ -norm pruning Li et al. (2017). It obtains the mask $M$ by $L _ { 1 }$ -norm sorting but adjusts the kept weights via regularization. Specifically, given a pre-trained model w and layer pruning ratio $r _ { l }$ , we sort the filters by $L _ { 1 }$ -norm and set the mask to zero for those with the least norms. Then, unlike Li et al. (2017) which removes the unimportant weights immediately (i.e., one-shot fashion), we impose a growing $L _ { 2 }$ penalty to drive them to zero first: ", + "bbox": [ + 173, + 535, + 825, + 606 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/51859927106d65252be86ad587592f9ca98b1e8e8cbba05681fd3829a57150f3.jpg", + "text": "$$\n\\lambda _ { j } = \\lambda _ { j } + \\delta \\lambda , j \\in \\{ j \\mid M [ j ] = 0 \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 380, + 607, + 614, + 625 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\lambda _ { j }$ is the penalty factor for $j$ -th weight; $\\delta \\lambda$ is the granularity in which we add up the penalty. Clearly, a smaller $\\delta \\lambda$ means this regularization process smoother. Besides, $\\lambda _ { j }$ is only updated every $K _ { u }$ iterations, which is a buffer time to let the network adapt to the new regularization. This algorithm is to explore whether the way we remove them (i.e., pruning schedule) leads to a difference given the same weights to prune. Simple as it is, the scheme can bring significant accuracy gains especially under a large pruning ratio (Tab. 1). Note that, we intentionally set $\\delta \\lambda$ the same for all the unimportant weights to keep the core idea simple. Natural extensions of using different penalty factors for different weights (such as those in Ding et al. (2018); Wang et al. (2019c)) may be worth exploring but out of the scope of this work. ", + "bbox": [ + 173, + 625, + 825, + 751 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "When $\\lambda _ { j }$ reaches a pre-set ceiling $\\tau$ , we terminate the training and prune those with the least $L _ { 1 }$ - norms, then fine-tune. Notably, the pruning will barely hurt the accuracy since the unimportant weights have been compressed to typically less than $\\frac { 1 } { 1 0 0 0 }$ the magnitude of remaining weights. ", + "bbox": [ + 174, + 757, + 825, + 801 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.3 IMPORTANCE CRITERION: GREG-2 ", + "text_level": 1, + "bbox": [ + 176, + 815, + 459, + 830 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our second algorithm is to further take advantage of the growing regularization scheme, not for pruning schedule but scoring. The training of neural networks is prone to overfitting, so regularization is normally employed. $L _ { 2 }$ regularization (or referred to as weight decay) is a standard technique for deep network training. Given a dataset $\\mathcal { D }$ , model parameters $\\mathbf { w }$ , the total loss will typically be ", + "bbox": [ + 174, + 840, + 825, + 897 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/64766edeea860dcfc7991fe775c58894666199785a4e91b42c201e9c3ee2f793.jpg", + "text": "$$\n\\mathcal { E } ( \\mathbf { w } , \\mathcal { D } ) = \\mathcal { L } ( \\mathbf { w } , \\mathcal { D } ) + \\frac { 1 } { 2 } \\lambda \\| \\mathbf { w } \\| _ { 2 } ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 387, + 898, + 609, + 928 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathcal { L }$ is the task loss function. When the training converges, there should be ", + "bbox": [ + 173, + 103, + 692, + 118 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/37e30984524a39872ef6c179119d2c7d2ca1c3877bfae90c6319ba6db782ec71.jpg", + "text": "$$\n\\lambda w _ { i } ^ { * } + \\frac { \\partial \\mathcal { L } } { \\partial w _ { i } } | _ { w _ { i } = w _ { i } ^ { * } } = 0 ,\n$$", + "text_format": "latex", + "bbox": [ + 416, + 122, + 581, + 155 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\boldsymbol { w } _ { i } ^ { * }$ indicates the $i$ -th weight at its local minimum. Eq. (4) shows that, for each specific weight element, its equilibrium position is determined by two forces: loss gradient (i.e., guidance from the task) and regularization gradient (i.e., guidance from our prior). Our idea is to slightly increase the $\\lambda$ to break the equilibrium and see how it results in a new one. A general impression is: If $\\lambda$ goes a little higher, the penalty force will drive the weights further towards origin and it will not stop unless proper loss gradient comes to halt it and then a new equilibrium is reached at $\\hat { w } _ { i } ^ { * }$ . Considering different weights have different scales, we define a ratio $r _ { i } = \\hat { w } _ { i } ^ { * } / w _ { i } ^ { * }$ to describe how much the weight magnitude changes after increasing the penalty factor. Our interest lies in how the $r _ { i }$ differs from one another and how it relates to the underlying Hessian information. ", + "bbox": [ + 173, + 159, + 825, + 285 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Deep neural networks are well-known over-parameterized and highly non-convex. To obtain a feasible analysis, we adopt a local quadratic approximation of the loss function based on Taylor series expansion Strang (1991) following common practices LeCun et al. (1990); Hassibi & Stork (1993); Wang et al. (2019a). Then when the model is converged, the error $\\mathcal { E }$ can be described by the converged weights $\\mathbf { w } ^ { * }$ and the underlying Hessian matrix $\\mathbf { H }$ (note $\\mathbf { H }$ is p.s.d. since the model is converged). After increasing the penalty $\\lambda$ by $\\delta \\lambda$ , the new converged weights can be proved to be ", + "bbox": [ + 173, + 290, + 825, + 375 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/ec0c9681aea3b66f1d6a5d0c4e5b60e3fd3e51a6428a3d01aaa644fb9fa1e46b.jpg", + "text": "$$\n\\hat { \\mathbf { w } } ^ { * } = ( \\mathbf { H } + \\delta \\lambda \\mathbf { I } ) ^ { - 1 } \\mathbf { H } \\mathbf { w } ^ { * } ,\n$$", + "text_format": "latex", + "bbox": [ + 408, + 378, + 588, + 397 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\mathbf { I }$ stands for the identity matrix. Here we meet with the common problem of estimating Hessian and its inverse, which are well-known to be intractable for deep neural networks. We explore two simplified cases to help us move forward. ", + "bbox": [ + 174, + 401, + 826, + 444 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "(1) $\\mathbf { H }$ is diagonal, which is a common simplification for Hessian LeCun et al. (1990), implying that the weights are independent of each other. For $w _ { i } ^ { * }$ with second derivative $h _ { i i }$ . With $L _ { 2 }$ penalty increased by $\\delta \\lambda$ $( \\delta \\lambda > 0 )$ , the new converged weights can be proved to be ", + "bbox": [ + 173, + 450, + 825, + 493 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/5b7b4dc32cb48b6b11396ba6bd41736ce3f582516dda7d1df22f73ab7eeaa507.jpg", + "text": "$$\n\\hat { w } _ { i } ^ { * } = \\frac { h _ { i i } } { h _ { i i } + \\delta \\lambda } w _ { i } ^ { * } , \\Rightarrow r _ { i } = \\frac { \\hat { w } _ { i } ^ { * } } { w _ { i } ^ { * } } = \\frac { 1 } { \\delta \\lambda / h _ { i i } + 1 } ,\n$$", + "text_format": "latex", + "bbox": [ + 339, + 497, + 658, + 531 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $r _ { i } \\in [ 0 , 1 )$ since $h _ { i i } \\geq 0$ and $\\delta \\lambda > 0$ . As seen, larger $h _ { i i }$ results in larger $r _ { i }$ (closer to 1), meaning that the weight is relatively less moved towards the origin. Our second algorithm primarily builds upon this finding, which implies when we add a penalty perturbation to the converged network, the way that different weights respond can reflect their underlying Hessian information. ", + "bbox": [ + 173, + 535, + 825, + 592 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "(2) In practice, we know $\\mathbf { H }$ is rarely diagonal. How the dependency among weights affects the finding abovecase, namely, $\\begin{array} { r } { \\mathbf { w } ^ { * } = \\binom { w _ { 1 } ^ { * } } { w _ { 2 } ^ { * } } , \\mathbf { H } = \\binom { h _ { 1 1 } h _ { 1 2 } } { h _ { 1 2 } h _ { 2 2 } } , \\hat { \\mathbf { H } } = \\binom { h _ { 1 1 } + \\delta \\lambda } { h _ { 1 2 } } _ { h _ { 2 2 } + \\delta \\lambda } } \\end{array}$ an in Eq. (5. The new c e explore the 2-derged weights can be analytically solved below, where the approximation equality is because that $\\delta \\lambda$ ", + "bbox": [ + 174, + 598, + 825, + 661 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/160c69c83de7c31885ca7bfc970c213f1558b16cc488e67cdb6564ab4671dcf1.jpg", + "text": "$$\n\\left\\{ \\hat { w } _ { 1 } ^ { * } \\right\\} = \\frac { 1 } { | \\hat { \\mathbf { H } } | } \\left\\{ \\begin{array} { l l } { ( h _ { 1 1 } h _ { 2 2 } + h _ { 1 1 } \\delta \\lambda - h _ { 1 2 } ^ { 2 } ) w _ { 1 } ^ { * } + \\delta \\lambda h _ { 1 2 } w _ { 2 } ^ { * } } \\\\ ( h _ { 1 1 } h _ { 2 2 } + h _ { 2 2 } \\delta \\lambda - h _ { 1 2 } ^ { 2 } ) w _ { 2 } ^ { * } + \\delta \\lambda h _ { 1 2 } w _ { 1 } ^ { * } \\right\\} \\approx \\frac { 1 } { | \\hat { \\mathbf { H } } | } \\left\\{ \\begin{array} { l l } { ( h _ { 1 1 } h _ { 2 2 } + h _ { 1 1 } \\delta \\lambda - h _ { 1 2 } ^ { 2 } ) w _ { 1 } ^ { * } } \\\\ { ( h _ { 1 1 } h _ { 2 2 } + h _ { 2 2 } \\delta \\lambda - h _ { 1 2 } ^ { 2 } ) w _ { 2 } ^ { * } + \\delta \\lambda h _ { 1 2 } w _ { 1 } ^ { * } } \\end{array} \\right\\} , \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 189, + 665, + 790, + 699 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/7e40e02521af9e22442c79b2bb12c69a4745526b9c968abf89e322d4dcacd3c5.jpg", + "text": "$$\n\\Rightarrow r _ { 1 } = \\frac { 1 } { \\left| \\hat { \\mathbf { H } } \\right| } ( h _ { 1 1 } h _ { 2 2 } + h _ { 1 1 } \\delta \\lambda - h _ { 1 2 } ^ { 2 } ) , r _ { 2 } = \\frac { 1 } { \\left| \\hat { \\mathbf { H } } \\right| } ( h _ { 1 1 } h _ { 2 2 } + h _ { 2 2 } \\delta \\lambda - h _ { 1 2 } ^ { 2 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 267, + 704, + 730, + 734 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As seen, $h _ { 1 1 } > h _ { 2 2 }$ also leads to $r _ { 1 } > r _ { 2 }$ , in line with the finding above. The existence of weight dependency (i.e., the $h _ { 1 2 }$ ) actually does not affect the conclusion since it is included in both ratios. ", + "bbox": [ + 174, + 734, + 825, + 763 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "These theoretical analyses show us that when the penalty is increased at the same pace, because of different local curvature structures, the weights actually respond differently – weights with larger curvature will be less moved. As such, the magnitude discrepancy among weights will be magnified as $\\lambda$ grows. Ultimately, the weights will naturally separate (see Fig. 1 for an empirical validation). When the discrepancy is large enough, even the simple $L _ { 1 }$ -norm can make an accurate criterion. Notably, the whole process happens itself with the uniformly rising $L _ { 2 }$ penalty, no need to know the Hessian values, thus not bothered by any issue arising from Hessian approximation in relevant prior arts LeCun et al. (1990); Hassibi & Stork (1993); Wang et al. (2019a); Singh & Alistarh (2020). ", + "bbox": [ + 173, + 770, + 826, + 882 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In terms of the specific algorithm, all the penalty factor is increased at the same pace, ", + "bbox": [ + 174, + 888, + 735, + 904 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/b1c23e2087c4a9132169d390922ee0e7cc07aa30b582e40deda35092999eb2cd.jpg", + "text": "$$\n\\lambda _ { j } = \\lambda _ { j } + \\delta \\lambda , { \\mathrm { ~ f o r ~ a l l ~ } } j .\n$$", + "text_format": "latex", + "bbox": [ + 418, + 909, + 578, + 926 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Algorithm 1 GReg-1 and GReg-2 Algorithms ", + "text_level": 1, + "bbox": [ + 176, + 103, + 475, + 118 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "1: Input: Pre-trained model w, pruning ratio for $l$ -th layer $r _ { l } , l = 1 \\sim L$ , original weight decay $\\gamma$ \n2: Input: Regularization ceiling $\\tau$ , ceiling for picking $\\cdot$ , interval $K _ { u } , K _ { s }$ , granularity $\\delta \\lambda$ . \n3: Init: Iteration $i = 0$ . $\\lambda _ { j } = 0$ for all filter $j$ . Set kept filter indexes $S _ { l } ^ { k }$ to $\\mathcal { D }$ for each layer $l$ . \n4: Init: Set pruned filter indexes $S _ { l } ^ { p }$ by $L _ { 1 }$ -norm sorting, set $S _ { l } ^ { p }$ to full set, for each layer $l$ . \n5: while $\\lambda _ { j } \\overset { \\cdot } { \\leq } \\tau , j \\in S _ { l } ^ { p }$ do \n6: if i % $K _ { u } = 0$ then \n7: if $\\cdot$ and $\\lambda _ { j } > \\tau ^ { \\prime } , j \\in S _ { l } ^ { p }$ then \n8: Set $S _ { l } ^ { p }$ by $\\cdot$ -norm scoring, $S _ { l } ^ { k }$ as the complementary set of $S _ { l } ^ { p }$ , for each layer $\\cdot$ . \n9: end if \n10: $\\lambda _ { j } = \\lambda _ { j } + \\delta \\lambda$ for $j \\in S _ { l } ^ { p }$ , $\\lambda _ { j } = - \\gamma$ for $j \\in S _ { l } ^ { k }$ , for each layer $l$ \n11: end if \n12: Weight update by stochastic gradient descent (where the regularization is enforced). \n13: $i = i + 1$ . \n14: end while \n15: Train for another $K _ { s }$ iterations to stabilize. Then prune by $L _ { 1 }$ -norms and get model $\\mathbf { w } _ { 1 }$ . \n16: Fine-tune $\\mathbf { w } _ { 1 }$ to regain accuracy. \n17: Output: Pruned model $\\mathbf { w } _ { 2 }$ . ", + "bbox": [ + 176, + 119, + 820, + 361 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "When $\\lambda _ { j }$ reaches some ceiling $\\tau ^ { \\prime }$ , the magnitude gap turns large enough to let $L _ { 1 }$ -norm do scoring faithfully. After this, the procedures are similar to those in GReg-1: $\\lambda$ for the unimportant weights are further increased. One extra step is to bring back the kept weights to the normal magnitude. Although they are the “survivors” during the previous competition under a large penalty, their expressivity are also hurt. To be exact, we adopt negative penalty factor for the kept weights to encourage them to recover. When the $\\lambda$ for unimportant weights reaches the threshold $\\tau$ (akin to that of GReg-1), the training is terminated. $L _ { 1 }$ -pruning is conducted and then fine-tune to regain accuracy. To this end, the proposed two algorithms can be summarized in Algorithm 1. ", + "bbox": [ + 173, + 387, + 825, + 500 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Pruning ratios. We employ pre-specified pruning ratios in this work to keep the core method neat (see Appendix for more discussion). Exploring layer-wise sensitivity is out of the scope of this work, but clearly any method that finds more proper pruning ratios can readily work with our approaches. ", + "bbox": [ + 174, + 506, + 825, + 547 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Discussion: differences from IncReg. Although our work shares a general spirit of growing regularization with IncReg Wang et al. (2019c;b), our work is actually starkly different from theirs in many axes: ", + "bbox": [ + 174, + 554, + 821, + 597 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "• Motivation. The motivations for using the growing regularization are different. Wang et al. (2019c;b) adopt growing regularization to select the unimportant weights by training. Namely, they focus on the importance criterion problem. In contrast, we use growing regularization to investigate the pruning schedule problem (for GReg-1) or exploit the underlying Hessian information (for GReg-2). The importance criterion is simply $L _ { 1 }$ -norm. \nAlgorithm design. Wang et al. (2019c;b) assign different regularization factors to different weight groups based on their relative importance, while we assign them with the same factors. For GReg-1, this may not be a substantial difference, while for GReg-2, the difference is fundamental because the theoretical analysis of GReg-2 (Sec. 3.3) relies on the fact that regularization factors are kept the same for different weights. \nTheoretical analysis. The algorithm in Wang et al. (2019c;b) is generally heuristic-based, while our work provides rigorous theoretical analyses (Sec. 3.3) to support the proposed algorithm GReg-2. \n• Empirical performance. Both our methods are significantly better than Wang et al. (2019c;b) on the large-scale ImageNet dataset, which will be shown in the experiment section (Tab. 3). ", + "bbox": [ + 215, + 608, + 825, + 843 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Discussion: other regularization forms. The proposed methods in this work adopts $L _ { 2 }$ regularization. Here we discuss the possibility to generalize the method to other regularization forms ( $L 1$ and $L _ { 0 }$ ). (1) For GReg-1, it can be easily generalized to other regularization forms like $L _ { 1 }$ . For GReg-2, since the theoretical basis in Sec. 3.3 relies on the local quadratic approximation, $L _ { 2 }$ regularization meets this requirement while $L _ { 1 }$ does not. Therefore, GReg-2 cannot be (easily) generalized to the ", + "bbox": [ + 174, + 854, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/12c8257961985aef474210918ea018a6e7ed1d45ba8c1c0a0bebc51d926e616c.jpg", + "table_caption": [ + "Table 1: Comparison between pruning schedules: one-shot pruning vs. our proposed GReg-1. Each setting is randomly run for 3 times, mean and std accuracies reported. " + ], + "table_footnote": [], + "table_body": "
ResNet56 + CIFAR10: Baseline accuracy 93.36%, #Params: 0.8530M,FLOPs: 0.1255G
Pruning ratio r (%)50709092.595
Sparsity (%)/ Speedup49.82/1.99×70.57/3.59×90.39/11.41×93.43/14.76×95.19/19.31×
Acc.(%,L1+one-shot)92.97±0.1591.88±0.0987.34±0.2187.31±0.2882.79±0.22
Acc.(%,GReg-1,ours)93.06±0.0992.23±0.2189.49±0.2388.39±0.1585.97±0.16
Acc. gain (%)0.090.352.151.083.18
VGG19 + CIFAR100: Baseline accuracy 74.02%,#Params: 20.0812M,FLOPs: 0.3982G
Pruning ratio r (%)5060708090
Sparsity(%)/Speedup74.87/3.60×84.00/5.41×90.98/8.84×95.95/17.30×98.96/44.22×
Acc.(%,L1+one-shot)71.49±0.1470.27±0.1266.05±0.0461.59±0.0351.36±0.11
Acc.(%,GReg-1,ours)71.50±0.1270.33±0.1267.35±0.1563.55±0.2957.09±0.03
Acc. gain (%)0.010.061.301.965.73
", + "bbox": [ + 171, + 145, + 821, + 313 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "$L _ { 1 }$ regularization as far as we can see currently. (2) For $L _ { 0 }$ regularization, it is well-known NP-hard. \nIn practice, it is typically converted to the $L _ { 1 }$ regularization case, which we just discussed. ", + "bbox": [ + 176, + 328, + 820, + 356 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 377, + 419, + 393 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Datasets and networks. We first conduct analyses on the CIFAR10/100 datasets Krizhevsky (2009) with ResNet56 He et al. (2016)/VGG19 Simonyan & Zisserman (2015). Then we evaluate our methods on the large-scale ImageNet dataset Deng et al. (2009) with ResNet34 and 50 He et al. (2016). For CIFAR datasets, we train our baseline models with accuracies comparable to those in the original papers. For ImageNet, we take the official PyTorch Paszke et al. (2019) pre-trained models2 as baseline to maintain comparability with other methods. ", + "bbox": [ + 174, + 409, + 825, + 492 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Training settings. To control the irrelevant factors as we can, for comparison methods that release their pruning ratios, we will adopt their ratios; otherwise, we will use our specified ones. We compare the speedup (measured by FLOPs reduction) since we mainly target model acceleration rather than compression. Detailed training settings (e.g., hyper-parameters and layer pruning ratios) are summarized in the Appendix. ", + "bbox": [ + 174, + 500, + 825, + 569 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 RESNET56/VGG19 ON CIFAR-10/100 ", + "text_level": 1, + "bbox": [ + 176, + 587, + 486, + 602 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Pruning schedule: GReg-1. First, we explore the effect of different pruning schedules on the performance of pruning. Specifically, we conduct two sets of experiments for comparison: (1) prune by $L _ { 1 }$ -norm sorting and fine-tune Li et al. (2017) (shorted as $^ { \\cdot \\cdot } L _ { 1 } +$ one-shot”); (2) employ the proposed growing regularization scheme (“GReg-1”) and fine-tune. We use a uniform pruning ratio scheme here: Pruning ratio $r$ is the same for all $l$ -th conv layer (the first layer is not pruned following common practice Gale et al. (2019)). For ResNet56, since it has the residual addition restriction, we only prune the first conv layer in a block as previous works do Li et al. (2017). For comprehensive comparisons, the pruning ratios vary in a large spectrum, covering acceleration ratios from around $2 \\times$ to $4 4 \\times$ . Note that we do not intend to obtain the best performance here but systematically explore the effect of different pruning schedules, so we employ relatively simple settings (e.g., the uniform pruning ratios). For fair comparisons, the fine-tuning scheme (e.g., number of epochs, learning rate schedule, etc.) is the same for different methods. Therefore, the key comparison here is to see which method can deliver a better base model before fine-tuning. ", + "bbox": [ + 173, + 613, + 825, + 794 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The results are shown in Tab. 1. We have the following observations: (1) On the whole, the proposed GReg-1 consistently outperforms $L _ { 1 }$ +one-shot. It is important to reiterate that the two settings have exactly the same pruned weights, so the only difference is how they are removed. The accuracy gaps show that apart from importance scoring, pruning schedule is also a critical factor. In the Appendix D, we present more results to demonstrate this finding actually is general, not merely limited to the case of $L _ { 1 }$ -norm criterion. The proposed regularization-based pruning schedule is consistently more favorable than the one-shot counterpart. (2) The larger pruning ratio, the more pronounced of the gain. This is reasonable since when more weights are pruned, the network cannot recover by its inherent plasticity Mittal et al. (2018), then the regularization-based way is more helpful because it helps the model transfer its expressive power to the remaining part. When the pruning ratio is relatively small (such as ResNet56, $r = 5 0 \\%$ ) , the plasticity of the model is enough to heal, so the benefit from GReg-1 is less significant compared with the one-shot counterpart. ", + "bbox": [ + 173, + 801, + 825, + 898 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/ac428ab6cf517902459262b7ec3fb69d47cf04a2e15dd1400507d10a01bda824.jpg", + "image_caption": [ + "Figure 1: Row 1: Illustration of weight separation as $L _ { 2 }$ penalty grows. Row 2: Normalized filter $L _ { 1 }$ -norm over iterations for ResNet50 layer2.3.conv1 (please see the Appendix for VGG19 plots). " + ], + "image_footnote": [], + "bbox": [ + 176, + 102, + 820, + 294 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/d907f6083b53bd48fe53b4c66cdc0b64e4a781cce80e1134643c84b63857284a.jpg", + "table_caption": [ + "Table 2: Comparison of different methods on the CIFAR10 and CIFAR100 datasets. " + ], + "table_footnote": [], + "table_body": "
MethodNetwork/DatasetBase acc.(%) Pruned acc. (%) Acc. drop Speedup
CP He et al. (2017)ResNet56/CIFAR1092.8091.801.002.00×
AMC He et al. (2018b)92.8091.900.902.00×
SFP He et al. (2018a)93.5993.360.232.11×
AFP Ding et al. (2018)93.9392.940.992.56×
C-SGD Ding et al. (2019a)93.3993.44-0.052.55×
GReg-1 (ours)93.3693.180.182.55×
GReg-2 (ours)93.3693.360.002.55×
Kron-OBD Wang et al. (2019a)73.3460.7012.645.73×
Kron-OBS Wang et al. (2019a)73.3460.6612.686.09×
EigenDamage Wang et al. (2019a） VGG19/CIFAR10073.3465.188.168.80×
GReg-1 (ours)74.0267.556.678.84×
GReg-2 (ours)74.0267.756.478.84×
", + "bbox": [ + 173, + 367, + 823, + 549 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 569, + 825, + 638 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Importance criterion: GReg-2. Here we empirically validate our finding in Sec. 3.3, that is, with uniformly rising $L _ { 2 }$ penalty, the weights should naturally separate. We claim, if $h _ { 1 1 } > h _ { 2 2 }$ , there should be $r _ { 1 } ~ > ~ r _ { 2 }$ , where $\\begin{array} { r } { r _ { 1 } = \\frac { \\tilde { \\hat { w _ { 1 } } } } { w _ { 1 } } , r _ { 2 } = \\frac { \\hat { w _ { 2 } } } { w _ { 2 } } } \\end{array}$ (the \\* mark indicating the local minimum is omitted here for readability). $r _ { 1 } ~ > ~ r _ { 2 }$ leads to $\\frac { \\bar { w } _ { 1 } } { w _ { 1 } } ~ > ~ \\frac { \\bar { w } _ { 2 } } { w _ { 2 } }$ , namely, $\\begin{array} { r } { r _ { 1 } \\ = \\ \\frac { \\hat { w _ { 1 } } } { \\hat { w _ { 2 } } } \\ > \\ \\frac { w _ { 1 } } { w _ { 2 } } } \\end{array}$ This shows that, after the $L _ { 2 }$ penalty grows a little, the new magnitude ratio of weight 1 over weight 2 will be magnified if $h _ { 1 1 } > h _ { 2 2 }$ $( w _ { 1 } , w _ { 2 }$ are positive in the analysis here, while the conclusion still holds if either of them is negative). In Fig. 1 (Row 1), we plot the standard deviation (divided by the means for normalization since the magnitude varies over iterations) of filter $L _ { 1 }$ -norms as the regularization grows. As seen, the normalized $L _ { 1 }$ -norm stddev grows larger and larger as $\\lambda$ grows. This phenomenon consistently appears across different models and datasets. To figuratively understand how the increasing penalty affects the relative magnitude over time, in Fig. 1 (Row 2), we plot the relative $L _ { 1 }$ -norms (divided by the max $L _ { 1 }$ -norm for normalization) at different iterations. As shown, it is hard to tell which filters are really important by the initial filter magnitude (Iter 0), but under a large penalty later, their discrepancy turns more and more obvious and finally it is very easy to identify which filters are more important. Since the magnitude gap is so large, the simple $L _ { 1 }$ -norm can make a sufficiently faithful criterion. ", + "bbox": [ + 173, + 645, + 825, + 875 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "CIFAR benchmarks. Finally, we compare the proposed algorithms with existing methods on the CIFAR datasets (Tab. 2). Here we adopt non-uniform pruning ratios (see the Appendix for specific numbers) for the best accuracy-FLOPs trade-off. On CIFAR10, compared with AMC He et al. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/a37e76fa341991da35b036954a7fb022287fc9ac19b9a1f8f740b9ea64331cb0.jpg", + "table_caption": [ + "Table 3: Acceleration comparison on ImageNet. FLOPs: ResNet34: 3.66G, ResNet50: 4.09G. " + ], + "table_footnote": [ + "Since the base models of C-SGD and AOFP have a much lower accuracy than ours, for fair comparison, we rain our own base models with similar accuracy. " + ], + "table_body": "
MethodNetworkBase top-1(%） Pruned top-1(%） Top-1 drop Speedup
L1 (pruned-B) Li et al. (2017) Taylor-FO Molchanov et al. (2019)ResNet3473.23 73.3172.17 72.831.06 0.481.32× 1.29×
GReg-1 (ours)73.3173.54-0.231.32×
GReg-2 (ours)73.3173.61-0.301.32×
ProvableFPLiebenwein et al. (2020) GReg-1 (ours)ResNet5076.13 76.1375.21 76.270.92 -0.141.43× 1.49×
AOFP Ding et al. (2019b) GReg-1 (ours)*ResNet5075.3475.63-0.291.49×
75.4076.13-0.731.49×
IncReg Wang et al. (2019b) SFP He et al. (2018a)75.60 76.1572.47 74.613.13 1.542.00× 1.72×
HRank Lin et al. (2020a) Taylor-FO Molchanov et al. (2019)ResNet5076.1574.981.171.78×
Factorized Li et al. (2019)76.18 76.1574.501.681.82×
74.551.602.33×
DCP Zhuang et al. (2018) CCP-AC Peng et al. (2019)76.0174.951.062.25×
76.1575.320.832.18×
GReg-1 (ours)76.1375.160.972.31×
GReg-2 (ours) C-SGD-50 Ding et al. (2019a)ResNet5076.1375.360.772.31×
75.3474.540.802.26×
AOFP Ding et al. (2019b)75.3475.110.232.31×
GReg-2 (ours)*ResNet5075.4075.220.182.31×
LFPC He et al. (2020)76.1574.461.692.55×
GReg-1 (ours)76.1374.851.282.56×
GReg-2 (ours)76.1374.931.202.56×
IncReg Wang et al. (2019b)75.6071.074.533.00×
Taylor-FO Molchanov et al. (2019)76.1871.693.05×
GReg-1 (ours)ResNet504.49
76.1373.752.383.06×
GReg-2 (ours)76.1373.902.233.06×
", + "bbox": [ + 173, + 132, + 818, + 501 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/15777f30c8f15e6704869b37f5118341cc015ffd6a4e2e618396b70cd11a8c78.jpg", + "table_caption": [ + "Table 4: Compression comparison on ImageNet with ResNet50. #Parameters: 25.56M. " + ], + "table_footnote": [], + "table_body": "
MethodBase top-1 (%)Pruned top-1(%） Top-1 dropSparsity (%)
GSM Ding et al. (2019c)75.7274.301.4280.00
Variational Dropout Molchanov et al. (2O17a)76.6975.281.4180.00
DPF Lin et al. (2020b)75.9574.551.4082.60
WoodFisher Singh & Alistarh (2020)75.9875.200.7882.70
GReg-1 (ours)76.1375.450.6882.70
GReg-2 (ours)76.1375.270.8682.70
", + "bbox": [ + 174, + 547, + 825, + 652 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "(2018b), though it adopts better layer-wise pruning ratios via reinforcement-learning, our algorithms can still deliver more favorable performance using sub-optimal human-specified ratios. AFP Ding et al. (2018) is another work exploring large regularization, while they do not adopt the growing scheme as we do. Its performance is also less favorable on CIFAR10 as shown in the table. Although our methods perform a little worse than C-SGD Ding et al. (2019a) on CIFAR10, on the large-scale ImageNet dataset, we will show our methods are significantly better than C-SGD. ", + "bbox": [ + 173, + 655, + 825, + 738 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Notably, on CIFAR100, Kron-OBD/OBS (an extension by Wang et al. (2019a) of the original OBD/OBS from unstructured pruning to structured pruning) are believed to be more accurate than $L _ { 1 }$ -norm in terms of capturing relative weight importance LeCun et al. (1990); Hassibi & Stork (1993); Wang et al. (2019a). Yet, they are significantly outperformed by our GReg-1 based on the simple $L _ { 1 }$ -norm scoring. This may inspire us that an average pruning schedule (like the one-shot fashion) can offset the gain from a more advanced importance scoring scheme. ", + "bbox": [ + 174, + 744, + 825, + 829 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 RESNET34/50 ON IMAGENET ", + "text_level": 1, + "bbox": [ + 176, + 845, + 419, + 859 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Then we evaluate our methods on the standard large-scale ImageNet benchmarks with ResNets He et al. (2016). We refer to the official PyTorch ImageNet training example3 to make sure the implementation (such as data augmentation, weight decay, momentum, etc.) is standard. Please refer to the summarized training setting in the Appendix for details. ", + "bbox": [ + 176, + 872, + 823, + 900 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 821, + 132 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The results are shown in Tab. 3. Methods with similar speedup are grouped together for easy comparison. In general, our method achieves comparable or better performance across various speedups on ResNet34 and 50. Concretely, (1) On both ResNet34 and 50, when the speedup is small (less than $2 \\times$ ), only our methods (and AOFP Ding et al. (2019b) for ResNet50) can even improve the top-1 accuracy. This phenomenon is broadly found by previous works Wen et al. (2016); Wang et al. (2018); He et al. (2017) but mainly on small datasets like CIFAR, while we make it on the much challenging ImageNet benchmark. (2) Similar to the results on CIFAR (Tab. 1), when the speedup is larger, the advantage of our method is more obvious. For example, ours GReg-2 only outperforms Taylor-FO Molchanov et al. (2019) by $0 . 8 6 \\%$ top-1 accuracy at the $\\sim 2 \\times$ setting, while at $\\sim 3 \\times$ , GReg-2 is better by $2 . 2 1 \\%$ top-1 accuracy. (3) Many methods work on the weight importance criterion problem, including some very recent ones (ProvableFP Liebenwein et al. (2020), LFPC He et al. (2020)). Yet as shown, our simple variant of $L _ { 1 }$ -norm pruning can still be a strong competitor in terms of accuracy-FLOPs trade-off. This reiterates one of our key ideas in this work that the pruning schedule may be as important as weight importance scoring and worth more research attention. ", + "bbox": [ + 173, + 138, + 825, + 333 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Unstructured pruning. Although we mainly target filter pruning in this work, the proposed methods actually can be applied to unstructured pruning as effectively. In Tab. 4, we present the results of unstructured pruning on ResNet50. WoodFisher Singh & Alistarh (2020) is the state-of-the-art Hessian-based unstructured pruning approach. Notably, without any Hessian approximation, our GReg-2 can achieve comparable performance with it (better absolute accuracy, yet slightly worse accuracy drop). Besides, the simple magnitude pruning variant GReg-1 delivers more favorable result, implying that a better pruning schedule also matters in the unstructured pruning case. ", + "bbox": [ + 174, + 340, + 825, + 438 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 459, + 318, + 476 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Regularization is long deemed as a sparsity-learning tool in neural network pruning, which usually works in the small strength regime. In this work, we present two algorithms that exploit regularization in a new fashion that the penalty factor is uniformly raised to a large level. Two central problems regarding deep neural pruning are tackled by the proposed methods, pruning schedule and weight importance criterion. The proposed approaches rely on few impractical assumptions, have a sound theoretical basis, and are scalable to large datasets and networks. Apart from the methodology itself, the encouraging results on CIFAR and ImageNet also justify our general ideas in this paper: (1) In addition to weight importance scoring, pruning schedule is another pivotal factor in deep neural pruning which may deserve more research attention. (2) Without any Hessian approximation, we can still tap into its power for pruning with the help of growing $L _ { 2 }$ regularization. ", + "bbox": [ + 174, + 491, + 825, + 631 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENTS ", + "text_level": 1, + "bbox": [ + 176, + 654, + 366, + 669 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The work is supported by the National Science Foundation Award ECCS-1916839 and the U.S. Army Research Office Award W911NF-17-1-0367. 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In ICLR, 2017. 3 ", + "bbox": [ + 176, + 887, + 823, + 915 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 102, + 297, + 117 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.1 EXPERIMENTAL SETTING DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 133, + 460, + 148 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Training setting summary. About the networks evaluated, we intentionally avoid AlexNet and VGG on the ImageNet benchmark because the single-branch architecture is no longer representative of the modern deep network architectures with residuals (but still keep VGG19 on the CIFAR analysis to make sure the findings are not limited to one specific architecture). Apart from some key settings stated in the paper, a more detailed training setting summary is shown as Tab. 5. ", + "bbox": [ + 174, + 159, + 825, + 229 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/850c677b90dbfd948de7c102f6c87f86dbaff4275eac0749e6340b7515e0809c.jpg", + "table_caption": [ + "Table 5: Training setting summary. For the SGD solver, in the parentheses are the momentum and weight decay. For ImageNet, batch size 64 is used for pruning instead of the standard 256, which is because we want to save the training time. " + ], + "table_footnote": [], + "table_body": "
DatasetCIFARImageNet
SolverSGD (0.9, 5e-4)SGD (0.9, 1e-4)
LR policy (prune)Fixed (1e-3)
LR policy (finetune)Multi-step (0:1e-2,60:1e-3,90:1e-4)Multi-step (0:1e-2, 60:1e-3,90:1e-4)Multi-step (0:1e-2, 30:1e-3, 60:1e-4,75:1e-5)
Total epoch (finetune)12090
Batch size (prune)25664
Batch size (finetune)256
", + "bbox": [ + 178, + 300, + 815, + 395 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Pruning ratios. Although several recent methods Ding et al. (2019b); Singh & Alistarh (2020) can automatically decide pruning ratios, in this paper we opt to consider pruning independent with the pruning ratio choosing. The main consideration is that pruning ratio is broadly believed to reflect the redundancy of different layers LeCun et al. (1990); Wen et al. (2016); He et al. (2017), which is an inherent characteristic of the model, thus should not be coupled with the subsequent pruning algorithms. ", + "bbox": [ + 173, + 416, + 825, + 501 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Before we list the specific pruning ratios, we explain how we set them. (1) For a ResNet, if it has $N$ stages, we will use a list of $N$ floats to represent its pruning ratios for the $N$ stages. For example, ResNet56 has 4 stages in conv layers, then “[0, 0.5, 0.5, 0.5]” means “for the first stage (which is also the first conv layer), the pruning ratio is 0; the other three stages have pruning ratio of $0 . 5 '$ . Besides, since we do not prune the last conv in a residual block, which means for a two-layer residual block (for ResNet56), we only prune the first layer; for a three-layer bottleneck block (for ResNet34 and 50), we only prune the first and second layers. (2) For VGG19, we use the following pruning ratio setting. For example, “[0:0, 1-9:0.3, 10-15:0.5]” means “for the first layer (index starting from 0), the pruning ratio is 0; for layer 1 to 9, the pruning ratio is 0.3; for layer 10 to 15, the pruning ratio is $0 . 5 '$ . ", + "bbox": [ + 173, + 507, + 825, + 646 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "With these, the specific pruning ratio for each of our experiments in the paper are listed in Tab. 6. We do not have strong rules to set them, except one, which is setting the pruning ratios of higher stages smaller, because the FLOPs of higher layers are relatively smaller (due to the fact that the spatial feature map sizes are smaller) and we are targeting more acceleration than compression. Of course, this scheme only is quite crude, yet as our results (Tab. 3 and 4) show, even with these crude settings, the performances are still competitive. ", + "bbox": [ + 174, + 652, + 825, + 737 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B PROOF OF EQ. 5 ", + "text_level": 1, + "bbox": [ + 176, + 756, + 344, + 773 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "When a quadratic function $\\mathcal { E }$ converges at $\\mathbf { w } ^ { * }$ with Hessian matrix $\\mathbf { H }$ , it can be formulated as ", + "bbox": [ + 169, + 787, + 784, + 803 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/a354779b1fe11cc9f5eb7262eb5daa29a2ba4a2657bb0a027f50705a65c78136.jpg", + "text": "$$\n\\mathcal { E } = ( \\mathbf { w } - \\mathbf { w } ^ { * } ) ^ { T } \\mathbf { H } ( \\mathbf { w } - \\mathbf { w } ^ { * } ) + C ,\n$$", + "text_format": "latex", + "bbox": [ + 382, + 809, + 614, + 829 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where $C$ is a constant. Now a new function is made by increasing the $L _ { 2 }$ penalty by small amount $\\delta \\lambda$ , namely, ", + "bbox": [ + 171, + 834, + 825, + 863 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/c34edfcfabc00cf2e1f4718de0c9ceeab070010b0548d9b2aac3a501a35144b7.jpg", + "text": "$$\n\\begin{array} { r } { \\hat { \\mathcal { E } } = \\mathcal { E } + \\delta \\lambda \\mathbf { w } ^ { T } \\mathbf { I } \\mathbf { w } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 431, + 861, + 565, + 878 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Let the new converged values be $\\hat { \\mathbf { w } } ^ { * }$ , then similar to Eq. 10, $\\hat { \\mathcal { E } }$ can be formulated as ", + "bbox": [ + 171, + 883, + 720, + 900 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/13e8756e6a8f3576944c1fa9c0c0737ff995ca8cb27e477bd45861524b23d6fa.jpg", + "text": "$$\n\\hat { \\mathcal { E } } = ( \\mathbf { w } - \\hat { \\mathbf { w } } ^ { * } ) ^ { T } \\hat { \\mathbf { H } } ( \\mathbf { w } - \\hat { \\mathbf { w } } ^ { * } ) + \\hat { C } , \\mathrm { w h e r e } \\hat { \\mathbf { H } } = \\mathbf { H } + \\delta \\lambda \\mathbf { I } .\n$$", + "text_format": "latex", + "bbox": [ + 305, + 905, + 691, + 924 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/9eb51b52ad79a8b97c1232d77d2bbb3f90a797f47cf9598a94a131052800195c.jpg", + "table_caption": [ + "Table 6: Pruning ratio summary. " + ], + "table_footnote": [ + "\\* In addition to the pruning ratios, several layers are skipped, following the setting of $L _ { 1 }$ (pruned-B) Li et al. (2017). Specifically, we refer to the implementation of Liu et al. (2019) at https://github.com/Ericmingjie/rethinking-network-pruning/tree/master/imagenet/l1-norm-pruning. " + ], + "table_body": "
DatasetNetworkSpeedupPruned top-1 accuracy (%)Pruning ratio
CIFAR10CIFAR100ResNet56VGG192.55×8.84×93.3667.56[0, 0.75, 0.75, 0.32, 0][1-15:0.7]
ImageNetImageNetImageNetImageNetImageNetResNet34ResNet50ResNet501.32×73.4476.2475.16[0, 0.50, 0.60, 0.40, 0, 0]*[0, 0.30, 0.30, 0.30, 0.14, 0][0, 0.60, 0.60, 0.60, 0.21, 0][0, 0.74, 0.74, 0.60, 0.21, 0][0, 0.68, 0.68, 0.68, 0.50, 0]
1.49×2.31×
ResNet50ResNet502.56×74.7573.50
3.06×
", + "bbox": [ + 173, + 131, + 854, + 251 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Meanwhile, combine Eq. 10 and Eq. 11, we can obtain ", + "bbox": [ + 173, + 313, + 532, + 328 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/3e92a4b74763a03469816778e023f02f170c348be759a20be4a6ae37b7713ea9.jpg", + "text": "$$\n\\begin{array} { r } { \\hat { \\mathcal { E } } = ( \\mathbf { w } - \\mathbf { w } ^ { * } ) ^ { T } \\mathbf { H } ( \\mathbf { w } - \\mathbf { w } ^ { * } ) + \\delta \\lambda \\mathbf { w } ^ { T } \\mathbf { I } \\mathbf { w } + C . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 341, + 330, + 656, + 351 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Compare Eq. 13 with Eq. 12, we have ", + "bbox": [ + 173, + 361, + 424, + 376 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/5e5c2915d39bb703757f0810d89c822d24d94c22f481c25d5e2a659ee95f70b9.jpg", + "text": "$$\n( \\mathbf { H } + \\delta \\lambda \\mathbf { I } ) { \\hat { \\mathbf { w } } } ^ { * } = \\mathbf { H } \\mathbf { w } ^ { * } \\Rightarrow { \\hat { \\mathbf { w } } } ^ { * } = ( \\mathbf { H } + \\delta \\lambda \\mathbf { I } ) ^ { - 1 } \\mathbf { H } \\mathbf { w } ^ { * } .\n$$", + "text_format": "latex", + "bbox": [ + 321, + 377, + 674, + 396 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C PROOF OF EQ. 7 ", + "bbox": [ + 174, + 414, + 344, + 430 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/ea9dc30c0730797c1c511ebc3edb22e5cea7c1db75ba0e57f8f864a49be11f03.jpg", + "text": "$$\n\\hat { \\mathbf { H } } = \\left\\{ \\begin{array} { c c } { h _ { 1 1 } + \\delta \\lambda } & { h _ { 1 2 } } \\\\ { h _ { 1 2 } } & { h _ { 2 2 } + \\delta \\lambda } \\end{array} \\right\\} \\Rightarrow \\hat { \\mathbf { H } } ^ { - 1 } = \\frac { 1 } { \\vert \\hat { \\mathbf { H } } \\vert } \\left\\{ \\begin{array} { c c } { h _ { 2 2 } + \\delta \\lambda } & { - h _ { 1 2 } } \\\\ { - h _ { 1 2 } } & { h _ { 1 1 } + \\delta \\lambda } \\end{array} \\right\\}\n$$", + "text_format": "latex", + "bbox": [ + 259, + 441, + 735, + 479 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Therefore, $\\hat { \\mathbf { w } } ^ { * } = \\hat { \\mathbf { H } } ^ { - 1 } \\mathbf { H } \\mathbf { w } ^ { * } \\Rightarrow$ ", + "bbox": [ + 174, + 488, + 380, + 503 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/51ddb2df0ebfaba7fefc2a8c3c7138da365237bf1ff5c5ce6187997a1c7278c3.jpg", + "text": "$$\n\\begin{array} { r l r } & { } & { \\left\\{ \\hat { w } _ { 1 } ^ { * } \\right\\} = \\hat { \\bf H } ^ { - 1 } { \\bf H } \\left\\{ w _ { 1 } ^ { * } \\right\\} = \\frac { 1 } { \\left| \\hat { \\bf H } \\right| } \\left\\{ \\begin{array} { c c } { h _ { 2 2 } + \\delta \\lambda } & { - h _ { 1 2 } } \\\\ { - h _ { 1 1 } } & { h _ { 1 1 } + \\delta \\lambda } \\end{array} \\right\\} \\left\\{ \\begin{array} { c c } { h _ { 1 1 } } & { h _ { 1 2 } } \\\\ { h _ { 1 2 } } & { h _ { 2 2 } } \\end{array} \\right\\} \\left\\{ \\begin{array} { c } { w _ { 1 } ^ { * } } \\\\ { w _ { 2 } ^ { * } } \\end{array} \\right\\} } \\\\ & { } & { = \\frac { 1 } { \\left| \\hat { \\bf H } \\right| } \\left\\{ \\begin{array} { c c } { \\left( h _ { 1 1 } h _ { 2 2 } + h _ { 1 1 } \\delta \\lambda - h _ { 1 2 } ^ { 2 } \\right) w _ { 1 } ^ { * } + \\delta \\lambda h _ { 1 2 } w _ { 2 } ^ { * } } \\\\ { \\left( h _ { 1 1 } h _ { 2 2 } + h _ { 2 2 } \\delta \\lambda - h _ { 1 2 } ^ { 2 } \\right) w _ { 2 } ^ { * } + \\delta \\lambda h _ { 1 2 } w _ { 1 } ^ { * } } \\end{array} \\right\\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 258, + 507, + 736, + 575 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "D GREG- $1 + \\mathrm { O B D }$ ", + "text_level": 1, + "bbox": [ + 174, + 588, + 346, + 604 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In Sec. 4.1, we show when pruning the same weights, GReg-1 is significantly better than the oneshot counterpart, where the pruned weights are selected by the $L _ { 1 }$ -norm criterion. Here we conduct the same comparison just with a different pruning criterion introduced in OBD LeCun et al. (1990). OBD is also an one-shot pruning method, using a Hessian-based criterion which is believed to be more advanced than $L _ { 1 }$ -norm. ", + "bbox": [ + 173, + 619, + 825, + 689 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Results are shown in Tab. 7. As seen, using this more advanced importance criterion, our pruning scheme based on growing regularization is still consistently better than the one-shot counterpart. Besides, it is also verified here that a better pruning schedule can bring more accuracy gain when the speedup is larger. ", + "bbox": [ + 173, + 695, + 825, + 753 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "E FILTER L1-NORM CHANGE OF VGG19 ", + "text_level": 1, + "bbox": [ + 174, + 771, + 529, + 787 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In Fig. 1 (Row 2), we plot the filter $L _ { 1 }$ -norm change over time for ResNet50 on ImageNet. Here we plot the case of VGG19 on CIFAR100 to show the weight separation phenomenon under growing regularization is a general one across different datasets and networks. ", + "bbox": [ + 174, + 803, + 823, + 844 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "F HYPER-PARAMETERS AND SENSITIVITY ANALYSIS ", + "text_level": 1, + "bbox": [ + 173, + 866, + 625, + 881 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "There are five introduced values in our methods: regularization ceiling $\\tau$ , ceiling for picking $\\tau ^ { \\prime }$ interval $K _ { u } , K _ { s }$ , granularity $\\delta \\lambda$ . Their settings are summarized in Tab. 8. Among them, the ceilings ", + "bbox": [ + 173, + 895, + 820, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Table 7: Comparison between pruning schedules: one-shot pruning vs. our proposed GReg-1 using the Hessian-based criterion introduced in OBD LeCun et al. (1990). Each setting is randomly run for 3 times, mean and std accuracies reported. We vary the global pruning ratio from 0.7 to 0.95 so as to cover the major speedup spectrum of interest. Same as Tab. 1, the pruned weights here are exactly the same for the two methods under each speedup ratio. The finetuning processes (number of epochs, LR schedules, etc.) are also the same to keep fair comparison. ", + "bbox": [ + 173, + 112, + 825, + 195 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/0f2d6935b0744fbb2ed80e72482e888c222e79ac5a2f589b70d6f4f3caae6562.jpg", + "image_caption": [ + "Figure 2: Normalized filter $L _ { 1 }$ -norm over iterations for VGG19 layer3. " + ], + "image_footnote": [], + "bbox": [ + 178, + 202, + 818, + 430 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "are set through validation: $\\tau = 1$ is set to make sure the unimportant weights are pushed down enough (as stated in the main paper, normally after the regularization training, their magnitudes are too small to cause significant accuracy degradation if they are completely removed). $\\tau ^ { \\prime } = 0 . 0 1$ is set generally for the same goal as $\\tau$ , but since it is applied to all the weight (not just the unimportant ones), we only expect it to be moderately large (thus smaller than $\\tau$ ) so that the important and unimportant can be differentiated with a clear boundary. For the $\\delta \\lambda$ , we use a very small regularization granularity $\\delta \\lambda$ , which our theoretical analysis is based on. We set its value to 1e-4 for GReg-1 and 1e-5 for GReg-2 with reference to the original weight decay value $5 \\times 1 0 ^ { - 4 }$ (for CIFAR models) and $1 0 ^ { - 4 }$ (for ImageNet models). Note that, these values come from our methods per se, not directly related to datasets and networks, thus are invariant to them. This is why we can employ the same setting of these three hyper-parameters in all our experiments, freeing practitioners from heavy tuning when dealing with different networks or datasets. ", + "bbox": [ + 173, + 479, + 825, + 647 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/5a7832f84ca81539a440aa31f97e6d3280db63e05a411adb11d23d2f9c302079.jpg", + "table_caption": [ + "Table 8: Hyper-parameters of our methods. " + ], + "table_footnote": [], + "table_body": "
NotationDefault value (CIFAR)Default value (ImageNet)
8入GReg-1: 1e-4, GReg-2: 1e-5
T1
T0.01
Ku10 iterations5 iterations
Ks5k iterations40k iterations
", + "bbox": [ + 289, + 690, + 705, + 771 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A little bit of change is for $K _ { u } , K _ { s }$ . Both are generally to let the network have enough time to converge to the new equilibrium. Generally, we prefer large update intervals, yet we also need to consider the time complexity: Too large of them will bring too many iterations, which may be unnecessary. Among them, $K _ { s }$ is less important since it is to stabilize the large regularization $\\mathit { \\Psi } _ { \\tau } = 1 \\mathit { \\Psi } _ { . }$ ). We introduce it simply to make sure the training is fully converged. Therefore, the possibly more sensitive hyper-parameter is the $K _ { u }$ (set to 5 for ImageNet and 10 for CIFAR). Here we will show the performance is insensitive to the varying $K _ { u }$ . As shown in Tab. 9, the peak performance appears at around $K _ { u } = 1 5$ for ResNet56 and $K _ { u } = 1 0$ for VGG19. We simply adopt 10 for a uniform setting in our paper. We did not heavily tune these hyper-parameters, yet as seen, they work pretty well across different networks and datasets. Notably, even for the worst cases in Tab. 9 (in blue color), they are still significantly better than those of the “ $L _ { 1 } +$ one-shot” scheme, demonstrating the robustness of the proposed algorithm. ", + "bbox": [ + 173, + 784, + 825, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/b487c3997dc43bf81e0de6e0e03adf10913f6f2638ad3242ff024ba42e14d307.jpg", + "table_caption": [ + "Table 9: Sensitivity analysis of $K _ { u }$ on CIFAR10/100 datasets with the proposed GReg-1 algorithm. $K _ { u } = 1 0$ is the default setting. Pruning ratio $9 0 \\%$ (ResNet56) and $7 0 \\%$ (VGG19) are explored here. Experiments are randomly run for 3 times with mean accuracy and standard deviation reported. The best is highlighted with bold and the worst is highlighted with blue color. " + ], + "table_footnote": [], + "table_body": "
Ku15101520L1+one-shot
Acc. (%,ResNet56)89.40±0.0489.38±0.1389.49±0.2389.69±0.0589.62±0.1387.34±0.21
Acc. (%,VGG19)67.22±0.3367.32±0.2467.35±0.1567.06±0.4066.93±0.2266.05±0.04
", + "bbox": [ + 174, + 217, + 825, + 262 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "G MORE RESULTS OF PRUNING SCHEDULE COMPARISON ", + "text_level": 1, + "bbox": [ + 176, + 292, + 658, + 308 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In Tab. 1, we show using $L _ { 1 }$ -norm sorting, our proposed GReg-1 can consistently surpass the oneshot schedule even pruning the same weights. Here we ask a more general question: Can the benefits from a regularization-based schedule consistently appear, agnostic to the weight importance scoring criterion? This question is important because it will show if the gain from a better pruning schedule is only a bonus concurrent with the $L _ { 1 }$ criterion or a really universal phenomenon. Since there are literally so many weight importance criteria, we cannot ablate them one by one. Nevertheless, given a pre-trained model and a pruning ratio $r$ , no matter what criterion, its role is to select a filter subset. For example, if there are 100 filters in a layer and $r = 0 . 5$ , then they are at most $\\binom { 1 0 0 } { 5 0 }$ importance criteria in theory for this layer. We can simply randomly pick a subset of filters (which corresponds to certain criterion, albeit unknown) and compare the one-shot way with regularization-based way on the subset. Based on this idea, we conduct five random runs on the ResNet56 and VGG19 to explore this. The pruning ratio is chosen as $9 0 \\%$ for ResNet56 and $7 0 \\%$ for VGG19 because under this ratio the compression (or acceleration) ratio is about 10 times, neither too large nor too small (where the network can heal itself regardless of pruning methods). ", + "bbox": [ + 173, + 321, + 825, + 518 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The results are shown in Tab. 10. Here is a sanity check: Compared with Tab. 1, the mean accuracy of pruning randomly picked filters should be less than pruning those picked by $L _ { 1 }$ -norm, confirmed by $8 6 . 8 5 \\%$ vs. $8 7 . 3 4 \\%$ for ResNet56 and $6 5 . 0 4 \\%$ vs. $6 6 . 0 5 \\%$ for VGG19. As seen, in each run, the regularization-based way also significantly surpasses its one-shot counterpart. Although five random runs are still too few given the exploding potential combinations, yet as shown by the accuracy standard deviations, the results are stable and thus qualified to support our finding that the regularization-based pruning schedule is better to the one-shot counterpart. ", + "bbox": [ + 173, + 525, + 825, + 623 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/cc24bcea8291b721861a734d53a0fd1a3a1c92c0bb1b0e8cee80e42150e4117a.jpg", + "table_caption": [ + "Table 10: Comparison between pruning schedules: one-shot vs. GReg-1. Pruning ratio is $90 \\%$ for ResNet56 and $70 \\%$ for VGG19. In each run, the weights to prune are picked randomly before the training starts. " + ], + "table_footnote": [], + "table_body": "
ResNet56 + CIFAR10Run #1Run #2Run #3Run #4Run #5Mean±std
Acc. (%, one-shot)Acc. (%, GReg-1, ours)87.5789.2687.0088.9886.2788.7886.7589.4286.6788.9686.85±0.4389.08±0.23
VGG19 + CIFAR100Run #1Run #2Run #3Run #4Run #5Mean±std
Acc. (%,one-shot)Acc. (%, GReg-1, ours)64.5666.6365.0666.5765.0766.8065.0566.8065.4867.1665.04±0.2966.79±0.21
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However, its role is mainly explored in the small penalty strength regime.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 235, + 469, + 248 + ], + "spans": [ + { + "bbox": [ + 141, + 235, + 469, + 248 + ], + "score": 1.0, + "content": "In this work, we extend its application to a new scenario where the regularization", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 246, + 470, + 259 + ], + "spans": [ + { + "bbox": [ + 141, + 246, + 470, + 259 + ], + "score": 1.0, + "content": "grows large gradually to tackle two central problems of pruning: pruning sched-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 257, + 470, + 270 + ], + "spans": [ + { + "bbox": [ + 141, + 257, + 470, + 270 + ], + "score": 1.0, + "content": "ule and weight importance scoring. (1) The former topic is newly brought up in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 267, + 469, + 280 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 469, + 280 + ], + "score": 1.0, + "content": "this work, which we find critical to the pruning performance while receives lit-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 279, + 469, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 279, + 346, + 291 + ], + "score": 1.0, + "content": "tle research attention. 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Their effectiveness is demon-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 366, + 469, + 379 + ], + "spans": [ + { + "bbox": [ + 141, + 366, + 469, + 379 + ], + "score": 1.0, + "content": "strated with modern deep neural networks on the CIFAR and ImageNet datasets,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 142, + 378, + 469, + 390 + ], + "spans": [ + { + "bbox": [ + 142, + 378, + 469, + 390 + ], + "score": 1.0, + "content": "achieving competitive results compared to many state-of-the-art algorithms. 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(2015); Schmidhuber (2015), their", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "remarkable effectiveness comes at a cost of rising storage, memory footprint, computing resources", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 478, + 504, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 504, + 490 + ], + "score": 1.0, + "content": "and energy consumption Cheng et al. (2017); Deng et al. (2020). Neural network pruning Han et al.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "(2015; 2016); Li et al. (2017); Wen et al. (2016); He et al. (2017); Gale et al. 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(1990); Hassibi & Stork (1993);", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 533, + 388, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 388, + 545 + ], + "score": 1.0, + "content": "Molchanov et al. (2017b; 2019); Wang et al. (2019a); He et al. (2020).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "The approaches to the scoring problem generally fall into two groups: importance-based and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "regularization-based Reed (1993). The former focuses on directly proposing certain theoretically", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "sound importance criterion so that we can prune the unimportant weights once for all. Thus, the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "pruning process is typically one-shot. In contrast, regularization-based approaches typically select", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "score": 1.0, + "content": "unimportant weights through training with a penalty term Han et al. (2015); Wen et al. (2016); Liu", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 603, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 104, + 603, + 505, + 618 + ], + "score": 1.0, + "content": "et al. (2017). However, the penalty strength is usually maintained in a small regime to avoid dam-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "aging the model expressivity. Whereas, a large penalty strength can be helpful, specifically in two", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "aspects. (1) A large penalty can push unimportant weights rather close to zero, then the pruning", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "later barely hurts the performance even if the simple weight magnitude is adopted as criterion. (2)", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "It is well-known that different weights of a neural network lie on the regions with different local", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "quadratic structures, i.e., Hessian information. Many methods try to tap into this to build a more", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 670, + 504, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 504, + 682 + ], + "score": 1.0, + "content": "accurate scoring LeCun et al. (1990); Hassibi & Stork (1993); Wang et al. (2019a); Singh & Al-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "istarh (2020). However, for deep networks, it is especially hard to estimate Hessian. 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However, its role is mainly explored in the small penalty strength regime.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 235, + 469, + 248 + ], + "spans": [ + { + "bbox": [ + 141, + 235, + 469, + 248 + ], + "score": 1.0, + "content": "In this work, we extend its application to a new scenario where the regularization", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 246, + 470, + 259 + ], + "spans": [ + { + "bbox": [ + 141, + 246, + 470, + 259 + ], + "score": 1.0, + "content": "grows large gradually to tackle two central problems of pruning: pruning sched-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 257, + 470, + 270 + ], + "spans": [ + { + "bbox": [ + 141, + 257, + 470, + 270 + ], + "score": 1.0, + "content": "ule and weight importance scoring. 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Specifically, we propose an", + "type": "text" + }, + { + "bbox": [ + 346, + 279, + 359, + 290 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 279, + 469, + 291 + ], + "score": 1.0, + "content": "regularization variant with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 291, + 469, + 302 + ], + "spans": [ + { + "bbox": [ + 141, + 291, + 469, + 302 + ], + "score": 1.0, + "content": "rising penalty factors and show it can bring significant accuracy gains compared", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 301, + 469, + 313 + ], + "spans": [ + { + "bbox": [ + 142, + 301, + 469, + 313 + ], + "score": 1.0, + "content": "with its one-shot counterpart, even when the same weights are removed. 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Empirically, the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 345, + 469, + 357 + ], + "spans": [ + { + "bbox": [ + 141, + 345, + 469, + 357 + ], + "score": 1.0, + "content": "proposed algorithms are easy to implement and scalable to large datasets and net-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 356, + 469, + 368 + ], + "spans": [ + { + "bbox": [ + 142, + 356, + 469, + 368 + ], + "score": 1.0, + "content": "works in both structured and unstructured pruning. Their effectiveness is demon-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 366, + 469, + 379 + ], + "spans": [ + { + "bbox": [ + 141, + 366, + 469, + 379 + ], + "score": 1.0, + "content": "strated with modern deep neural networks on the CIFAR and ImageNet datasets,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 142, + 378, + 469, + 390 + ], + "spans": [ + { + "bbox": [ + 142, + 378, + 469, + 390 + ], + "score": 1.0, + "content": "achieving competitive results compared to many state-of-the-art algorithms. Our", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 388, + 469, + 401 + ], + "spans": [ + { + "bbox": [ + 141, + 388, + 469, + 401 + ], + "score": 1.0, + "content": "code and trained models are publicly available at https://github.com/mingsun-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 399, + 250, + 412 + ], + "spans": [ + { + "bbox": [ + 141, + 399, + 250, + 412 + ], + "score": 1.0, + "content": "tse/regularization-pruning.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 14.5, + "bbox_fs": [ + 141, + 214, + 470, + 412 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 431, + 206, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 208, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 208, + 447 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "As deep neural networks advance in recent years LeCun et al. (2015); Schmidhuber (2015), their", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "remarkable effectiveness comes at a cost of rising storage, memory footprint, computing resources", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 478, + 504, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 504, + 490 + ], + "score": 1.0, + "content": "and energy consumption Cheng et al. (2017); Deng et al. (2020). Neural network pruning Han et al.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "(2015; 2016); Li et al. (2017); Wen et al. (2016); He et al. (2017); Gale et al. (2019) is deemed as a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "promising force to alleviate this problem. Since its early debut Mozer & Smolensky (1989); Reed", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "(1993), the central problem of neural network pruning has been (arguably) how to choose weights", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "to discard, i.e., the weight importance scoring problem LeCun et al. (1990); Hassibi & Stork (1993);", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 533, + 388, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 388, + 545 + ], + "score": 1.0, + "content": "Molchanov et al. (2017b; 2019); Wang et al. (2019a); He et al. (2020).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 456, + 506, + 545 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "The approaches to the scoring problem generally fall into two groups: importance-based and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "regularization-based Reed (1993). The former focuses on directly proposing certain theoretically", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "sound importance criterion so that we can prune the unimportant weights once for all. Thus, the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "pruning process is typically one-shot. In contrast, regularization-based approaches typically select", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "score": 1.0, + "content": "unimportant weights through training with a penalty term Han et al. (2015); Wen et al. (2016); Liu", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 603, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 104, + 603, + 505, + 618 + ], + "score": 1.0, + "content": "et al. (2017). However, the penalty strength is usually maintained in a small regime to avoid dam-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "aging the model expressivity. Whereas, a large penalty strength can be helpful, specifically in two", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "aspects. (1) A large penalty can push unimportant weights rather close to zero, then the pruning", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "later barely hurts the performance even if the simple weight magnitude is adopted as criterion. (2)", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "It is well-known that different weights of a neural network lie on the regions with different local", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "quadratic structures, i.e., Hessian information. Many methods try to tap into this to build a more", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 670, + 504, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 504, + 682 + ], + "score": 1.0, + "content": "accurate scoring LeCun et al. (1990); Hassibi & Stork (1993); Wang et al. (2019a); Singh & Al-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "istarh (2020). However, for deep networks, it is especially hard to estimate Hessian. Sometimes,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 692, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 505, + 705 + ], + "score": 1.0, + "content": "even the computing itself can be intractable without resorting to proper approximation Wang et al.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 702, + 505, + 717 + ], + "spans": [ + { + "bbox": [ + 106, + 702, + 505, + 717 + ], + "score": 1.0, + "content": "(2019a). On this problem, we ask: Is it possible to exploit the Hessian information without knowing", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "their specific values? This is the second scenario where a growing regularization can help. We will", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "show under a growing regularization, the weight magnitude will naturally separate because of their", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "different underlying local quadratic structure, therein we can pick the unimportant weights more", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "faithfully even using the simple magnitude-based criterion. Corresponding to these two aspects,", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 316, + 140 + ], + "score": 1.0, + "content": "we will present two algorithms based on a growing", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 317, + 127, + 329, + 137 + ], + "score": 0.84, + "content": "L _ { 2 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 330, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "regularization paradigm, in which the first", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 458, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 458, + 150 + ], + "score": 1.0, + "content": "highlights a better pruning schedule1 and the second explores a better pruning criterion.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 40, + "bbox_fs": [ + 104, + 550, + 506, + 717 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "their specific values? This is the second scenario where a growing regularization can help. We will", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "show under a growing regularization, the weight magnitude will naturally separate because of their", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "different underlying local quadratic structure, therein we can pick the unimportant weights more", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "faithfully even using the simple magnitude-based criterion. Corresponding to these two aspects,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 316, + 140 + ], + "score": 1.0, + "content": "we will present two algorithms based on a growing", + "type": "text" + }, + { + "bbox": [ + 317, + 127, + 329, + 137 + ], + "score": 0.84, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "regularization paradigm, in which the first", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 458, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 458, + 150 + ], + "score": 1.0, + "content": "highlights a better pruning schedule1 and the second explores a better pruning criterion.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 506, + 168 + ], + "score": 1.0, + "content": "Our contributions. (1) We propose a simple yet effective growing regularization scheme, which", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 163, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 104, + 163, + 506, + 180 + ], + "score": 1.0, + "content": "can help transfer the model expressivity to the remaining part during pruning. The encouraging", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "performance inspires us that the pruning schedule may be as critical as the weight importance cri-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "terion and deserve more research attention. (2) We further adopt growing regularization to exploit", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "Hessian implicitly, without knowing their specific values. The method can help choose the unim-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 210, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 506, + 222 + ], + "score": 1.0, + "content": "portant weights more faithfully with a theoretically sound basis. In this regard, our paper is the first", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "score": 1.0, + "content": "to show the connection between magnitude-based pruning and Hessian-based pruning, pointing out", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "that the latter can be turned into the first one through our proposed growing regularization scheme.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "(3) The proposed two algorithms are easy to implement and scalable to large-scale datasets and net-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "score": 1.0, + "content": "works. We show their effectiveness compared with many state-of-the-arts. Especially, the methods", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 386, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 386, + 278 + ], + "score": 1.0, + "content": "can work seamlessly for both filter pruning and unstructured pruning.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 108, + 293, + 211, + 306 + ], + "lines": [ + { + "bbox": [ + 104, + 291, + 213, + 308 + ], + "spans": [ + { + "bbox": [ + 104, + 291, + 213, + 308 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 319, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 319, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 505, + 332 + ], + "score": 1.0, + "content": "Regularization-based pruning. The first group of relevant works is those applying regularization", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 320, + 343 + ], + "score": 1.0, + "content": "to learn sparsity. The most famous probably is to use", + "type": "text" + }, + { + "bbox": [ + 320, + 331, + 333, + 342 + ], + "score": 0.88, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 330, + 345, + 343 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 345, + 331, + 357, + 342 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "regularization Louizos et al. (2018);", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "score": 1.0, + "content": "Liu et al. (2017); Ye et al. (2018) due to their sparsity-inducing nature. In addition, the common", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 352, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 119, + 364 + ], + "score": 0.86, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 352, + 505, + 366 + ], + "score": 1.0, + "content": "regularization is also explored for approximated sparsity Han et al. (2015; 2016). The early", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 104, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "papers focus more on unstructured pruning, which is beneficial to model compression yet not to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 375, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 386 + ], + "score": 1.0, + "content": "acceleration. For structured pruning in favor of acceleration, Group-wise Brain Damage Lebedev &", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "Lempitsky (2016) and SSL Wen et al. (2016) propose to use Group LASSO Yuan & Lin (2006) to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "learn regular sparsity, where the penalty strength is still kept in small scale because the penalty is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "uniformly applied to all the weights. To resolve this, Ding et al. (2018) and Wang et al. (2019c)", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 418, + 487, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 487, + 430 + ], + "score": 1.0, + "content": "propose to employ different penalty factors for different weights, enabling large regularization.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "Importance-based pruning. Importance-based pruning tries to establish certain advanced impor-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "tance criteria that can reflect the true relative importance among weights as faithfully as possible.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 456, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 470 + ], + "score": 1.0, + "content": "The pruned weights are usually decided immediately by some proposed formula instead of by train-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "ing (although the whole pruning process can involve training, e.g., iterative pruning). The most", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "widely used criterion is the magnitude-based: weight absolute value for unstructured pruningHan", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 196, + 501 + ], + "score": 1.0, + "content": "et al. (2015; 2016) or", + "type": "text" + }, + { + "bbox": [ + 196, + 490, + 225, + 502 + ], + "score": 0.93, + "content": "L _ { 1 } / L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 491, + 505, + 501 + ], + "score": 1.0, + "content": "-norm for structured pruning Li et al. (2017). This heuristic criterion", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "was proposed a long time ago Reed (1993) and has been argued to be inaccurate. In this respect,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "improvement mainly comes from using Hessian information to obtain a more accurate approxima-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "tion of the increased loss when a weight is removed LeCun et al. (1990); Hassibi & Stork (1993).", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "Hessian is intractable to compute for large networks, so some methods (e.g., EigenDamage Wang", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "et al. (2019a), WoodFisher Singh & Alistarh (2020)) employ cheap approximation (such as K-FAC", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 554, + 475, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 475, + 568 + ], + "score": 1.0, + "content": "Fisher Martens & Grosse (2015)) to make the 2nd-order criteria tractable on deep networks.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "score": 1.0, + "content": "Note that, there is no a hard boundary between the importance-based and regularization-based.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 583, + 504, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 504, + 596 + ], + "score": 1.0, + "content": "Many papers present their schemes in the combination of the two Ding et al. (2018); Wang et al.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "(2019c). The difference mainly lies in their emphasis: Regularization-based method focuses more", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 505, + 618 + ], + "score": 1.0, + "content": "on an advanced penalty scheme so that the subsequent pruning criterion can be simple; while the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "importance-based one focus more on an advanced importance criterion itself. Meanwhile, regular-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 627, + 504, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 504, + 639 + ], + "score": 1.0, + "content": "ization paradigm always involves iterative training, while the importance-based can be one-shot Le-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 639, + 504, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 504, + 650 + ], + "score": 1.0, + "content": "Cun et al. (1990); Hassibi & Stork (1993); Wang et al. (2019a) (no training for picking weights to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 649, + 465, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 465, + 661 + ], + "score": 1.0, + "content": "prune) or involve iterative training Molchanov et al. (2017b; 2019); Ding et al. (2019a;b).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43.5 + }, + { + "type": "text", + "bbox": [ + 108, + 666, + 504, + 699 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "Other model compression methods. Apart from pruning, there are also many other model com-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "pression approaches, e.g., quantization Courbariaux & Bengio (2016); Courbariaux et al. (2016);", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Rastegari et al. (2016), knowledge distillation Bucilua et al. ˇ (2006); Hinton et al. (2014), low-", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 712, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 119, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "1By pruning schedule, we mean the way to remove weights (e.g., removing all weights in a single step or", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 720, + 368, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 368, + 733 + ], + "score": 1.0, + "content": "multi-steps), not the training schedule such as learning rate settings, etc.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 105, + 83, + 506, + 150 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 506, + 168 + ], + "score": 1.0, + "content": "Our contributions. (1) We propose a simple yet effective growing regularization scheme, which", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 163, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 104, + 163, + 506, + 180 + ], + "score": 1.0, + "content": "can help transfer the model expressivity to the remaining part during pruning. The encouraging", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "performance inspires us that the pruning schedule may be as critical as the weight importance cri-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "terion and deserve more research attention. (2) We further adopt growing regularization to exploit", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "Hessian implicitly, without knowing their specific values. The method can help choose the unim-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 210, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 506, + 222 + ], + "score": 1.0, + "content": "portant weights more faithfully with a theoretically sound basis. In this regard, our paper is the first", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 233 + ], + "score": 1.0, + "content": "to show the connection between magnitude-based pruning and Hessian-based pruning, pointing out", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "that the latter can be turned into the first one through our proposed growing regularization scheme.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "(3) The proposed two algorithms are easy to implement and scalable to large-scale datasets and net-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 266 + ], + "score": 1.0, + "content": "works. We show their effectiveness compared with many state-of-the-arts. Especially, the methods", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 386, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 386, + 278 + ], + "score": 1.0, + "content": "can work seamlessly for both filter pruning and unstructured pruning.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11, + "bbox_fs": [ + 104, + 154, + 506, + 278 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 293, + 211, + 306 + ], + "lines": [ + { + "bbox": [ + 104, + 291, + 213, + 308 + ], + "spans": [ + { + "bbox": [ + 104, + 291, + 213, + 308 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 319, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 319, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 505, + 332 + ], + "score": 1.0, + "content": "Regularization-based pruning. The first group of relevant works is those applying regularization", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 320, + 343 + ], + "score": 1.0, + "content": "to learn sparsity. The most famous probably is to use", + "type": "text" + }, + { + "bbox": [ + 320, + 331, + 333, + 342 + ], + "score": 0.88, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 330, + 345, + 343 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 345, + 331, + 357, + 342 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "regularization Louizos et al. (2018);", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "score": 1.0, + "content": "Liu et al. (2017); Ye et al. (2018) due to their sparsity-inducing nature. In addition, the common", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 352, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 119, + 364 + ], + "score": 0.86, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 352, + 505, + 366 + ], + "score": 1.0, + "content": "regularization is also explored for approximated sparsity Han et al. (2015; 2016). The early", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 104, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "papers focus more on unstructured pruning, which is beneficial to model compression yet not to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 375, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 505, + 386 + ], + "score": 1.0, + "content": "acceleration. For structured pruning in favor of acceleration, Group-wise Brain Damage Lebedev &", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "Lempitsky (2016) and SSL Wen et al. (2016) propose to use Group LASSO Yuan & Lin (2006) to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "learn regular sparsity, where the penalty strength is still kept in small scale because the penalty is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "uniformly applied to all the weights. To resolve this, Ding et al. (2018) and Wang et al. (2019c)", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 418, + 487, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 487, + 430 + ], + "score": 1.0, + "content": "propose to employ different penalty factors for different weights, enabling large regularization.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22.5, + "bbox_fs": [ + 104, + 319, + 506, + 430 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "Importance-based pruning. Importance-based pruning tries to establish certain advanced impor-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "tance criteria that can reflect the true relative importance among weights as faithfully as possible.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 456, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 470 + ], + "score": 1.0, + "content": "The pruned weights are usually decided immediately by some proposed formula instead of by train-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "ing (although the whole pruning process can involve training, e.g., iterative pruning). The most", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "widely used criterion is the magnitude-based: weight absolute value for unstructured pruningHan", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 196, + 501 + ], + "score": 1.0, + "content": "et al. (2015; 2016) or", + "type": "text" + }, + { + "bbox": [ + 196, + 490, + 225, + 502 + ], + "score": 0.93, + "content": "L _ { 1 } / L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 491, + 505, + 501 + ], + "score": 1.0, + "content": "-norm for structured pruning Li et al. (2017). This heuristic criterion", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "was proposed a long time ago Reed (1993) and has been argued to be inaccurate. In this respect,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "improvement mainly comes from using Hessian information to obtain a more accurate approxima-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "tion of the increased loss when a weight is removed LeCun et al. (1990); Hassibi & Stork (1993).", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "Hessian is intractable to compute for large networks, so some methods (e.g., EigenDamage Wang", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "et al. (2019a), WoodFisher Singh & Alistarh (2020)) employ cheap approximation (such as K-FAC", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 554, + 475, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 475, + 568 + ], + "score": 1.0, + "content": "Fisher Martens & Grosse (2015)) to make the 2nd-order criteria tractable on deep networks.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 435, + 505, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "score": 1.0, + "content": "Note that, there is no a hard boundary between the importance-based and regularization-based.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 583, + 504, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 504, + 596 + ], + "score": 1.0, + "content": "Many papers present their schemes in the combination of the two Ding et al. (2018); Wang et al.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "(2019c). The difference mainly lies in their emphasis: Regularization-based method focuses more", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 505, + 618 + ], + "score": 1.0, + "content": "on an advanced penalty scheme so that the subsequent pruning criterion can be simple; while the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "importance-based one focus more on an advanced importance criterion itself. Meanwhile, regular-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 627, + 504, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 504, + 639 + ], + "score": 1.0, + "content": "ization paradigm always involves iterative training, while the importance-based can be one-shot Le-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 639, + 504, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 504, + 650 + ], + "score": 1.0, + "content": "Cun et al. (1990); Hassibi & Stork (1993); Wang et al. (2019a) (no training for picking weights to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 649, + 465, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 465, + 661 + ], + "score": 1.0, + "content": "prune) or involve iterative training Molchanov et al. (2017b; 2019); Ding et al. (2019a;b).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 572, + 505, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 666, + 504, + 699 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "Other model compression methods. Apart from pruning, there are also many other model com-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "pression approaches, e.g., quantization Courbariaux & Bengio (2016); Courbariaux et al. (2016);", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Rastegari et al. (2016), knowledge distillation Bucilua et al. ˇ (2006); Hinton et al. (2014), low-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "rank decomposition Denton et al. (2014); Jaderberg et al. (2014); Lebedev et al. (2014); Zhang", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "et al. (2015), and efficient architecture design or search Howard et al. (2017); Sandler et al. (2018);", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 117 + ], + "score": 1.0, + "content": "Howard et al. (2019); Zhang et al. (2018); Tan & Le (2019); Zoph & Le (2017); Elsken et al. 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The difference lies in how we define the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "weight group: For filter pruning, a 4-d tensor convolutional filter (or 2-d tensor for fully-connected", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 378, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 394 + ], + "score": 1.0, + "content": "layers) is regarded as a weight group, while for unstructured pruning, a single weight makes a group.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 336, + 506, + 394 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 403, + 261, + 415 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 263, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 263, + 416 + ], + "score": 1.0, + "content": "3.2 PRUNING SCHEDULE: GREG-1", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 424, + 505, + 480 + ], + "lines": [ + { + "bbox": [ + 106, + 424, + 504, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 273, + 436 + ], + "score": 1.0, + "content": "Our first method (GReg-1) is a variant of", + "type": "text" + }, + { + "bbox": [ + 274, + 425, + 286, + 435 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 424, + 492, + 436 + ], + "score": 1.0, + "content": "-norm pruning Li et al. 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This", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 527, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 542 + ], + "score": 1.0, + "content": "algorithm is to explore whether the way we remove them (i.e., pruning schedule) leads to a difference", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "given the same weights to prune. Simple as it is, the scheme can bring significant accuracy gains", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 425, + 563 + ], + "score": 1.0, + "content": "especially under a large pruning ratio (Tab. 1). 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The training of neural networks is prone to overfitting, so regulariza-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 216, + 701 + ], + "score": 1.0, + "content": "tion is normally employed.", + "type": "text" + }, + { + "bbox": [ + 216, + 689, + 228, + 700 + ], + "score": 0.89, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "regularization (or referred to as weight decay) is a standard technique", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 497, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 277, + 713 + ], + "score": 1.0, + "content": "for deep network training. 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Eq. (4) shows that, for each specific weight", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "element, its equilibrium position is determined by two forces: loss gradient (i.e., guidance from the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "task) and regularization gradient (i.e., guidance from our prior). Our idea is to slightly increase the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 158, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 107, + 160, + 114, + 169 + ], + "score": 0.8, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 158, + 469, + 172 + ], + "score": 1.0, + "content": "to break the equilibrium and see how it results in a new one. A general impression is: If", + "type": "text" + }, + { + "bbox": [ + 469, + 160, + 476, + 169 + ], + "score": 0.77, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 158, + 506, + 172 + ], + "score": 1.0, + "content": "goes a", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 169, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 184 + ], + "score": 1.0, + "content": "little higher, the penalty force will drive the weights further towards origin and it will not stop unless", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 180, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 433, + 195 + ], + "score": 1.0, + "content": "proper loss gradient comes to halt it and then a new equilibrium is reached at", + "type": "text" + }, + { + "bbox": [ + 433, + 182, + 446, + 193 + ], + "score": 0.9, + "content": "\\hat { w } _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 180, + 505, + 195 + ], + "score": 1.0, + "content": ". Considering", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 340, + 204 + ], + "score": 1.0, + "content": "different weights have different scales, we define a ratio", + "type": "text" + }, + { + "bbox": [ + 340, + 192, + 394, + 204 + ], + "score": 0.93, + "content": "r _ { i } = \\hat { w } _ { i } ^ { * } / w _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "to describe how much the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 203, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 466, + 215 + ], + "score": 1.0, + "content": "weight magnitude changes after increasing the penalty factor. Our interest lies in how the", + "type": "text" + }, + { + "bbox": [ + 466, + 204, + 475, + 214 + ], + "score": 0.85, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 203, + 505, + 215 + ], + "score": 1.0, + "content": "differs", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 214, + 406, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 406, + 226 + ], + "score": 1.0, + "content": "from one another and how it relates to the underlying Hessian information.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 230, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "score": 1.0, + "content": "Deep neural networks are well-known over-parameterized and highly non-convex. To obtain a fea-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "sible analysis, we adopt a local quadratic approximation of the loss function based on Taylor se-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "score": 1.0, + "content": "ries expansion Strang (1991) following common practices LeCun et al. (1990); Hassibi & Stork", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 412, + 277 + ], + "score": 1.0, + "content": "(1993); Wang et al. (2019a). Then when the model is converged, the error", + "type": "text" + }, + { + "bbox": [ + 412, + 264, + 420, + 274 + ], + "score": 0.79, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "can be described by", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 197, + 287 + ], + "score": 1.0, + "content": "the converged weights", + "type": "text" + }, + { + "bbox": [ + 198, + 275, + 212, + 285 + ], + "score": 0.79, + "content": "\\mathbf { w } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 274, + 352, + 287 + ], + "score": 1.0, + "content": "and the underlying Hessian matrix", + "type": "text" + }, + { + "bbox": [ + 352, + 275, + 363, + 285 + ], + "score": 0.55, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 274, + 386, + 287 + ], + "score": 1.0, + "content": "(note", + "type": "text" + }, + { + "bbox": [ + 387, + 275, + 397, + 285 + ], + "score": 0.43, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "is p.s.d. since the model is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 285, + 497, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 270, + 298 + ], + "score": 1.0, + "content": "converged). After increasing the penalty", + "type": "text" + }, + { + "bbox": [ + 270, + 286, + 277, + 295 + ], + "score": 0.83, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 285, + 290, + 298 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 291, + 286, + 302, + 296 + ], + "score": 0.85, + "content": "\\delta \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 285, + 497, + 298 + ], + "score": 1.0, + "content": ", the new converged weights can be proved to be", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 300, + 360, + 315 + ], + "lines": [ + { + "bbox": [ + 250, + 300, + 360, + 315 + ], + "spans": [ + { + "bbox": [ + 250, + 300, + 360, + 315 + ], + "score": 0.92, + "content": "\\hat { \\mathbf { w } } ^ { * } = ( \\mathbf { H } + \\delta \\lambda \\mathbf { I } ) ^ { - 1 } \\mathbf { H } \\mathbf { w } ^ { * } ,", + "type": "interline_equation", + "image_path": "ec0c9681aea3b66f1d6a5d0c4e5b60e3fd3e51a6428a3d01aaa644fb9fa1e46b.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 250, + 300, + 360, + 315 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 318, + 506, + 352 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 134, + 332 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 319, + 141, + 328 + ], + "score": 0.26, + "content": "\\mathbf { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 316, + 506, + 332 + ], + "score": 1.0, + "content": "stands for the identity matrix. Here we meet with the common problem of estimating", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "Hessian and its inverse, which are well-known to be intractable for deep neural networks. We", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 340, + 324, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 324, + 352 + ], + "score": 1.0, + "content": "explore two simplified cases to help us move forward.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 357, + 505, + 391 + ], + "lines": [ + { + "bbox": [ + 106, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 120, + 370 + ], + "score": 1.0, + "content": "(1)", + "type": "text" + }, + { + "bbox": [ + 120, + 357, + 131, + 367 + ], + "score": 0.36, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "is diagonal, which is a common simplification for Hessian LeCun et al. (1990), implying that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 304, + 381 + ], + "score": 1.0, + "content": "the weights are independent of each other. For", + "type": "text" + }, + { + "bbox": [ + 304, + 369, + 317, + 380 + ], + "score": 0.9, + "content": "w _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 367, + 414, + 381 + ], + "score": 1.0, + "content": "with second derivative", + "type": "text" + }, + { + "bbox": [ + 415, + 369, + 428, + 379 + ], + "score": 0.88, + "content": "h _ { i i }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 367, + 458, + 381 + ], + "score": 1.0, + "content": ". With", + "type": "text" + }, + { + "bbox": [ + 459, + 369, + 471, + 380 + ], + "score": 0.87, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "penalty", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 379, + 403, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 159, + 391 + ], + "score": 1.0, + "content": "increased by", + "type": "text" + }, + { + "bbox": [ + 159, + 379, + 171, + 389 + ], + "score": 0.54, + "content": "\\delta \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 379, + 207, + 390 + ], + "score": 0.71, + "content": "( \\delta \\lambda > 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 379, + 403, + 391 + ], + "score": 1.0, + "content": ", the new converged weights can be proved to be", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 394, + 403, + 421 + ], + "lines": [ + { + "bbox": [ + 208, + 394, + 403, + 421 + ], + "spans": [ + { + "bbox": [ + 208, + 394, + 403, + 421 + ], + "score": 0.95, + "content": "\\hat { w } _ { i } ^ { * } = \\frac { h _ { i i } } { h _ { i i } + \\delta \\lambda } w _ { i } ^ { * } , \\Rightarrow r _ { i } = \\frac { \\hat { w } _ { i } ^ { * } } { w _ { i } ^ { * } } = \\frac { 1 } { \\delta \\lambda / h _ { i i } + 1 } ,", + "type": "interline_equation", + "image_path": "5b7b4dc32cb48b6b11396ba6bd41736ce3f582516dda7d1df22f73ab7eeaa507.jpg" + } + ] + } + ], + "index": 24.5, + "virtual_lines": [ + { + "bbox": [ + 208, + 394, + 403, + 407.5 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 208, + 407.5, + 403, + 421.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 424, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 424, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 133, + 438 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 425, + 177, + 437 + ], + "score": 0.92, + "content": "r _ { i } \\in [ 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 424, + 201, + 438 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 202, + 425, + 235, + 436 + ], + "score": 0.91, + "content": "h _ { i i } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 424, + 253, + 438 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 253, + 425, + 285, + 435 + ], + "score": 0.91, + "content": "\\delta \\lambda > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 424, + 354, + 438 + ], + "score": 1.0, + "content": ". As seen, larger", + "type": "text" + }, + { + "bbox": [ + 354, + 425, + 367, + 436 + ], + "score": 0.88, + "content": "h _ { i i }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 424, + 438, + 438 + ], + "score": 1.0, + "content": "results in larger", + "type": "text" + }, + { + "bbox": [ + 439, + 426, + 448, + 436 + ], + "score": 0.84, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 424, + 505, + 438 + ], + "score": 1.0, + "content": "(closer to 1),", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 436, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 505, + 448 + ], + "score": 1.0, + "content": "meaning that the weight is relatively less moved towards the origin. Our second algorithm pri-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "marily builds upon this finding, which implies when we add a penalty perturbation to the converged", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 458, + 495, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 495, + 470 + ], + "score": 1.0, + "content": "network, the way that different weights respond can reflect their underlying Hessian information.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 474, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 212, + 488 + ], + "score": 1.0, + "content": "(2) In practice, we know", + "type": "text" + }, + { + "bbox": [ + 212, + 475, + 223, + 484 + ], + "score": 0.55, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 474, + 505, + 488 + ], + "score": 1.0, + "content": "is rarely diagonal. How the dependency among weights affects the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 484, + 511, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 161, + 521 + ], + "score": 1.0, + "content": "finding abovecase, namely,", + "type": "text" + }, + { + "bbox": [ + 162, + 497, + 373, + 514 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathbf { w } ^ { * } = \\binom { w _ { 1 } ^ { * } } { w _ { 2 } ^ { * } } , \\mathbf { H } = \\binom { h _ { 1 1 } h _ { 1 2 } } { h _ { 1 2 } h _ { 2 2 } } , \\hat { \\mathbf { H } } = \\binom { h _ { 1 1 } + \\delta \\lambda } { h _ { 1 2 } } _ { h _ { 2 2 } + \\delta \\lambda } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 484, + 421, + 521 + ], + "score": 1.0, + "content": "an in Eq. (5. The new c", + "type": "text" + }, + { + "bbox": [ + 434, + 484, + 511, + 521 + ], + "score": 1.0, + "content": "e explore the 2-derged weights can", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 513, + 433, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 421, + 525 + ], + "score": 1.0, + "content": "be analytically solved below, where the approximation equality is because that", + "type": "text" + }, + { + "bbox": [ + 422, + 513, + 433, + 523 + ], + "score": 0.83, + "content": "\\delta \\lambda", + "type": "inline_equation" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 116, + 527, + 484, + 554 + ], + "lines": [ + { + "bbox": [ + 116, + 527, + 484, + 554 + ], + "spans": [ + { + "bbox": [ + 116, + 527, + 484, + 554 + ], + "score": 0.91, + "content": "\\left\\{ \\hat { w } _ { 1 } ^ { * } \\right\\} = \\frac { 1 } { | \\hat { \\mathbf { H } } | } \\left\\{ \\begin{array} { l l } { ( h _ { 1 1 } h _ { 2 2 } + h _ { 1 1 } \\delta \\lambda - h _ { 1 2 } ^ { 2 } ) w _ { 1 } ^ { * } + \\delta \\lambda h _ { 1 2 } w _ { 2 } ^ { * } } \\\\ ( h _ { 1 1 } h _ { 2 2 } + h _ { 2 2 } \\delta \\lambda - h _ { 1 2 } ^ { 2 } ) w _ { 2 } ^ { * } + \\delta \\lambda h _ { 1 2 } w _ { 1 } ^ { * } \\right\\} \\approx \\frac { 1 } { | \\hat { \\mathbf { H } } | } \\left\\{ \\begin{array} { l l } { ( h _ { 1 1 } h _ { 2 2 } + h _ { 1 1 } \\delta \\lambda - h _ { 1 2 } ^ { 2 } ) w _ { 1 } ^ { * } } \\\\ { ( h _ { 1 1 } h _ { 2 2 } + h _ { 2 2 } \\delta \\lambda - h _ { 1 2 } ^ { 2 } ) w _ { 2 } ^ { * } + \\delta \\lambda h _ { 1 2 } w _ { 1 } ^ { * } } \\end{array} \\right\\} , \\end{array}", + "type": "interline_equation", + "image_path": "160c69c83de7c31885ca7bfc970c213f1558b16cc488e67cdb6564ab4671dcf1.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 116, + 527, + 484, + 536.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 116, + 536.0, + 484, + 545.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 116, + 545.0, + 484, + 554.0 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 558, + 447, + 582 + ], + "lines": [ + { + "bbox": [ + 164, + 558, + 447, + 582 + ], + "spans": [ + { + "bbox": [ + 164, + 558, + 447, + 582 + ], + "score": 0.92, + "content": "\\Rightarrow r _ { 1 } = \\frac { 1 } { \\left| \\hat { \\mathbf { H } } \\right| } ( h _ { 1 1 } h _ { 2 2 } + h _ { 1 1 } \\delta \\lambda - h _ { 1 2 } ^ { 2 } ) , r _ { 2 } = \\frac { 1 } { \\left| \\hat { \\mathbf { H } } \\right| } ( h _ { 1 1 } h _ { 2 2 } + h _ { 2 2 } \\delta \\lambda - h _ { 1 2 } ^ { 2 } ) .", + "type": "interline_equation", + "image_path": "7e40e02521af9e22442c79b2bb12c69a4745526b9c968abf89e322d4dcacd3c5.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 164, + 558, + 447, + 582 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 581, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 144, + 595 + ], + "score": 1.0, + "content": "As seen,", + "type": "text" + }, + { + "bbox": [ + 144, + 583, + 188, + 594 + ], + "score": 0.92, + "content": "h _ { 1 1 } > h _ { 2 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 581, + 242, + 595 + ], + "score": 1.0, + "content": "also leads to", + "type": "text" + }, + { + "bbox": [ + 243, + 584, + 276, + 594 + ], + "score": 0.9, + "content": "r _ { 1 } > r _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 581, + 505, + 595 + ], + "score": 1.0, + "content": ", in line with the finding above. The existence of weight", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 593, + 502, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 191, + 606 + ], + "score": 1.0, + "content": "dependency (i.e., the", + "type": "text" + }, + { + "bbox": [ + 191, + 594, + 207, + 605 + ], + "score": 0.87, + "content": "h _ { 1 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 593, + 502, + 606 + ], + "score": 1.0, + "content": ") actually does not affect the conclusion since it is included in both ratios.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 506, + 699 + ], + "lines": [ + { + "bbox": [ + 106, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "These theoretical analyses show us that when the penalty is increased at the same pace, because of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "score": 1.0, + "content": "different local curvature structures, the weights actually respond differently – weights with larger", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 104, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "curvature will be less moved. As such, the magnitude discrepancy among weights will be magnified", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 644, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 117, + 655 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 644, + 125, + 654 + ], + "score": 0.73, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 644, + 505, + 655 + ], + "score": 1.0, + "content": "grows. Ultimately, the weights will naturally separate (see Fig. 1 for an empirical validation).", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 653, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 335, + 668 + ], + "score": 1.0, + "content": "When the discrepancy is large enough, even the simple", + "type": "text" + }, + { + "bbox": [ + 336, + 655, + 348, + 666 + ], + "score": 0.89, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 653, + 505, + 668 + ], + "score": 1.0, + "content": "-norm can make an accurate criterion.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 664, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 104, + 664, + 376, + 678 + ], + "score": 1.0, + "content": "Notably, the whole process happens itself with the uniformly rising", + "type": "text" + }, + { + "bbox": [ + 376, + 666, + 388, + 677 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 664, + 506, + 678 + ], + "score": 1.0, + "content": "penalty, no need to know the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "Hessian values, thus not bothered by any issue arising from Hessian approximation in relevant prior", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 687, + 489, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 489, + 699 + ], + "score": 1.0, + "content": "arts LeCun et al. (1990); Hassibi & Stork (1993); Wang et al. 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Eq. (4) shows that, for each specific weight", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "element, its equilibrium position is determined by two forces: loss gradient (i.e., guidance from the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "task) and regularization gradient (i.e., guidance from our prior). Our idea is to slightly increase the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 158, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 107, + 160, + 114, + 169 + ], + "score": 0.8, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 158, + 469, + 172 + ], + "score": 1.0, + "content": "to break the equilibrium and see how it results in a new one. 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Considering", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 340, + 204 + ], + "score": 1.0, + "content": "different weights have different scales, we define a ratio", + "type": "text" + }, + { + "bbox": [ + 340, + 192, + 394, + 204 + ], + "score": 0.93, + "content": "r _ { i } = \\hat { w } _ { i } ^ { * } / w _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "to describe how much the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 203, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 466, + 215 + ], + "score": 1.0, + "content": "weight magnitude changes after increasing the penalty factor. Our interest lies in how the", + "type": "text" + }, + { + "bbox": [ + 466, + 204, + 475, + 214 + ], + "score": 0.85, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 203, + 505, + 215 + ], + "score": 1.0, + "content": "differs", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 214, + 406, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 406, + 226 + ], + "score": 1.0, + "content": "from one another and how it relates to the underlying Hessian information.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 127, + 506, + 226 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 230, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "score": 1.0, + "content": "Deep neural networks are well-known over-parameterized and highly non-convex. To obtain a fea-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "sible analysis, we adopt a local quadratic approximation of the loss function based on Taylor se-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "score": 1.0, + "content": "ries expansion Strang (1991) following common practices LeCun et al. (1990); Hassibi & Stork", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 412, + 277 + ], + "score": 1.0, + "content": "(1993); Wang et al. (2019a). Then when the model is converged, the error", + "type": "text" + }, + { + "bbox": [ + 412, + 264, + 420, + 274 + ], + "score": 0.79, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "can be described by", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 197, + 287 + ], + "score": 1.0, + "content": "the converged weights", + "type": "text" + }, + { + "bbox": [ + 198, + 275, + 212, + 285 + ], + "score": 0.79, + "content": "\\mathbf { w } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 274, + 352, + 287 + ], + "score": 1.0, + "content": "and the underlying Hessian matrix", + "type": "text" + }, + { + "bbox": [ + 352, + 275, + 363, + 285 + ], + "score": 0.55, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 274, + 386, + 287 + ], + "score": 1.0, + "content": "(note", + "type": "text" + }, + { + "bbox": [ + 387, + 275, + 397, + 285 + ], + "score": 0.43, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "is p.s.d. since the model is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 285, + 497, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 270, + 298 + ], + "score": 1.0, + "content": "converged). 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Here we meet with the common problem of estimating", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "Hessian and its inverse, which are well-known to be intractable for deep neural networks. We", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 340, + 324, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 324, + 352 + ], + "score": 1.0, + "content": "explore two simplified cases to help us move forward.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 316, + 506, + 352 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 357, + 505, + 391 + ], + "lines": [ + { + "bbox": [ + 106, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 120, + 370 + ], + "score": 1.0, + "content": "(1)", + "type": "text" + }, + { + "bbox": [ + 120, + 357, + 131, + 367 + ], + "score": 0.36, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "is diagonal, which is a common simplification for Hessian LeCun et al. (1990), implying that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 304, + 381 + ], + "score": 1.0, + "content": "the weights are independent of each other. For", + "type": "text" + }, + { + "bbox": [ + 304, + 369, + 317, + 380 + ], + "score": 0.9, + "content": "w _ { i } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 367, + 414, + 381 + ], + "score": 1.0, + "content": "with second derivative", + "type": "text" + }, + { + "bbox": [ + 415, + 369, + 428, + 379 + ], + "score": 0.88, + "content": "h _ { i i }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 367, + 458, + 381 + ], + "score": 1.0, + "content": ". 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As seen, larger", + "type": "text" + }, + { + "bbox": [ + 354, + 425, + 367, + 436 + ], + "score": 0.88, + "content": "h _ { i i }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 424, + 438, + 438 + ], + "score": 1.0, + "content": "results in larger", + "type": "text" + }, + { + "bbox": [ + 439, + 426, + 448, + 436 + ], + "score": 0.84, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 424, + 505, + 438 + ], + "score": 1.0, + "content": "(closer to 1),", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 436, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 505, + 448 + ], + "score": 1.0, + "content": "meaning that the weight is relatively less moved towards the origin. 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The existence of weight", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 593, + 502, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 191, + 606 + ], + "score": 1.0, + "content": "dependency (i.e., the", + "type": "text" + }, + { + "bbox": [ + 191, + 594, + 207, + 605 + ], + "score": 0.87, + "content": "h _ { 1 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 593, + 502, + 606 + ], + "score": 1.0, + "content": ") actually does not affect the conclusion since it is included in both ratios.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 581, + 505, + 606 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 506, + 699 + ], + "lines": [ + { + "bbox": [ + 106, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "These theoretical analyses show us that when the penalty is increased at the same pace, because of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "score": 1.0, + "content": "different local curvature structures, the weights actually respond differently – weights with larger", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 104, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "curvature will be less moved. 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After this, the procedures are similar to those in GReg-1:", + "type": "text" + }, + { + "bbox": [ + 413, + 318, + 420, + 328 + ], + "score": 0.52, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 318, + 506, + 330 + ], + "score": 1.0, + "content": "for the unimportant", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 329, + 504, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 504, + 342 + ], + "score": 1.0, + "content": "weights are further increased. One extra step is to bring back the kept weights to the normal mag-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "nitude. Although they are the “survivors” during the previous competition under a large penalty,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 352, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 363 + ], + "score": 1.0, + "content": "their expressivity are also hurt. To be exact, we adopt negative penalty factor for the kept weights", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 274, + 374 + ], + "score": 1.0, + "content": "to encourage them to recover. When the", + "type": "text" + }, + { + "bbox": [ + 275, + 362, + 282, + 372 + ], + "score": 0.74, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 362, + 473, + 374 + ], + "score": 1.0, + "content": "for unimportant weights reaches the threshold", + "type": "text" + }, + { + "bbox": [ + 473, + 363, + 480, + 372 + ], + "score": 0.7, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "(akin", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 291, + 385 + ], + "score": 1.0, + "content": "to that of GReg-1), the training is terminated.", + "type": "text" + }, + { + "bbox": [ + 291, + 373, + 304, + 384 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "-pruning is conducted and then fine-tune to regain", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 384, + 456, + 396 + ], + "spans": [ + { + "bbox": [ + 104, + 384, + 456, + 396 + ], + "score": 1.0, + "content": "accuracy. To this end, the proposed two algorithms can be summarized in Algorithm 1.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 401, + 505, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 414 + ], + "score": 1.0, + "content": "Pruning ratios. We employ pre-specified pruning ratios in this work to keep the core method neat", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "(see Appendix for more discussion). Exploring layer-wise sensitivity is out of the scope of this work,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 422, + 504, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 504, + 436 + ], + "score": 1.0, + "content": "but clearly any method that finds more proper pruning ratios can readily work with our approaches.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 439, + 503, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 454 + ], + "score": 1.0, + "content": "Discussion: differences from IncReg. Although our work shares a general spirit of growing reg-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "ularization with IncReg Wang et al. (2019c;b), our work is actually starkly different from theirs in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 463, + 154, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 154, + 474 + ], + "score": 1.0, + "content": "many axes:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 132, + 482, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 132, + 480, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 132, + 480, + 506, + 495 + ], + "score": 1.0, + "content": "• Motivation. The motivations for using the growing regularization are different. Wang", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 491, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 141, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "et al. (2019c;b) adopt growing regularization to select the unimportant weights by train-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 141, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "ing. Namely, they focus on the importance criterion problem. In contrast, we use growing", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 141, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "regularization to investigate the pruning schedule problem (for GReg-1) or exploit the un-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 142, + 525, + 501, + 538 + ], + "spans": [ + { + "bbox": [ + 142, + 525, + 459, + 538 + ], + "score": 1.0, + "content": "derlying Hessian information (for GReg-2). The importance criterion is simply", + "type": "text" + }, + { + "bbox": [ + 460, + 526, + 472, + 537 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 525, + 501, + 538 + ], + "score": 1.0, + "content": "-norm.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 138, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 138, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "Algorithm design. Wang et al. (2019c;b) assign different regularization factors to different", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 142, + 551, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 142, + 551, + 505, + 563 + ], + "score": 1.0, + "content": "weight groups based on their relative importance, while we assign them with the same fac-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 142, + 562, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 142, + 562, + 505, + 573 + ], + "score": 1.0, + "content": "tors. For GReg-1, this may not be a substantial difference, while for GReg-2, the difference", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 142, + 573, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 142, + 573, + 505, + 584 + ], + "score": 1.0, + "content": "is fundamental because the theoretical analysis of GReg-2 (Sec. 3.3) relies on the fact that", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 141, + 583, + 388, + 596 + ], + "spans": [ + { + "bbox": [ + 141, + 583, + 388, + 596 + ], + "score": 1.0, + "content": "regularization factors are kept the same for different weights.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 139, + 597, + 504, + 611 + ], + "spans": [ + { + "bbox": [ + 139, + 597, + 504, + 611 + ], + "score": 1.0, + "content": "Theoretical analysis. The algorithm in Wang et al. (2019c;b) is generally heuristic-based,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 141, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 141, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "while our work provides rigorous theoretical analyses (Sec. 3.3) to support the proposed", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 142, + 621, + 219, + 632 + ], + "spans": [ + { + "bbox": [ + 142, + 621, + 219, + 632 + ], + "score": 1.0, + "content": "algorithm GReg-2.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 135, + 633, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 135, + 633, + 505, + 647 + ], + "score": 1.0, + "content": "• Empirical performance. Both our methods are significantly better than Wang et al.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 142, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 142, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "(2019c;b) on the large-scale ImageNet dataset, which will be shown in the experiment", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 141, + 655, + 209, + 669 + ], + "spans": [ + { + "bbox": [ + 141, + 655, + 209, + 669 + ], + "score": 1.0, + "content": "section (Tab. 3).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 446, + 689 + ], + "score": 1.0, + "content": "Discussion: other regularization forms. The proposed methods in this work adopts", + "type": "text" + }, + { + "bbox": [ + 446, + 677, + 459, + 688 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "regulariza-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 474, + 700 + ], + "score": 1.0, + "content": "tion. 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When the", + "type": "text" + }, + { + "bbox": [ + 275, + 362, + 282, + 372 + ], + "score": 0.74, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 362, + 473, + 374 + ], + "score": 1.0, + "content": "for unimportant weights reaches the threshold", + "type": "text" + }, + { + "bbox": [ + 473, + 363, + 480, + 372 + ], + "score": 0.7, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "(akin", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 291, + 385 + ], + "score": 1.0, + "content": "to that of GReg-1), the training is terminated.", + "type": "text" + }, + { + "bbox": [ + 291, + 373, + 304, + 384 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "-pruning is conducted and then fine-tune to regain", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 384, + 456, + 396 + ], + "spans": [ + { + "bbox": [ + 104, + 384, + 456, + 396 + ], + "score": 1.0, + "content": "accuracy. To this end, the proposed two algorithms can be summarized in Algorithm 1.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 306, + 506, + 396 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 401, + 505, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 414 + ], + "score": 1.0, + "content": "Pruning ratios. We employ pre-specified pruning ratios in this work to keep the core method neat", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "(see Appendix for more discussion). Exploring layer-wise sensitivity is out of the scope of this work,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 422, + 504, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 504, + 436 + ], + "score": 1.0, + "content": "but clearly any method that finds more proper pruning ratios can readily work with our approaches.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 399, + 506, + 436 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 439, + 503, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 454 + ], + "score": 1.0, + "content": "Discussion: differences from IncReg. Although our work shares a general spirit of growing reg-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "ularization with IncReg Wang et al. (2019c;b), our work is actually starkly different from theirs in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 463, + 154, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 154, + 474 + ], + "score": 1.0, + "content": "many axes:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 437, + 505, + 474 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 482, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 132, + 480, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 132, + 480, + 506, + 495 + ], + "score": 1.0, + "content": "• Motivation. The motivations for using the growing regularization are different. Wang", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 491, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 141, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "et al. (2019c;b) adopt growing regularization to select the unimportant weights by train-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 141, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "ing. Namely, they focus on the importance criterion problem. In contrast, we use growing", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 141, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "regularization to investigate the pruning schedule problem (for GReg-1) or exploit the un-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 142, + 525, + 501, + 538 + ], + "spans": [ + { + "bbox": [ + 142, + 525, + 459, + 538 + ], + "score": 1.0, + "content": "derlying Hessian information (for GReg-2). The importance criterion is simply", + "type": "text" + }, + { + "bbox": [ + 460, + 526, + 472, + 537 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 525, + 501, + 538 + ], + "score": 1.0, + "content": "-norm.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 138, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 138, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "Algorithm design. Wang et al. (2019c;b) assign different regularization factors to different", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 551, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 142, + 551, + 505, + 563 + ], + "score": 1.0, + "content": "weight groups based on their relative importance, while we assign them with the same fac-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 142, + 562, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 142, + 562, + 505, + 573 + ], + "score": 1.0, + "content": "tors. For GReg-1, this may not be a substantial difference, while for GReg-2, the difference", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 142, + 573, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 142, + 573, + 505, + 584 + ], + "score": 1.0, + "content": "is fundamental because the theoretical analysis of GReg-2 (Sec. 3.3) relies on the fact that", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 141, + 583, + 388, + 596 + ], + "spans": [ + { + "bbox": [ + 141, + 583, + 388, + 596 + ], + "score": 1.0, + "content": "regularization factors are kept the same for different weights.", + "type": "text" + } + ], + "index": 41, + "is_list_end_line": true + }, + { + "bbox": [ + 139, + 597, + 504, + 611 + ], + "spans": [ + { + "bbox": [ + 139, + 597, + 504, + 611 + ], + "score": 1.0, + "content": "Theoretical analysis. The algorithm in Wang et al. (2019c;b) is generally heuristic-based,", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 141, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "while our work provides rigorous theoretical analyses (Sec. 3.3) to support the proposed", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 142, + 621, + 219, + 632 + ], + "spans": [ + { + "bbox": [ + 142, + 621, + 219, + 632 + ], + "score": 1.0, + "content": "algorithm GReg-2.", + "type": "text" + } + ], + "index": 44, + "is_list_end_line": true + }, + { + "bbox": [ + 135, + 633, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 135, + 633, + 505, + 647 + ], + "score": 1.0, + "content": "• Empirical performance. Both our methods are significantly better than Wang et al.", + "type": "text" + } + ], + "index": 45, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 142, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "(2019c;b) on the large-scale ImageNet dataset, which will be shown in the experiment", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 141, + 655, + 209, + 669 + ], + "spans": [ + { + "bbox": [ + 141, + 655, + 209, + 669 + ], + "score": 1.0, + "content": "section (Tab. 3).", + "type": "text" + } + ], + "index": 47, + "is_list_end_line": true + } + ], + "index": 39.5, + "bbox_fs": [ + 132, + 480, + 506, + 669 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 446, + 689 + ], + "score": 1.0, + "content": "Discussion: other regularization forms. The proposed methods in this work adopts", + "type": "text" + }, + { + "bbox": [ + 446, + 677, + 459, + 688 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "regulariza-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 474, + 700 + ], + "score": 1.0, + "content": "tion. Here we discuss the possibility to generalize the method to other regularization forms (", + "type": "text" + }, + { + "bbox": [ + 474, + 688, + 487, + 698 + ], + "score": 0.64, + "content": "L 1", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 697, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 119, + 710 + ], + "score": 0.79, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 697, + 435, + 713 + ], + "score": 1.0, + "content": "). (1) For GReg-1, it can be easily generalized to other regularization forms like", + "type": "text" + }, + { + "bbox": [ + 436, + 699, + 448, + 710 + ], + "score": 0.87, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 697, + 505, + 713 + ], + "score": 1.0, + "content": ". For GReg-2,", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 433, + 722 + ], + "score": 1.0, + "content": "since the theoretical basis in Sec. 3.3 relies on the local quadratic approximation,", + "type": "text" + }, + { + "bbox": [ + 433, + 710, + 446, + 721 + ], + "score": 0.87, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "regularization", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 225, + 734 + ], + "score": 1.0, + "content": "meets this requirement while", + "type": "text" + }, + { + "bbox": [ + 225, + 721, + 237, + 732 + ], + "score": 0.89, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "does not. Therefore, GReg-2 cannot be (easily) generalized to the", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 50, + "bbox_fs": [ + 105, + 677, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 105, + 115, + 503, + 248 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 89, + 503, + 111 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 88, + 505, + 102 + ], + "spans": [ + { + "bbox": [ + 106, + 88, + 505, + 102 + ], + "score": 1.0, + "content": "Table 1: Comparison between pruning schedules: one-shot pruning vs. our proposed GReg-1. Each", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 100, + 387, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 387, + 113 + ], + "score": 1.0, + "content": "setting is randomly run for 3 times, mean and std accuracies reported.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 105, + 115, + 503, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 115, + 503, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 503, + 248 + ], + "score": 0.983, + "html": "
ResNet56 + CIFAR10: Baseline accuracy 93.36%, #Params: 0.8530M,FLOPs: 0.1255G
Pruning ratio r (%)50709092.595
Sparsity (%)/ Speedup49.82/1.99×70.57/3.59×90.39/11.41×93.43/14.76×95.19/19.31×
Acc.(%,L1+one-shot)92.97±0.1591.88±0.0987.34±0.2187.31±0.2882.79±0.22
Acc.(%,GReg-1,ours)93.06±0.0992.23±0.2189.49±0.2388.39±0.1585.97±0.16
Acc. gain (%)0.090.352.151.083.18
VGG19 + CIFAR100: Baseline accuracy 74.02%,#Params: 20.0812M,FLOPs: 0.3982G
Pruning ratio r (%)5060708090
Sparsity(%)/Speedup74.87/3.60×84.00/5.41×90.98/8.84×95.95/17.30×98.96/44.22×
Acc.(%,L1+one-shot)71.49±0.1470.27±0.1266.05±0.0461.59±0.0351.36±0.11
Acc.(%,GReg-1,ours)71.50±0.1270.33±0.1267.35±0.1563.55±0.2957.09±0.03
Acc. gain (%)0.010.061.301.965.73
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(2) For", + "type": "text" + }, + { + "bbox": [ + 327, + 261, + 339, + 271 + ], + "score": 0.87, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 260, + 504, + 272 + ], + "score": 1.0, + "content": "regularization, it is well-known NP-hard.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 271, + 469, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 274, + 284 + ], + "score": 1.0, + "content": "In practice, it is typically converted to the", + "type": "text" + }, + { + "bbox": [ + 274, + 271, + 287, + 282 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 271, + 469, + 284 + ], + "score": 1.0, + "content": "regularization case, which we just discussed.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 108, + 299, + 257, + 312 + ], + "lines": [ + { + "bbox": [ + 105, + 299, + 258, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 258, + 313 + ], + "score": 1.0, + "content": "4 EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 324, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 105, + 323, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 336 + ], + "score": 1.0, + "content": "Datasets and networks. We first conduct analyses on the CIFAR10/100 datasets Krizhevsky (2009)", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "with ResNet56 He et al. (2016)/VGG19 Simonyan & Zisserman (2015). Then we evaluate our", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "methods on the large-scale ImageNet dataset Deng et al. (2009) with ResNet34 and 50 He et al.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "(2016). For CIFAR datasets, we train our baseline models with accuracies comparable to those in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "the original papers. For ImageNet, we take the official PyTorch Paszke et al. (2019) pre-trained", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 378, + 375, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 375, + 392 + ], + "score": 1.0, + "content": "models2 as baseline to maintain comparability with other methods.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 410 + ], + "score": 1.0, + "content": "Training settings. To control the irrelevant factors as we can, for comparison methods that release", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 408, + 504, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 504, + 420 + ], + "score": 1.0, + "content": "their pruning ratios, we will adopt their ratios; otherwise, we will use our specified ones. We com-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 417, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 432 + ], + "score": 1.0, + "content": "pare the speedup (measured by FLOPs reduction) since we mainly target model acceleration rather", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 428, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 443 + ], + "score": 1.0, + "content": "than compression. Detailed training settings (e.g., hyper-parameters and layer pruning ratios) are", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 441, + 226, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 226, + 452 + ], + "score": 1.0, + "content": "summarized in the Appendix.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 108, + 465, + 298, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 300, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 300, + 479 + ], + "score": 1.0, + "content": "4.1 RESNET56/VGG19 ON CIFAR-10/100", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 486, + 505, + 629 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 498 + ], + "score": 1.0, + "content": "Pruning schedule: GReg-1. First, we explore the effect of different pruning schedules on the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "performance of pruning. Specifically, we conduct two sets of experiments for comparison: (1)", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 509, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 144, + 521 + ], + "score": 1.0, + "content": "prune by", + "type": "text" + }, + { + "bbox": [ + 144, + 509, + 156, + 519 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 509, + 377, + 521 + ], + "score": 1.0, + "content": "-norm sorting and fine-tune Li et al. 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For ResNet56, since it has the residual addition restriction, we", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 552, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 564 + ], + "score": 1.0, + "content": "only prune the first conv layer in a block as previous works do Li et al. (2017). 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Note that we do not intend to obtain the best performance here but systematically explore", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "the effect of different pruning schedules, so we employ relatively simple settings (e.g., the uniform", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "pruning ratios). For fair comparisons, the fine-tuning scheme (e.g., number of epochs, learning rate", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "schedule, etc.) is the same for different methods. Therefore, the key comparison here is to see which", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 617, + 341, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 341, + 631 + ], + "score": 1.0, + "content": "method can deliver a better base model before fine-tuning.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 635, + 505, + 712 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 648 + ], + "score": 1.0, + "content": "The results are shown in Tab. 1. We have the following observations: (1) On the whole, the proposed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 241, + 658 + ], + "score": 1.0, + "content": "GReg-1 consistently outperforms", + "type": "text" + }, + { + "bbox": [ + 241, + 646, + 253, + 657 + ], + "score": 0.78, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "+one-shot. It is important to reiterate that the two settings have", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 656, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 670 + ], + "score": 1.0, + "content": "exactly the same pruned weights, so the only difference is how they are removed. The accuracy", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "gaps show that apart from importance scoring, pruning schedule is also a critical factor. 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ResNet56 + CIFAR10: Baseline accuracy 93.36%, #Params: 0.8530M,FLOPs: 0.1255G
Pruning ratio r (%)50709092.595
Sparsity (%)/ Speedup49.82/1.99×70.57/3.59×90.39/11.41×93.43/14.76×95.19/19.31×
Acc.(%,L1+one-shot)92.97±0.1591.88±0.0987.34±0.2187.31±0.2882.79±0.22
Acc.(%,GReg-1,ours)93.06±0.0992.23±0.2189.49±0.2388.39±0.1585.97±0.16
Acc. gain (%)0.090.352.151.083.18
VGG19 + CIFAR100: Baseline accuracy 74.02%,#Params: 20.0812M,FLOPs: 0.3982G
Pruning ratio r (%)5060708090
Sparsity(%)/Speedup74.87/3.60×84.00/5.41×90.98/8.84×95.95/17.30×98.96/44.22×
Acc.(%,L1+one-shot)71.49±0.1470.27±0.1266.05±0.0461.59±0.0351.36±0.11
Acc.(%,GReg-1,ours)71.50±0.1270.33±0.1267.35±0.1563.55±0.2957.09±0.03
Acc. gain (%)0.010.061.301.965.73
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(2) For", + "type": "text" + }, + { + "bbox": [ + 327, + 261, + 339, + 271 + ], + "score": 0.87, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 260, + 504, + 272 + ], + "score": 1.0, + "content": "regularization, it is well-known NP-hard.", + "type": "text" + } + ], + "index": 5, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 271, + 469, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 274, + 284 + ], + "score": 1.0, + "content": "In practice, it is typically converted to the", + "type": "text" + }, + { + "bbox": [ + 274, + 271, + 287, + 282 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 271, + 469, + 284 + ], + "score": 1.0, + "content": "regularization case, which we just discussed.", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 260, + 504, + 284 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 299, + 257, + 312 + ], + "lines": [ + { + "bbox": [ + 105, + 299, + 258, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 258, + 313 + ], + "score": 1.0, + "content": "4 EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 324, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 105, + 323, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 336 + ], + "score": 1.0, + "content": "Datasets and networks. We first conduct analyses on the CIFAR10/100 datasets Krizhevsky (2009)", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "with ResNet56 He et al. (2016)/VGG19 Simonyan & Zisserman (2015). Then we evaluate our", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "methods on the large-scale ImageNet dataset Deng et al. (2009) with ResNet34 and 50 He et al.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "(2016). For CIFAR datasets, we train our baseline models with accuracies comparable to those in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "the original papers. For ImageNet, we take the official PyTorch Paszke et al. (2019) pre-trained", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 378, + 375, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 375, + 392 + ], + "score": 1.0, + "content": "models2 as baseline to maintain comparability with other methods.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 323, + 505, + 392 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 410 + ], + "score": 1.0, + "content": "Training settings. To control the irrelevant factors as we can, for comparison methods that release", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 408, + 504, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 504, + 420 + ], + "score": 1.0, + "content": "their pruning ratios, we will adopt their ratios; otherwise, we will use our specified ones. We com-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 417, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 432 + ], + "score": 1.0, + "content": "pare the speedup (measured by FLOPs reduction) since we mainly target model acceleration rather", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 428, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 443 + ], + "score": 1.0, + "content": "than compression. Detailed training settings (e.g., hyper-parameters and layer pruning ratios) are", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 441, + 226, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 226, + 452 + ], + "score": 1.0, + "content": "summarized in the Appendix.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 395, + 506, + 452 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 465, + 298, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 300, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 300, + 479 + ], + "score": 1.0, + "content": "4.1 RESNET56/VGG19 ON CIFAR-10/100", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 486, + 505, + 629 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 498 + ], + "score": 1.0, + "content": "Pruning schedule: GReg-1. First, we explore the effect of different pruning schedules on the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "performance of pruning. Specifically, we conduct two sets of experiments for comparison: (1)", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 509, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 144, + 521 + ], + "score": 1.0, + "content": "prune by", + "type": "text" + }, + { + "bbox": [ + 144, + 509, + 156, + 519 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 509, + 377, + 521 + ], + "score": 1.0, + "content": "-norm sorting and fine-tune Li et al. 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For ResNet56, since it has the residual addition restriction, we", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 552, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 564 + ], + "score": 1.0, + "content": "only prune the first conv layer in a block as previous works do Li et al. (2017). For comprehensive", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "comparisons, the pruning ratios vary in a large spectrum, covering acceleration ratios from around", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 573, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 120, + 585 + ], + "score": 0.86, + "content": "2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 573, + 131, + 587 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 131, + 574, + 150, + 585 + ], + "score": 0.88, + "content": "4 4 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 573, + 505, + 587 + ], + "score": 1.0, + "content": ". Note that we do not intend to obtain the best performance here but systematically explore", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "the effect of different pruning schedules, so we employ relatively simple settings (e.g., the uniform", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "pruning ratios). For fair comparisons, the fine-tuning scheme (e.g., number of epochs, learning rate", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "schedule, etc.) is the same for different methods. Therefore, the key comparison here is to see which", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 617, + 341, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 341, + 631 + ], + "score": 1.0, + "content": "method can deliver a better base model before fine-tuning.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 487, + 506, + 631 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 635, + 505, + 712 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 648 + ], + "score": 1.0, + "content": "The results are shown in Tab. 1. We have the following observations: (1) On the whole, the proposed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 241, + 658 + ], + "score": 1.0, + "content": "GReg-1 consistently outperforms", + "type": "text" + }, + { + "bbox": [ + 241, + 646, + 253, + 657 + ], + "score": 0.78, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "+one-shot. It is important to reiterate that the two settings have", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 656, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 670 + ], + "score": 1.0, + "content": "exactly the same pruned weights, so the only difference is how they are removed. The accuracy", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "gaps show that apart from importance scoring, pruning schedule is also a critical factor. In the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 679, + 504, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 504, + 690 + ], + "score": 1.0, + "content": "Appendix D, we present more results to demonstrate this finding actually is general, not merely", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 688, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 197, + 703 + ], + "score": 1.0, + "content": "limited to the case of", + "type": "text" + }, + { + "bbox": [ + 198, + 690, + 210, + 701 + ], + "score": 0.89, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 688, + 505, + 703 + ], + "score": 1.0, + "content": "-norm criterion. The proposed regularization-based pruning schedule is", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "consistently more favorable than the one-shot counterpart. (2) The larger pruning ratio, the more", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 451, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 462 + ], + "score": 1.0, + "content": "pronounced of the gain. This is reasonable since when more weights are pruned, the network cannot", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "recover by its inherent plasticity Mittal et al. (2018), then the regularization-based way is more", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "helpful because it helps the model transfer its expressive power to the remaining part. When the", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 311, + 496 + ], + "score": 1.0, + "content": "pruning ratio is relatively small (such as ResNet56,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 311, + 484, + 350, + 495 + ], + "score": 0.89, + "content": "r = 5 0 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 350, + 483, + 505, + 496 + ], + "score": 1.0, + "content": ") , the plasticity of the model is enough", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 493, + 484, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 484, + 508 + ], + "score": 1.0, + "content": "to heal, so the benefit from GReg-1 is less significant compared with the one-shot counterpart.", + "type": "text", + "cross_page": true + } + ], + "index": 13 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 633, + 506, + 713 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 81, + 502, + 233 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 81, + 502, + 233 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 81, + 502, + 233 + ], + "spans": [ + { + "bbox": [ + 108, + 81, + 502, + 233 + ], + "score": 0.974, + "type": "image", + "image_path": "ac428ab6cf517902459262b7ec3fb69d47cf04a2e15dd1400507d10a01bda824.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 81, + 502, + 131.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 131.66666666666666, + 502, + 182.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 182.33333333333331, + 502, + 232.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 104, + 241, + 505, + 264 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 323, + 254 + ], + "score": 1.0, + "content": "Figure 1: Row 1: Illustration of weight separation as", + "type": "text" + }, + { + "bbox": [ + 323, + 242, + 336, + 253 + ], + "score": 0.85, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "penalty grows. 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MethodNetwork/DatasetBase acc.(%) Pruned acc. (%) Acc. drop Speedup
CP He et al. (2017)ResNet56/CIFAR1092.8091.801.002.00×
AMC He et al. (2018b)92.8091.900.902.00×
SFP He et al. (2018a)93.5993.360.232.11×
AFP Ding et al. (2018)93.9392.940.992.56×
C-SGD Ding et al. (2019a)93.3993.44-0.052.55×
GReg-1 (ours)93.3693.180.182.55×
GReg-2 (ours)93.3693.360.002.55×
Kron-OBD Wang et al. (2019a)73.3460.7012.645.73×
Kron-OBS Wang et al. (2019a)73.3460.6612.686.09×
EigenDamage Wang et al. (2019a） VGG19/CIFAR10073.3465.188.168.80×
GReg-1 (ours)74.0267.556.678.84×
GReg-2 (ours)74.0267.756.478.84×
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(2018), then the regularization-based way is more", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "helpful because it helps the model transfer its expressive power to the remaining part. When the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 311, + 496 + ], + "score": 1.0, + "content": "pruning ratio is relatively small (such as ResNet56,", + "type": "text" + }, + { + "bbox": [ + 311, + 484, + 350, + 495 + ], + "score": 0.89, + "content": "r = 5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 483, + 505, + 496 + ], + "score": 1.0, + "content": ") , the plasticity of the model is enough", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 493, + 484, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 484, + 508 + ], + "score": 1.0, + "content": "to heal, so the benefit from GReg-1 is less significant compared with the one-shot counterpart.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 511, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "Importance criterion: GReg-2. Here we empirically validate our finding in Sec. 3.3, that is,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 196, + 535 + ], + "score": 1.0, + "content": "with uniformly rising", + "type": "text" + }, + { + "bbox": [ + 196, + 523, + 209, + 534 + ], + "score": 0.86, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 522, + 456, + 535 + ], + "score": 1.0, + "content": "penalty, the weights should naturally separate. We claim, if", + "type": "text" + }, + { + "bbox": [ + 456, + 523, + 501, + 534 + ], + "score": 0.91, + "content": "h _ { 1 1 } > h _ { 2 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 522, + 505, + 535 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 533, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 173, + 547 + ], + "score": 1.0, + "content": "there should be", + "type": "text" + }, + { + "bbox": [ + 173, + 536, + 210, + 546 + ], + "score": 0.9, + "content": "r _ { 1 } ~ > ~ r _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 534, + 243, + 547 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 244, + 533, + 327, + 549 + ], + "score": 0.93, + "content": "\\begin{array} { r } { r _ { 1 } = \\frac { \\tilde { \\hat { w _ { 1 } } } } { w _ { 1 } } , r _ { 2 } = \\frac { \\hat { w _ { 2 } } } { w _ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 533, + 506, + 547 + ], + "score": 1.0, + "content": "(the * mark indicating the local minimum", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 547, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 243, + 561 + ], + "score": 1.0, + "content": "is omitted here for readability).", + "type": "text" + }, + { + "bbox": [ + 243, + 550, + 281, + 560 + ], + "score": 0.89, + "content": "r _ { 1 } ~ > ~ r _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 547, + 320, + 561 + ], + "score": 1.0, + "content": "leads to", + "type": "text" + }, + { + "bbox": [ + 321, + 547, + 363, + 563 + ], + "score": 0.93, + "content": "\\frac { \\bar { w } _ { 1 } } { w _ { 1 } } ~ > ~ \\frac { \\bar { w } _ { 2 } } { w _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 547, + 365, + 565 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 367, + 548, + 405, + 562 + ], + "score": 1.0, + "content": "namely,", + "type": "text" + }, + { + "bbox": [ + 405, + 547, + 476, + 563 + ], + "score": 0.93, + "content": "\\begin{array} { r } { r _ { 1 } \\ = \\ \\frac { \\hat { w _ { 1 } } } { \\hat { w _ { 2 } } } \\ > \\ \\frac { w _ { 1 } } { w _ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 547, + 506, + 561 + ], + "score": 1.0, + "content": "This", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 193, + 574 + ], + "score": 1.0, + "content": "shows that, after the", + "type": "text" + }, + { + "bbox": [ + 193, + 561, + 206, + 572 + ], + "score": 0.87, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "penalty grows a little, the new magnitude ratio of weight 1 over weight", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 199, + 585 + ], + "score": 1.0, + "content": "2 will be magnified if", + "type": "text" + }, + { + "bbox": [ + 199, + 573, + 246, + 583 + ], + "score": 0.74, + "content": "h _ { 1 1 } > h _ { 2 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 573, + 280, + 583 + ], + "score": 0.51, + "content": "( w _ { 1 } , w _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "are positive in the analysis here, while the conclusion", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "still holds if either of them is negative). In Fig. 1 (Row 1), we plot the standard deviation (divided", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 451, + 606 + ], + "score": 1.0, + "content": "by the means for normalization since the magnitude varies over iterations) of filter", + "type": "text" + }, + { + "bbox": [ + 451, + 594, + 463, + 605 + ], + "score": 0.9, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "-norms as", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 605, + 504, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 317, + 617 + ], + "score": 1.0, + "content": "the regularization grows. As seen, the normalized", + "type": "text" + }, + { + "bbox": [ + 317, + 605, + 329, + 616 + ], + "score": 0.89, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 606, + 496, + 617 + ], + "score": 1.0, + "content": "-norm stddev grows larger and larger as", + "type": "text" + }, + { + "bbox": [ + 497, + 605, + 504, + 615 + ], + "score": 0.68, + "content": "\\lambda", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "grows. This phenomenon consistently appears across different models and datasets. To figuratively", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "understand how the increasing penalty affects the relative magnitude over time, in Fig. 1 (Row 2),", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 638, + 504, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 183, + 649 + ], + "score": 1.0, + "content": "we plot the relative", + "type": "text" + }, + { + "bbox": [ + 184, + 639, + 196, + 649 + ], + "score": 0.89, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 638, + 305, + 649 + ], + "score": 1.0, + "content": "-norms (divided by the max", + "type": "text" + }, + { + "bbox": [ + 306, + 639, + 318, + 649 + ], + "score": 0.89, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 638, + 504, + 649 + ], + "score": 1.0, + "content": "-norm for normalization) at different iterations.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "As shown, it is hard to tell which filters are really important by the initial filter magnitude (Iter 0),", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 658, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 104, + 658, + 506, + 673 + ], + "score": 1.0, + "content": "but under a large penalty later, their discrepancy turns more and more obvious and finally it is very", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "easy to identify which filters are more important. Since the magnitude gap is so large, the simple", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 681, + 309, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 119, + 693 + ], + "score": 0.87, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 681, + 309, + 694 + ], + "score": 1.0, + "content": "-norm can make a sufficiently faithful criterion.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "CIFAR benchmarks. Finally, we compare the proposed algorithms with existing methods on the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "CIFAR datasets (Tab. 2). Here we adopt non-uniform pruning ratios (see the Appendix for specific", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "numbers) for the best accuracy-FLOPs trade-off. On CIFAR10, compared with AMC He et al.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 81, + 502, + 233 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 81, + 502, + 233 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 81, + 502, + 233 + ], + "spans": [ + { + "bbox": [ + 108, + 81, + 502, + 233 + ], + "score": 0.974, + "type": "image", + "image_path": "ac428ab6cf517902459262b7ec3fb69d47cf04a2e15dd1400507d10a01bda824.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 81, + 502, + 131.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 131.66666666666666, + 502, + 182.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 182.33333333333331, + 502, + 232.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 104, + 241, + 505, + 264 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 323, + 254 + ], + "score": 1.0, + "content": "Figure 1: Row 1: Illustration of weight separation as", + "type": "text" + }, + { + "bbox": [ + 323, + 242, + 336, + 253 + ], + "score": 0.85, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "penalty grows. Row 2: Normalized filter", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 252, + 499, + 265 + ], + "spans": [ + { + "bbox": [ + 107, + 253, + 119, + 263 + ], + "score": 0.85, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 252, + 499, + 265 + ], + "score": 1.0, + "content": "-norm over iterations for ResNet50 layer2.3.conv1 (please see the Appendix for VGG19 plots).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "table", + "bbox": [ + 106, + 291, + 504, + 435 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 134, + 276, + 474, + 289 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 136, + 276, + 475, + 290 + ], + "spans": [ + { + "bbox": [ + 136, + 276, + 475, + 290 + ], + "score": 1.0, + "content": "Table 2: Comparison of different methods on the CIFAR10 and CIFAR100 datasets.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 291, + 504, + 435 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 291, + 504, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 504, + 435 + ], + "score": 0.98, + "html": "
MethodNetwork/DatasetBase acc.(%) Pruned acc. (%) Acc. drop Speedup
CP He et al. (2017)ResNet56/CIFAR1092.8091.801.002.00×
AMC He et al. (2018b)92.8091.900.902.00×
SFP He et al. (2018a)93.5993.360.232.11×
AFP Ding et al. (2018)93.9392.940.992.56×
C-SGD Ding et al. (2019a)93.3993.44-0.052.55×
GReg-1 (ours)93.3693.180.182.55×
GReg-2 (ours)93.3693.360.002.55×
Kron-OBD Wang et al. (2019a)73.3460.7012.645.73×
Kron-OBS Wang et al. (2019a)73.3460.6612.686.09×
EigenDamage Wang et al. (2019a） VGG19/CIFAR10073.3465.188.168.80×
GReg-1 (ours)74.0267.556.678.84×
GReg-2 (ours)74.0267.756.478.84×
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Here we empirically validate our finding in Sec. 3.3, that is,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 196, + 535 + ], + "score": 1.0, + "content": "with uniformly rising", + "type": "text" + }, + { + "bbox": [ + 196, + 523, + 209, + 534 + ], + "score": 0.86, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 522, + 456, + 535 + ], + "score": 1.0, + "content": "penalty, the weights should naturally separate. We claim, if", + "type": "text" + }, + { + "bbox": [ + 456, + 523, + 501, + 534 + ], + "score": 0.91, + "content": "h _ { 1 1 } > h _ { 2 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 522, + 505, + 535 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 533, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 173, + 547 + ], + "score": 1.0, + "content": "there should be", + "type": "text" + }, + { + "bbox": [ + 173, + 536, + 210, + 546 + ], + "score": 0.9, + "content": "r _ { 1 } ~ > ~ r _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 534, + 243, + 547 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 244, + 533, + 327, + 549 + ], + "score": 0.93, + "content": "\\begin{array} { r } { r _ { 1 } = \\frac { \\tilde { \\hat { w _ { 1 } } } } { w _ { 1 } } , r _ { 2 } = \\frac { \\hat { w _ { 2 } } } { w _ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 533, + 506, + 547 + ], + "score": 1.0, + "content": "(the * mark indicating the local minimum", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 547, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 243, + 561 + ], + "score": 1.0, + "content": "is omitted here for readability).", + "type": "text" + }, + { + "bbox": [ + 243, + 550, + 281, + 560 + ], + "score": 0.89, + "content": "r _ { 1 } ~ > ~ r _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 547, + 320, + 561 + ], + "score": 1.0, + "content": "leads to", + "type": "text" + }, + { + "bbox": [ + 321, + 547, + 363, + 563 + ], + "score": 0.93, + "content": "\\frac { \\bar { w } _ { 1 } } { w _ { 1 } } ~ > ~ \\frac { \\bar { w } _ { 2 } } { w _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 547, + 365, + 565 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 367, + 548, + 405, + 562 + ], + "score": 1.0, + "content": "namely,", + "type": "text" + }, + { + "bbox": [ + 405, + 547, + 476, + 563 + ], + "score": 0.93, + "content": "\\begin{array} { r } { r _ { 1 } \\ = \\ \\frac { \\hat { w _ { 1 } } } { \\hat { w _ { 2 } } } \\ > \\ \\frac { w _ { 1 } } { w _ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 547, + 506, + 561 + ], + "score": 1.0, + "content": "This", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 193, + 574 + ], + "score": 1.0, + "content": "shows that, after the", + "type": "text" + }, + { + "bbox": [ + 193, + 561, + 206, + 572 + ], + "score": 0.87, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "penalty grows a little, the new magnitude ratio of weight 1 over weight", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 199, + 585 + ], + "score": 1.0, + "content": "2 will be magnified if", + "type": "text" + }, + { + "bbox": [ + 199, + 573, + 246, + 583 + ], + "score": 0.74, + "content": "h _ { 1 1 } > h _ { 2 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 573, + 280, + 583 + ], + "score": 0.51, + "content": "( w _ { 1 } , w _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "are positive in the analysis here, while the conclusion", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "still holds if either of them is negative). In Fig. 1 (Row 1), we plot the standard deviation (divided", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 451, + 606 + ], + "score": 1.0, + "content": "by the means for normalization since the magnitude varies over iterations) of filter", + "type": "text" + }, + { + "bbox": [ + 451, + 594, + 463, + 605 + ], + "score": 0.9, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "-norms as", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 605, + 504, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 317, + 617 + ], + "score": 1.0, + "content": "the regularization grows. As seen, the normalized", + "type": "text" + }, + { + "bbox": [ + 317, + 605, + 329, + 616 + ], + "score": 0.89, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 606, + 496, + 617 + ], + "score": 1.0, + "content": "-norm stddev grows larger and larger as", + "type": "text" + }, + { + "bbox": [ + 497, + 605, + 504, + 615 + ], + "score": 0.68, + "content": "\\lambda", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "grows. This phenomenon consistently appears across different models and datasets. To figuratively", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "understand how the increasing penalty affects the relative magnitude over time, in Fig. 1 (Row 2),", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 638, + 504, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 183, + 649 + ], + "score": 1.0, + "content": "we plot the relative", + "type": "text" + }, + { + "bbox": [ + 184, + 639, + 196, + 649 + ], + "score": 0.89, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 638, + 305, + 649 + ], + "score": 1.0, + "content": "-norms (divided by the max", + "type": "text" + }, + { + "bbox": [ + 306, + 639, + 318, + 649 + ], + "score": 0.89, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 638, + 504, + 649 + ], + "score": 1.0, + "content": "-norm for normalization) at different iterations.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "As shown, it is hard to tell which filters are really important by the initial filter magnitude (Iter 0),", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 658, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 104, + 658, + 506, + 673 + ], + "score": 1.0, + "content": "but under a large penalty later, their discrepancy turns more and more obvious and finally it is very", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "easy to identify which filters are more important. 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Finally, we compare the proposed algorithms with existing methods on the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "CIFAR datasets (Tab. 2). Here we adopt non-uniform pruning ratios (see the Appendix for specific", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "numbers) for the best accuracy-FLOPs trade-off. On CIFAR10, compared with AMC He et al.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 698, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 105, + 501, + 397 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 113, + 89, + 493, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 88, + 495, + 102 + ], + "spans": [ + { + "bbox": [ + 116, + 88, + 495, + 102 + ], + "score": 1.0, + "content": "Table 3: Acceleration comparison on ImageNet. FLOPs: ResNet34: 3.66G, ResNet50: 4.09G.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 106, + 105, + 501, + 397 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 105, + 501, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 501, + 397 + ], + "score": 0.985, + "html": "
MethodNetworkBase top-1(%） Pruned top-1(%） Top-1 drop Speedup
L1 (pruned-B) Li et al. (2017) Taylor-FO Molchanov et al. (2019)ResNet3473.23 73.3172.17 72.831.06 0.481.32× 1.29×
GReg-1 (ours)73.3173.54-0.231.32×
GReg-2 (ours)73.3173.61-0.301.32×
ProvableFPLiebenwein et al. (2020) GReg-1 (ours)ResNet5076.13 76.1375.21 76.270.92 -0.141.43× 1.49×
AOFP Ding et al. (2019b) GReg-1 (ours)*ResNet5075.3475.63-0.291.49×
75.4076.13-0.731.49×
IncReg Wang et al. (2019b) SFP He et al. (2018a)75.60 76.1572.47 74.613.13 1.542.00× 1.72×
HRank Lin et al. (2020a) Taylor-FO Molchanov et al. (2019)ResNet5076.1574.981.171.78×
Factorized Li et al. (2019)76.18 76.1574.501.681.82×
74.551.602.33×
DCP Zhuang et al. (2018) CCP-AC Peng et al. (2019)76.0174.951.062.25×
76.1575.320.832.18×
GReg-1 (ours)76.1375.160.972.31×
GReg-2 (ours) C-SGD-50 Ding et al. (2019a)ResNet5076.1375.360.772.31×
75.3474.540.802.26×
AOFP Ding et al. (2019b)75.3475.110.232.31×
GReg-2 (ours)*ResNet5075.4075.220.182.31×
LFPC He et al. (2020)76.1574.461.692.55×
GReg-1 (ours)76.1374.851.282.56×
GReg-2 (ours)76.1374.931.202.56×
IncReg Wang et al. (2019b)75.6071.074.533.00×
Taylor-FO Molchanov et al. (2019)76.1871.693.05×
GReg-1 (ours)ResNet504.49
76.1373.752.383.06×
GReg-2 (ours)76.1373.902.233.06×
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MethodBase top-1 (%)Pruned top-1(%） Top-1 dropSparsity (%)
GSM Ding et al. (2019c)75.7274.301.4280.00
Variational Dropout Molchanov et al. (2O17a)76.6975.281.4180.00
DPF Lin et al. (2020b)75.9574.551.4082.60
WoodFisher Singh & Alistarh (2020)75.9875.200.7882.70
GReg-1 (ours)76.1375.450.6882.70
GReg-2 (ours)76.1375.270.8682.70
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FLOPs: ResNet34: 3.66G, ResNet50: 4.09G.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 106, + 105, + 501, + 397 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 105, + 501, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 501, + 397 + ], + "score": 0.985, + "html": "
MethodNetworkBase top-1(%） Pruned top-1(%） Top-1 drop Speedup
L1 (pruned-B) Li et al. (2017) Taylor-FO Molchanov et al. (2019)ResNet3473.23 73.3172.17 72.831.06 0.481.32× 1.29×
GReg-1 (ours)73.3173.54-0.231.32×
GReg-2 (ours)73.3173.61-0.301.32×
ProvableFPLiebenwein et al. (2020) GReg-1 (ours)ResNet5076.13 76.1375.21 76.270.92 -0.141.43× 1.49×
AOFP Ding et al. (2019b) GReg-1 (ours)*ResNet5075.3475.63-0.291.49×
75.4076.13-0.731.49×
IncReg Wang et al. (2019b) SFP He et al. (2018a)75.60 76.1572.47 74.613.13 1.542.00× 1.72×
HRank Lin et al. (2020a) Taylor-FO Molchanov et al. (2019)ResNet5076.1574.981.171.78×
Factorized Li et al. (2019)76.18 76.1574.501.681.82×
74.551.602.33×
DCP Zhuang et al. (2018) CCP-AC Peng et al. (2019)76.0174.951.062.25×
76.1575.320.832.18×
GReg-1 (ours)76.1375.160.972.31×
GReg-2 (ours) C-SGD-50 Ding et al. (2019a)ResNet5076.1375.360.772.31×
75.3474.540.802.26×
AOFP Ding et al. (2019b)75.3475.110.232.31×
GReg-2 (ours)*ResNet5075.4075.220.182.31×
LFPC He et al. (2020)76.1574.461.692.55×
GReg-1 (ours)76.1374.851.282.56×
GReg-2 (ours)76.1374.931.202.56×
IncReg Wang et al. (2019b)75.6071.074.533.00×
Taylor-FO Molchanov et al. (2019)76.1871.693.05×
GReg-1 (ours)ResNet504.49
76.1373.752.383.06×
GReg-2 (ours)76.1373.902.233.06×
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MethodBase top-1 (%)Pruned top-1(%） Top-1 dropSparsity (%)
GSM Ding et al. (2019c)75.7274.301.4280.00
Variational Dropout Molchanov et al. (2O17a)76.6975.281.4180.00
DPF Lin et al. (2020b)75.9574.551.4082.60
WoodFisher Singh & Alistarh (2020)75.9875.200.7882.70
GReg-1 (ours)76.1375.450.6882.70
GReg-2 (ours)76.1375.270.8682.70
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(2019b) for ResNet50) can even improve the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "top-1 accuracy. This phenomenon is broadly found by previous works Wen et al. (2016); Wang et al.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 164, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 505, + 177 + ], + "score": 1.0, + "content": "(2018); He et al. (2017) but mainly on small datasets like CIFAR, while we make it on the much", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "score": 1.0, + "content": "challenging ImageNet benchmark. (2) Similar to the results on CIFAR (Tab. 1), when the speedup", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "is larger, the advantage of our method is more obvious. For example, ours GReg-2 only outperforms", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 264, + 211 + ], + "score": 1.0, + "content": "Taylor-FO Molchanov et al. 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(3) Many methods work on the weight importance crite-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "rion problem, including some very recent ones (ProvableFP Liebenwein et al. (2020), LFPC He et al.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 291, + 244 + ], + "score": 1.0, + "content": "(2020)). Yet as shown, our simple variant of", + "type": "text" + }, + { + "bbox": [ + 292, + 231, + 304, + 242 + ], + "score": 0.9, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "-norm pruning can still be a strong competitor in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 240, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 255 + ], + "score": 1.0, + "content": "terms of accuracy-FLOPs trade-off. This reiterates one of our key ideas in this work that the pruning", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 488, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 488, + 265 + ], + "score": 1.0, + "content": "schedule may be as important as weight importance scoring and worth more research attention.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 505, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "Unstructured pruning. Although we mainly target filter pruning in this work, the proposed meth-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "ods actually can be applied to unstructured pruning as effectively. In Tab. 4, we present the results", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "of unstructured pruning on ResNet50. WoodFisher Singh & Alistarh (2020) is the state-of-the-art", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "score": 1.0, + "content": "Hessian-based unstructured pruning approach. 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In this work, we present two algorithms that exploit regulariza-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "tion in a new fashion that the penalty factor is uniformly raised to a large level. Two central problems", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "regarding deep neural pruning are tackled by the proposed methods, pruning schedule and weight", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "importance criterion. The proposed approaches rely on few impractical assumptions, have a sound", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "theoretical basis, and are scalable to large datasets and networks. Apart from the methodology it-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "self, the encouraging results on CIFAR and ImageNet also justify our general ideas in this paper: (1)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "In addition to weight importance scoring, pruning schedule is another pivotal factor in deep neural", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "pruning which may deserve more research attention. (2) Without any Hessian approximation, we", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 489, + 434, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 360, + 501 + ], + "score": 1.0, + "content": "can still tap into its power for pruning with the help of growing", + "type": "text" + }, + { + "bbox": [ + 360, + 489, + 373, + 500 + ], + "score": 0.86, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 489, + 434, + 501 + ], + "score": 1.0, + "content": "regularization.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 108, + 518, + 224, + 530 + ], + "lines": [ + { + "bbox": [ + 107, + 517, + 226, + 533 + ], + "spans": [ + { + "bbox": [ + 107, + 517, + 226, + 533 + ], + "score": 1.0, + "content": "ACKNOWLEDGEMENTS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 543, + 504, + 565 + ], + "lines": [ + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "The work is supported by the National Science Foundation Award ECCS-1916839 and the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 554, + 334, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 334, + 565 + ], + "score": 1.0, + "content": "U.S. Army Research Office Award W911NF-17-1-0367.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "title", + "bbox": [ + 108, + 583, + 175, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 176, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 176, + 597 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 602, + 504, + 625 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 615 + ], + "score": 1.0, + "content": "Cristian Bucilua, Rich Caruana, and Alexandru Niculescu-Mizil. 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Methods with similar speedup are grouped together for easy com-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 120, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 505, + 135 + ], + "score": 1.0, + "content": "parison. In general, our method achieves comparable or better performance across various speedups", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "score": 1.0, + "content": "on ResNet34 and 50. 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(2016); Wang et al.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 164, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 505, + 177 + ], + "score": 1.0, + "content": "(2018); He et al. (2017) but mainly on small datasets like CIFAR, while we make it on the much", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "score": 1.0, + "content": "challenging ImageNet benchmark. (2) Similar to the results on CIFAR (Tab. 1), when the speedup", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "is larger, the advantage of our method is more obvious. For example, ours GReg-2 only outperforms", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 264, + 211 + ], + "score": 1.0, + "content": "Taylor-FO Molchanov et al. (2019) by", + "type": "text" + }, + { + "bbox": [ + 264, + 198, + 292, + 209 + ], + "score": 0.86, + "content": "0 . 8 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 197, + 380, + 211 + ], + "score": 1.0, + "content": "top-1 accuracy at the", + "type": "text" + }, + { + "bbox": [ + 381, + 199, + 407, + 209 + ], + "score": 0.89, + "content": "\\sim 2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 197, + 475, + 211 + ], + "score": 1.0, + "content": "setting, while at", + "type": "text" + }, + { + "bbox": [ + 476, + 199, + 501, + 209 + ], + "score": 0.88, + "content": "\\sim 3 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 197, + 505, + 211 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 186, + 222 + ], + "score": 1.0, + "content": "GReg-2 is better by", + "type": "text" + }, + { + "bbox": [ + 187, + 209, + 214, + 220 + ], + "score": 0.87, + "content": "2 . 2 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "top-1 accuracy. (3) Many methods work on the weight importance crite-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "rion problem, including some very recent ones (ProvableFP Liebenwein et al. (2020), LFPC He et al.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 291, + 244 + ], + "score": 1.0, + "content": "(2020)). Yet as shown, our simple variant of", + "type": "text" + }, + { + "bbox": [ + 292, + 231, + 304, + 242 + ], + "score": 0.9, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "-norm pruning can still be a strong competitor in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 240, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 255 + ], + "score": 1.0, + "content": "terms of accuracy-FLOPs trade-off. This reiterates one of our key ideas in this work that the pruning", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 488, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 488, + 265 + ], + "score": 1.0, + "content": "schedule may be as important as weight importance scoring and worth more research attention.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 109, + 505, + 265 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 505, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "Unstructured pruning. Although we mainly target filter pruning in this work, the proposed meth-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "ods actually can be applied to unstructured pruning as effectively. In Tab. 4, we present the results", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "of unstructured pruning on ResNet50. WoodFisher Singh & Alistarh (2020) is the state-of-the-art", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "score": 1.0, + "content": "Hessian-based unstructured pruning approach. 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Besides, the simple magnitude pruning variant GReg-1 delivers more favorable", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 475, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 475, + 349 + ], + "score": 1.0, + "content": "result, implying that a better pruning schedule also matters in the unstructured pruning case.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 269, + 506, + 349 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 364, + 195, + 377 + ], + "lines": [ + { + "bbox": [ + 104, + 363, + 197, + 380 + ], + "spans": [ + { + "bbox": [ + 104, + 363, + 197, + 380 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 505, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "Regularization is long deemed as a sparsity-learning tool in neural network pruning, which usually", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 401, + 504, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 504, + 414 + ], + "score": 1.0, + "content": "works in the small strength regime. In this work, we present two algorithms that exploit regulariza-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "tion in a new fashion that the penalty factor is uniformly raised to a large level. Two central problems", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "regarding deep neural pruning are tackled by the proposed methods, pruning schedule and weight", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "importance criterion. The proposed approaches rely on few impractical assumptions, have a sound", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "theoretical basis, and are scalable to large datasets and networks. Apart from the methodology it-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "self, the encouraging results on CIFAR and ImageNet also justify our general ideas in this paper: (1)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "In addition to weight importance scoring, pruning schedule is another pivotal factor in deep neural", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "pruning which may deserve more research attention. (2) Without any Hessian approximation, we", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 489, + 434, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 360, + 501 + ], + "score": 1.0, + "content": "can still tap into its power for pruning with the help of growing", + "type": "text" + }, + { + "bbox": [ + 360, + 489, + 373, + 500 + ], + "score": 0.86, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 489, + 434, + 501 + ], + "score": 1.0, + "content": "regularization.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 389, + 506, + 501 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 518, + 224, + 530 + ], + "lines": [ + { + "bbox": [ + 107, + 517, + 226, + 533 + ], + "spans": [ + { + "bbox": [ + 107, + 517, + 226, + 533 + ], + "score": 1.0, + "content": "ACKNOWLEDGEMENTS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 543, + 504, + 565 + ], + "lines": [ + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "The work is supported by the National Science Foundation Award ECCS-1916839 and the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 554, + 334, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 334, + 565 + ], + "score": 1.0, + "content": "U.S. Army Research Office Award W911NF-17-1-0367.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 106, + 543, + 505, + 565 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 583, + 175, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 176, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 176, + 597 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 602, + 504, + 625 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 615 + ], + "score": 1.0, + "content": "Cristian Bucilua, Rich Caruana, and Alexandru Niculescu-Mizil. 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In", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 116, + 680, + 190, + 691 + ], + "spans": [ + { + "bbox": [ + 116, + 680, + 190, + 691 + ], + "score": 1.0, + "content": "NeurIPS, 2018. 8", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 657, + 505, + 691 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 703, + 504, + 725 + ], + "lines": [ + { + "bbox": [ + 106, + 702, + 505, + 716 + ], + "spans": [ + { + "bbox": [ + 106, + 702, + 505, + 716 + ], + "score": 1.0, + "content": "Barret Zoph and Quoc Le. Neural architecture search with reinforcement learning. In ICLR, 2017.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 115, + 714, + 124, + 725 + ], + "spans": [ + { + "bbox": [ + 115, + 714, + 124, + 725 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 106, + 702, + 505, + 725 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 182, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 185, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 185, + 97 + ], + "score": 1.0, + "content": "A APPENDIX", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 108, + 106, + 282, + 118 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 283, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 283, + 119 + ], + "score": 1.0, + "content": "A.1 EXPERIMENTAL SETTING DETAILS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 126, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "Training setting summary. About the networks evaluated, we intentionally avoid AlexNet and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 136, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 505, + 151 + ], + "score": 1.0, + "content": "VGG on the ImageNet benchmark because the single-branch architecture is no longer representa-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "tive of the modern deep network architectures with residuals (but still keep VGG19 on the CIFAR", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "analysis to make sure the findings are not limited to one specific architecture). Apart from some key", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 171, + 460, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 460, + 183 + ], + "score": 1.0, + "content": "settings stated in the paper, a more detailed training setting summary is shown as Tab. 5.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4 + }, + { + "type": "table", + "bbox": [ + 109, + 238, + 499, + 313 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 201, + 504, + 235 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "Table 5: Training setting summary. For the SGD solver, in the parentheses are the momentum and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 225 + ], + "score": 1.0, + "content": "weight decay. For ImageNet, batch size 64 is used for pruning instead of the standard 256, which is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 223, + 277, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 277, + 236 + ], + "score": 1.0, + "content": "because we want to save the training time.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "table_body", + "bbox": [ + 109, + 238, + 499, + 313 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 238, + 499, + 313 + ], + "spans": [ + { + "bbox": [ + 109, + 238, + 499, + 313 + ], + "score": 0.981, + "html": "
DatasetCIFARImageNet
SolverSGD (0.9, 5e-4)SGD (0.9, 1e-4)
LR policy (prune)Fixed (1e-3)
LR policy (finetune)Multi-step (0:1e-2,60:1e-3,90:1e-4)Multi-step (0:1e-2, 60:1e-3,90:1e-4)Multi-step (0:1e-2, 30:1e-3, 60:1e-4,75:1e-5)
Total epoch (finetune)12090
Batch size (prune)25664
Batch size (finetune)256
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(2019b); Singh & Alistarh (2020) can", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 341, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 505, + 353 + ], + "score": 1.0, + "content": "automatically decide pruning ratios, in this paper we opt to consider pruning independent with the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "pruning ratio choosing. The main consideration is that pruning ratio is broadly believed to reflect", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 363, + 504, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 504, + 375 + ], + "score": 1.0, + "content": "the redundancy of different layers LeCun et al. (1990); Wen et al. (2016); He et al. (2017), which", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 372, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 104, + 372, + 506, + 388 + ], + "score": 1.0, + "content": "is an inherent characteristic of the model, thus should not be coupled with the subsequent pruning", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 385, + 155, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 155, + 398 + ], + "score": 1.0, + "content": "algorithms.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 402, + 505, + 512 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 504, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 493, + 415 + ], + "score": 1.0, + "content": "Before we list the specific pruning ratios, we explain how we set them. (1) For a ResNet, if it has", + "type": "text" + }, + { + "bbox": [ + 493, + 402, + 504, + 412 + ], + "score": 0.73, + "content": "N", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 412, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 104, + 412, + 219, + 426 + ], + "score": 1.0, + "content": "stages, we will use a list of", + "type": "text" + }, + { + "bbox": [ + 219, + 413, + 230, + 423 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 412, + 406, + 426 + ], + "score": 1.0, + "content": "floats to represent its pruning ratios for the", + "type": "text" + }, + { + "bbox": [ + 407, + 413, + 417, + 423 + ], + "score": 0.75, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 412, + 505, + 426 + ], + "score": 1.0, + "content": "stages. For example,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "ResNet56 has 4 stages in conv layers, then “[0, 0.5, 0.5, 0.5]” means “for the first stage (which is also", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 446, + 448 + ], + "score": 1.0, + "content": "the first conv layer), the pruning ratio is 0; the other three stages have pruning ratio of", + "type": "text" + }, + { + "bbox": [ + 447, + 435, + 465, + 446 + ], + "score": 0.63, + "content": "0 . 5 '", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 435, + 506, + 448 + ], + "score": 1.0, + "content": ". Besides,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "since we do not prune the last conv in a residual block, which means for a two-layer residual block", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "(for ResNet56), we only prune the first layer; for a three-layer bottleneck block (for ResNet34 and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 468, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 506, + 481 + ], + "score": 1.0, + "content": "50), we only prune the first and second layers. (2) For VGG19, we use the following pruning ratio", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "setting. For example, “[0:0, 1-9:0.3, 10-15:0.5]” means “for the first layer (index starting from 0),", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 490, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 506, + 503 + ], + "score": 1.0, + "content": "the pruning ratio is 0; for layer 1 to 9, the pruning ratio is 0.3; for layer 10 to 15, the pruning ratio is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 499, + 129, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 124, + 512 + ], + "score": 0.48, + "content": "0 . 5 '", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 499, + 129, + 513 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 517, + 505, + 584 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 506, + 531 + ], + "score": 1.0, + "content": "With these, the specific pruning ratio for each of our experiments in the paper are listed in Tab. 6.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 527, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 506, + 542 + ], + "score": 1.0, + "content": "We do not have strong rules to set them, except one, which is setting the pruning ratios of higher", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "stages smaller, because the FLOPs of higher layers are relatively smaller (due to the fact that the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 551, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 563 + ], + "score": 1.0, + "content": "spatial feature map sizes are smaller) and we are targeting more acceleration than compression. Of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 562, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 573 + ], + "score": 1.0, + "content": "course, this scheme only is quite crude, yet as our results (Tab. 3 and 4) show, even with these crude", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 573, + 297, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 297, + 585 + ], + "score": 1.0, + "content": "settings, the performances are still competitive.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 108, + 599, + 211, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 598, + 213, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 213, + 615 + ], + "score": 1.0, + "content": "B PROOF OF EQ. 5", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 104, + 624, + 480, + 636 + ], + "lines": [ + { + "bbox": [ + 106, + 624, + 482, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 215, + 638 + ], + "score": 1.0, + "content": "When a quadratic function", + "type": "text" + }, + { + "bbox": [ + 215, + 626, + 222, + 635 + ], + "score": 0.83, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 624, + 275, + 638 + ], + "score": 1.0, + "content": "converges at", + "type": "text" + }, + { + "bbox": [ + 276, + 626, + 289, + 635 + ], + "score": 0.86, + "content": "\\mathbf { w } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 624, + 374, + 638 + ], + "score": 1.0, + "content": "with Hessian matrix", + "type": "text" + }, + { + "bbox": [ + 374, + 626, + 384, + 635 + ], + "score": 0.68, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 624, + 482, + 638 + ], + "score": 1.0, + "content": ", it can be formulated as", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 641, + 376, + 657 + ], + "lines": [ + { + "bbox": [ + 234, + 641, + 376, + 657 + ], + "spans": [ + { + "bbox": [ + 234, + 641, + 376, + 657 + ], + "score": 0.92, + "content": "\\mathcal { E } = ( \\mathbf { w } - \\mathbf { w } ^ { * } ) ^ { T } \\mathbf { H } ( \\mathbf { w } - \\mathbf { w } ^ { * } ) + C ,", + "type": "interline_equation", + "image_path": "a354779b1fe11cc9f5eb7262eb5daa29a2ba4a2657bb0a027f50705a65c78136.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 234, + 641, + 376, + 657 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 661, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 660, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 133, + 674 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 662, + 143, + 672 + ], + "score": 0.85, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 660, + 389, + 674 + ], + "score": 1.0, + "content": "is a constant. Now a new function is made by increasing the", + "type": "text" + }, + { + "bbox": [ + 389, + 662, + 402, + 673 + ], + "score": 0.89, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 660, + 505, + 674 + ], + "score": 1.0, + "content": "penalty by small amount", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 671, + 156, + 686 + ], + "spans": [ + { + "bbox": [ + 107, + 673, + 118, + 682 + ], + "score": 0.84, + "content": "\\delta \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 671, + 156, + 686 + ], + "score": 1.0, + "content": ", namely,", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 682, + 346, + 696 + ], + "lines": [ + { + "bbox": [ + 264, + 682, + 346, + 696 + ], + "spans": [ + { + "bbox": [ + 264, + 682, + 346, + 696 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\hat { \\mathcal { E } } = \\mathcal { E } + \\delta \\lambda \\mathbf { w } ^ { T } \\mathbf { I } \\mathbf { w } . } \\end{array}", + "type": "interline_equation", + "image_path": "c34edfcfabc00cf2e1f4718de0c9ceeab070010b0548d9b2aac3a501a35144b7.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 264, + 682, + 346, + 696 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 700, + 441, + 713 + ], + "lines": [ + { + "bbox": [ + 104, + 699, + 443, + 715 + ], + "spans": [ + { + "bbox": [ + 104, + 699, + 239, + 715 + ], + "score": 1.0, + "content": "Let the new converged values be", + "type": "text" + }, + { + "bbox": [ + 239, + 702, + 252, + 712 + ], + "score": 0.83, + "content": "\\hat { \\mathbf { w } } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 699, + 347, + 715 + ], + "score": 1.0, + "content": ", then similar to Eq. 10,", + "type": "text" + }, + { + "bbox": [ + 348, + 700, + 355, + 711 + ], + "score": 0.8, + "content": "\\hat { \\mathcal { E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 699, + 443, + 715 + ], + "score": 1.0, + "content": "can be formulated as", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 717, + 423, + 732 + ], + "lines": [ + { + "bbox": [ + 187, + 717, + 423, + 732 + ], + "spans": [ + { + "bbox": [ + 187, + 717, + 423, + 732 + ], + "score": 0.87, + "content": "\\hat { \\mathcal { E } } = ( \\mathbf { w } - \\hat { \\mathbf { w } } ^ { * } ) ^ { T } \\hat { \\mathbf { H } } ( \\mathbf { w } - \\hat { \\mathbf { w } } ^ { * } ) + \\hat { C } , \\mathrm { w h e r e } \\hat { \\mathbf { H } } = \\mathbf { H } + \\delta \\lambda \\mathbf { I } .", + "type": "interline_equation", + "image_path": "13e8756e6a8f3576944c1fa9c0c0737ff995ca8cb27e477bd45861524b23d6fa.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 187, + 717, + 423, + 732 + ], + "spans": [], + "index": 42 + } + ] + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "13", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 182, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 185, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 185, + 97 + ], + "score": 1.0, + "content": "A APPENDIX", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 108, + 106, + 282, + 118 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 283, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 283, + 119 + ], + "score": 1.0, + "content": "A.1 EXPERIMENTAL SETTING DETAILS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 126, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "Training setting summary. About the networks evaluated, we intentionally avoid AlexNet and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 136, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 505, + 151 + ], + "score": 1.0, + "content": "VGG on the ImageNet benchmark because the single-branch architecture is no longer representa-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "tive of the modern deep network architectures with residuals (but still keep VGG19 on the CIFAR", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "analysis to make sure the findings are not limited to one specific architecture). Apart from some key", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 171, + 460, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 460, + 183 + ], + "score": 1.0, + "content": "settings stated in the paper, a more detailed training setting summary is shown as Tab. 5.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 127, + 505, + 183 + ] + }, + { + "type": "table", + "bbox": [ + 109, + 238, + 499, + 313 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 201, + 504, + 235 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "Table 5: Training setting summary. For the SGD solver, in the parentheses are the momentum and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 225 + ], + "score": 1.0, + "content": "weight decay. For ImageNet, batch size 64 is used for pruning instead of the standard 256, which is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 223, + 277, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 277, + 236 + ], + "score": 1.0, + "content": "because we want to save the training time.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "table_body", + "bbox": [ + 109, + 238, + 499, + 313 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 238, + 499, + 313 + ], + "spans": [ + { + "bbox": [ + 109, + 238, + 499, + 313 + ], + "score": 0.981, + "html": "
DatasetCIFARImageNet
SolverSGD (0.9, 5e-4)SGD (0.9, 1e-4)
LR policy (prune)Fixed (1e-3)
LR policy (finetune)Multi-step (0:1e-2,60:1e-3,90:1e-4)Multi-step (0:1e-2, 60:1e-3,90:1e-4)Multi-step (0:1e-2, 30:1e-3, 60:1e-4,75:1e-5)
Total epoch (finetune)12090
Batch size (prune)25664
Batch size (finetune)256
", + "type": "table", + "image_path": "850c677b90dbfd948de7c102f6c87f86dbaff4275eac0749e6340b7515e0809c.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 109, + 238, + 499, + 263.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 109, + 263.0, + 499, + 288.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 109, + 288.0, + 499, + 313.0 + ], + "spans": [], + "index": 12 + } + ] + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 505, + 397 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 504, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 504, + 342 + ], + "score": 1.0, + "content": "Pruning ratios. Although several recent methods Ding et al. (2019b); Singh & Alistarh (2020) can", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 341, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 505, + 353 + ], + "score": 1.0, + "content": "automatically decide pruning ratios, in this paper we opt to consider pruning independent with the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "pruning ratio choosing. The main consideration is that pruning ratio is broadly believed to reflect", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 363, + 504, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 504, + 375 + ], + "score": 1.0, + "content": "the redundancy of different layers LeCun et al. (1990); Wen et al. (2016); He et al. (2017), which", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 372, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 104, + 372, + 506, + 388 + ], + "score": 1.0, + "content": "is an inherent characteristic of the model, thus should not be coupled with the subsequent pruning", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 385, + 155, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 155, + 398 + ], + "score": 1.0, + "content": "algorithms.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 331, + 506, + 398 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 402, + 505, + 512 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 504, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 493, + 415 + ], + "score": 1.0, + "content": "Before we list the specific pruning ratios, we explain how we set them. (1) For a ResNet, if it has", + "type": "text" + }, + { + "bbox": [ + 493, + 402, + 504, + 412 + ], + "score": 0.73, + "content": "N", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 412, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 104, + 412, + 219, + 426 + ], + "score": 1.0, + "content": "stages, we will use a list of", + "type": "text" + }, + { + "bbox": [ + 219, + 413, + 230, + 423 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 412, + 406, + 426 + ], + "score": 1.0, + "content": "floats to represent its pruning ratios for the", + "type": "text" + }, + { + "bbox": [ + 407, + 413, + 417, + 423 + ], + "score": 0.75, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 412, + 505, + 426 + ], + "score": 1.0, + "content": "stages. For example,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "ResNet56 has 4 stages in conv layers, then “[0, 0.5, 0.5, 0.5]” means “for the first stage (which is also", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 446, + 448 + ], + "score": 1.0, + "content": "the first conv layer), the pruning ratio is 0; the other three stages have pruning ratio of", + "type": "text" + }, + { + "bbox": [ + 447, + 435, + 465, + 446 + ], + "score": 0.63, + "content": "0 . 5 '", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 435, + 506, + 448 + ], + "score": 1.0, + "content": ". Besides,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "since we do not prune the last conv in a residual block, which means for a two-layer residual block", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "(for ResNet56), we only prune the first layer; for a three-layer bottleneck block (for ResNet34 and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 468, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 506, + 481 + ], + "score": 1.0, + "content": "50), we only prune the first and second layers. (2) For VGG19, we use the following pruning ratio", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "setting. For example, “[0:0, 1-9:0.3, 10-15:0.5]” means “for the first layer (index starting from 0),", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 490, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 506, + 503 + ], + "score": 1.0, + "content": "the pruning ratio is 0; for layer 1 to 9, the pruning ratio is 0.3; for layer 10 to 15, the pruning ratio is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 499, + 129, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 124, + 512 + ], + "score": 0.48, + "content": "0 . 5 '", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 499, + 129, + 513 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 402, + 506, + 513 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 517, + 505, + 584 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 506, + 531 + ], + "score": 1.0, + "content": "With these, the specific pruning ratio for each of our experiments in the paper are listed in Tab. 6.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 527, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 506, + 542 + ], + "score": 1.0, + "content": "We do not have strong rules to set them, except one, which is setting the pruning ratios of higher", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "stages smaller, because the FLOPs of higher layers are relatively smaller (due to the fact that the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 551, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 563 + ], + "score": 1.0, + "content": "spatial feature map sizes are smaller) and we are targeting more acceleration than compression. Of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 562, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 573 + ], + "score": 1.0, + "content": "course, this scheme only is quite crude, yet as our results (Tab. 3 and 4) show, even with these crude", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 573, + 297, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 297, + 585 + ], + "score": 1.0, + "content": "settings, the performances are still competitive.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 516, + 506, + 585 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 599, + 211, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 598, + 213, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 213, + 615 + ], + "score": 1.0, + "content": "B PROOF OF EQ. 5", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 104, + 624, + 480, + 636 + ], + "lines": [ + { + "bbox": [ + 106, + 624, + 482, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 215, + 638 + ], + "score": 1.0, + "content": "When a quadratic function", + "type": "text" + }, + { + "bbox": [ + 215, + 626, + 222, + 635 + ], + "score": 0.83, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 624, + 275, + 638 + ], + "score": 1.0, + "content": "converges at", + "type": "text" + }, + { + "bbox": [ + 276, + 626, + 289, + 635 + ], + "score": 0.86, + "content": "\\mathbf { w } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 624, + 374, + 638 + ], + "score": 1.0, + "content": "with Hessian matrix", + "type": "text" + }, + { + "bbox": [ + 374, + 626, + 384, + 635 + ], + "score": 0.68, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 624, + 482, + 638 + ], + "score": 1.0, + "content": ", it can be formulated as", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 106, + 624, + 482, + 638 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 641, + 376, + 657 + ], + "lines": [ + { + "bbox": [ + 234, + 641, + 376, + 657 + ], + "spans": [ + { + "bbox": [ + 234, + 641, + 376, + 657 + ], + "score": 0.92, + "content": "\\mathcal { E } = ( \\mathbf { w } - \\mathbf { w } ^ { * } ) ^ { T } \\mathbf { H } ( \\mathbf { w } - \\mathbf { w } ^ { * } ) + C ,", + "type": "interline_equation", + "image_path": "a354779b1fe11cc9f5eb7262eb5daa29a2ba4a2657bb0a027f50705a65c78136.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 234, + 641, + 376, + 657 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 661, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 660, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 133, + 674 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 662, + 143, + 672 + ], + "score": 0.85, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 660, + 389, + 674 + ], + "score": 1.0, + "content": "is a constant. Now a new function is made by increasing the", + "type": "text" + }, + { + "bbox": [ + 389, + 662, + 402, + 673 + ], + "score": 0.89, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 660, + 505, + 674 + ], + "score": 1.0, + "content": "penalty by small amount", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 671, + 156, + 686 + ], + "spans": [ + { + "bbox": [ + 107, + 673, + 118, + 682 + ], + "score": 0.84, + "content": "\\delta \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 671, + 156, + 686 + ], + "score": 1.0, + "content": ", namely,", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 660, + 505, + 686 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 682, + 346, + 696 + ], + "lines": [ + { + "bbox": [ + 264, + 682, + 346, + 696 + ], + "spans": [ + { + "bbox": [ + 264, + 682, + 346, + 696 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\hat { \\mathcal { E } } = \\mathcal { E } + \\delta \\lambda \\mathbf { w } ^ { T } \\mathbf { I } \\mathbf { w } . } \\end{array}", + "type": "interline_equation", + "image_path": "c34edfcfabc00cf2e1f4718de0c9ceeab070010b0548d9b2aac3a501a35144b7.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 264, + 682, + 346, + 696 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 700, + 441, + 713 + ], + "lines": [ + { + "bbox": [ + 104, + 699, + 443, + 715 + ], + "spans": [ + { + "bbox": [ + 104, + 699, + 239, + 715 + ], + "score": 1.0, + "content": "Let the new converged values be", + "type": "text" + }, + { + "bbox": [ + 239, + 702, + 252, + 712 + ], + "score": 0.83, + "content": "\\hat { \\mathbf { w } } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 699, + 347, + 715 + ], + "score": 1.0, + "content": ", then similar to Eq. 10,", + "type": "text" + }, + { + "bbox": [ + 348, + 700, + 355, + 711 + ], + "score": 0.8, + "content": "\\hat { \\mathcal { E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 699, + 443, + 715 + ], + "score": 1.0, + "content": "can be formulated as", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 104, + 699, + 443, + 715 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 717, + 423, + 732 + ], + "lines": [ + { + "bbox": [ + 187, + 717, + 423, + 732 + ], + "spans": [ + { + "bbox": [ + 187, + 717, + 423, + 732 + ], + "score": 0.87, + "content": "\\hat { \\mathcal { E } } = ( \\mathbf { w } - \\hat { \\mathbf { w } } ^ { * } ) ^ { T } \\hat { \\mathbf { H } } ( \\mathbf { w } - \\hat { \\mathbf { w } } ^ { * } ) + \\hat { C } , \\mathrm { w h e r e } \\hat { \\mathbf { H } } = \\mathbf { H } + \\delta \\lambda \\mathbf { I } .", + "type": "interline_equation", + "image_path": "13e8756e6a8f3576944c1fa9c0c0737ff995ca8cb27e477bd45861524b23d6fa.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 187, + 717, + 423, + 732 + ], + "spans": [], + "index": 42 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 104, + 523, + 199 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 240, + 90, + 371, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 239, + 87, + 372, + 104 + ], + "spans": [ + { + "bbox": [ + 239, + 87, + 372, + 104 + ], + "score": 1.0, + "content": "Table 6: Pruning ratio summary.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 106, + 104, + 523, + 199 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 104, + 523, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 523, + 199 + ], + "score": 0.949, + "html": "
DatasetNetworkSpeedupPruned top-1 accuracy (%)Pruning ratio
CIFAR10CIFAR100ResNet56VGG192.55×8.84×93.3667.56[0, 0.75, 0.75, 0.32, 0][1-15:0.7]
ImageNetImageNetImageNetImageNetImageNetResNet34ResNet50ResNet501.32×73.4476.2475.16[0, 0.50, 0.60, 0.40, 0, 0]*[0, 0.30, 0.30, 0.30, 0.14, 0][0, 0.60, 0.60, 0.60, 0.21, 0][0, 0.74, 0.74, 0.60, 0.21, 0][0, 0.68, 0.68, 0.68, 0.50, 0]
1.49×2.31×
ResNet50ResNet502.56×74.7573.50
3.06×
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(2017). Specifically, we refer to the implementation of Liu et al. (2019) at https://github.com/Eric-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 220, + 381, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 381, + 232 + ], + "score": 1.0, + "content": "mingjie/rethinking-network-pruning/tree/master/imagenet/l1-norm-pruning.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 248, + 326, + 260 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 327, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 327, + 261 + ], + "score": 1.0, + "content": "Meanwhile, combine Eq. 10 and Eq. 11, we can obtain", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 262, + 402, + 278 + ], + "lines": [ + { + "bbox": [ + 209, + 262, + 402, + 278 + ], + "spans": [ + { + "bbox": [ + 209, + 262, + 402, + 278 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\hat { \\mathcal { E } } = ( \\mathbf { w } - \\mathbf { w } ^ { * } ) ^ { T } \\mathbf { H } ( \\mathbf { w } - \\mathbf { w } ^ { * } ) + \\delta \\lambda \\mathbf { w } ^ { T } \\mathbf { I } \\mathbf { w } + C . } \\end{array}", + "type": "interline_equation", + "image_path": "3e92a4b74763a03469816778e023f02f170c348be759a20be4a6ae37b7713ea9.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 209, + 262, + 402, + 278 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 286, + 260, + 298 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 261, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 261, + 299 + ], + "score": 1.0, + "content": "Compare Eq. 13 with Eq. 12, we have", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 299, + 413, + 314 + ], + "lines": [ + { + "bbox": [ + 197, + 299, + 413, + 314 + ], + "spans": [ + { + "bbox": [ + 197, + 299, + 413, + 314 + ], + "score": 0.87, + "content": "( \\mathbf { H } + \\delta \\lambda \\mathbf { I } ) { \\hat { \\mathbf { w } } } ^ { * } = \\mathbf { H } \\mathbf { w } ^ { * } \\Rightarrow { \\hat { \\mathbf { w } } } ^ { * } = ( \\mathbf { H } + \\delta \\lambda \\mathbf { I } ) ^ { - 1 } \\mathbf { H } \\mathbf { w } ^ { * } .", + "type": "interline_equation", + "image_path": "5e5c2915d39bb703757f0810d89c822d24d94c22f481c25d5e2a659ee95f70b9.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 197, + 299, + 413, + 314 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 328, + 211, + 341 + ], + "lines": [ + { + "bbox": [ + 106, + 327, + 213, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 213, + 344 + ], + "score": 1.0, + "content": "C PROOF OF EQ. 7", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 350, + 450, + 380 + ], + "lines": [ + { + "bbox": [ + 159, + 350, + 450, + 380 + ], + "spans": [ + { + "bbox": [ + 159, + 350, + 450, + 380 + ], + "score": 0.93, + "content": "\\hat { \\mathbf { H } } = \\left\\{ \\begin{array} { c c } { h _ { 1 1 } + \\delta \\lambda } & { h _ { 1 2 } } \\\\ { h _ { 1 2 } } & { h _ { 2 2 } + \\delta \\lambda } \\end{array} \\right\\} \\Rightarrow \\hat { \\mathbf { H } } ^ { - 1 } = \\frac { 1 } { \\vert \\hat { \\mathbf { H } } \\vert } \\left\\{ \\begin{array} { c c } { h _ { 2 2 } + \\delta \\lambda } & { - h _ { 1 2 } } \\\\ { - h _ { 1 2 } } & { h _ { 1 1 } + \\delta \\lambda } \\end{array} \\right\\}", + "type": "interline_equation", + "image_path": "ea9dc30c0730797c1c511ebc3edb22e5cea7c1db75ba0e57f8f864a49be11f03.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 159, + 350, + 450, + 360.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 159, + 360.0, + 450, + 370.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 159, + 370.0, + 450, + 380.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 387, + 233, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 386, + 233, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 150, + 402 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + }, + { + "bbox": [ + 150, + 387, + 233, + 399 + ], + "score": 0.85, + "content": "\\hat { \\mathbf { w } } ^ { * } = \\hat { \\mathbf { H } } ^ { - 1 } \\mathbf { H } \\mathbf { w } ^ { * } \\Rightarrow", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 402, + 451, + 456 + ], + "lines": [ + { + "bbox": [ + 158, + 402, + 451, + 456 + ], + "spans": [ + { + "bbox": [ + 158, + 402, + 451, + 456 + ], + "score": 0.93, + "content": "\\begin{array} { r l r } & { } & { \\left\\{ \\hat { w } _ { 1 } ^ { * } \\right\\} = \\hat { \\bf H } ^ { - 1 } { \\bf H } \\left\\{ w _ { 1 } ^ { * } \\right\\} = \\frac { 1 } { \\left| \\hat { \\bf H } \\right| } \\left\\{ \\begin{array} { c c } { h _ { 2 2 } + \\delta \\lambda } & { - h _ { 1 2 } } \\\\ { - h _ { 1 1 } } & { h _ { 1 1 } + \\delta \\lambda } \\end{array} \\right\\} \\left\\{ \\begin{array} { c c } { h _ { 1 1 } } & { h _ { 1 2 } } \\\\ { h _ { 1 2 } } & { h _ { 2 2 } } \\end{array} \\right\\} \\left\\{ \\begin{array} { c } { w _ { 1 } ^ { * } } \\\\ { w _ { 2 } ^ { * } } \\end{array} \\right\\} } \\\\ & { } & { = \\frac { 1 } { \\left| \\hat { \\bf H } \\right| } \\left\\{ \\begin{array} { c c } { \\left( h _ { 1 1 } h _ { 2 2 } + h _ { 1 1 } \\delta \\lambda - h _ { 1 2 } ^ { 2 } \\right) w _ { 1 } ^ { * } + \\delta \\lambda h _ { 1 2 } w _ { 2 } ^ { * } } \\\\ { \\left( h _ { 1 1 } h _ { 2 2 } + h _ { 2 2 } \\delta \\lambda - h _ { 1 2 } ^ { 2 } \\right) w _ { 2 } ^ { * } + \\delta \\lambda h _ { 1 2 } w _ { 1 } ^ { * } } \\end{array} \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "51ddb2df0ebfaba7fefc2a8c3c7138da365237bf1ff5c5ce6187997a1c7278c3.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 158, + 402, + 451, + 420.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 158, + 420.0, + 451, + 438.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 158, + 438.0, + 451, + 456.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 466, + 212, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 213, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 164, + 482 + ], + "score": 1.0, + "content": "D GREG-", + "type": "text" + }, + { + "bbox": [ + 164, + 466, + 213, + 479 + ], + "score": 0.31, + "content": "1 + \\mathrm { O B D }", + "type": "inline_equation" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 491, + 505, + 546 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "score": 1.0, + "content": "In Sec. 4.1, we show when pruning the same weights, GReg-1 is significantly better than the one-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 357, + 514 + ], + "score": 1.0, + "content": "shot counterpart, where the pruned weights are selected by the", + "type": "text" + }, + { + "bbox": [ + 357, + 502, + 370, + 513 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "-norm criterion. Here we conduct", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "the same comparison just with a different pruning criterion introduced in OBD LeCun et al. (1990).", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "OBD is also an one-shot pruning method, using a Hessian-based criterion which is believed to be", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 535, + 228, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 189, + 547 + ], + "score": 1.0, + "content": "more advanced than", + "type": "text" + }, + { + "bbox": [ + 189, + 535, + 201, + 546 + ], + "score": 0.89, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 535, + 228, + 547 + ], + "score": 1.0, + "content": "-norm.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 551, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 565 + ], + "score": 1.0, + "content": "Results are shown in Tab. 7. As seen, using this more advanced importance criterion, our pruning", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 563, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 505, + 575 + ], + "score": 1.0, + "content": "scheme based on growing regularization is still consistently better than the one-shot counterpart.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "Besides, it is also verified here that a better pruning schedule can bring more accuracy gain when", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 584, + 195, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 195, + 598 + ], + "score": 1.0, + "content": "the speedup is larger.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 107, + 611, + 324, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 324, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 324, + 626 + ], + "score": 1.0, + "content": "E FILTER L1-NORM CHANGE OF VGG19", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 636, + 504, + 669 + ], + "lines": [ + { + "bbox": [ + 104, + 635, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 104, + 635, + 248, + 649 + ], + "score": 1.0, + "content": "In Fig. 1 (Row 2), we plot the filter", + "type": "text" + }, + { + "bbox": [ + 248, + 637, + 260, + 647 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 635, + 505, + 649 + ], + "score": 1.0, + "content": "-norm change over time for ResNet50 on ImageNet. Here we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 646, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 660 + ], + "score": 1.0, + "content": "plot the case of VGG19 on CIFAR100 to show the weight separation phenomenon under growing", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 659, + 386, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 386, + 670 + ], + "score": 1.0, + "content": "regularization is a general one across different datasets and networks.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 106, + 686, + 383, + 698 + ], + "lines": [ + { + "bbox": [ + 104, + 684, + 384, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 684, + 384, + 700 + ], + "score": 1.0, + "content": "F HYPER-PARAMETERS AND SENSITIVITY ANALYSIS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 502, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 501, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 401, + 722 + ], + "score": 1.0, + "content": "There are five introduced values in our methods: regularization ceiling", + "type": "text" + }, + { + "bbox": [ + 401, + 712, + 407, + 720 + ], + "score": 0.68, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 709, + 491, + 722 + ], + "score": 1.0, + "content": ", ceiling for picking", + "type": "text" + }, + { + "bbox": [ + 492, + 710, + 501, + 720 + ], + "score": 0.82, + "content": "\\tau ^ { \\prime }", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 139, + 734 + ], + "score": 1.0, + "content": "interval", + "type": "text" + }, + { + "bbox": [ + 139, + 721, + 171, + 732 + ], + "score": 0.92, + "content": "K _ { u } , K _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 719, + 221, + 734 + ], + "score": 1.0, + "content": ", granularity", + "type": "text" + }, + { + "bbox": [ + 221, + 721, + 233, + 730 + ], + "score": 0.81, + "content": "\\delta \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 719, + 505, + 734 + ], + "score": 1.0, + "content": ". Their settings are summarized in Tab. 8. Among them, the ceilings", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "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": "14", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 104, + 523, + 199 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 240, + 90, + 371, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 239, + 87, + 372, + 104 + ], + "spans": [ + { + "bbox": [ + 239, + 87, + 372, + 104 + ], + "score": 1.0, + "content": "Table 6: Pruning ratio summary.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 106, + 104, + 523, + 199 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 104, + 523, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 523, + 199 + ], + "score": 0.949, + "html": "
DatasetNetworkSpeedupPruned top-1 accuracy (%)Pruning ratio
CIFAR10CIFAR100ResNet56VGG192.55×8.84×93.3667.56[0, 0.75, 0.75, 0.32, 0][1-15:0.7]
ImageNetImageNetImageNetImageNetImageNetResNet34ResNet50ResNet501.32×73.4476.2475.16[0, 0.50, 0.60, 0.40, 0, 0]*[0, 0.30, 0.30, 0.30, 0.14, 0][0, 0.60, 0.60, 0.60, 0.21, 0][0, 0.74, 0.74, 0.60, 0.21, 0][0, 0.68, 0.68, 0.68, 0.50, 0]
1.49×2.31×
ResNet50ResNet502.56×74.7573.50
3.06×
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(2017). Specifically, we refer to the implementation of Liu et al. (2019) at https://github.com/Eric-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 220, + 381, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 381, + 232 + ], + "score": 1.0, + "content": "mingjie/rethinking-network-pruning/tree/master/imagenet/l1-norm-pruning.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 248, + 326, + 260 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 327, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 327, + 261 + ], + "score": 1.0, + "content": "Meanwhile, combine Eq. 10 and Eq. 11, we can obtain", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 247, + 327, + 261 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 262, + 402, + 278 + ], + "lines": [ + { + "bbox": [ + 209, + 262, + 402, + 278 + ], + "spans": [ + { + "bbox": [ + 209, + 262, + 402, + 278 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\hat { \\mathcal { E } } = ( \\mathbf { w } - \\mathbf { w } ^ { * } ) ^ { T } \\mathbf { H } ( \\mathbf { w } - \\mathbf { w } ^ { * } ) + \\delta \\lambda \\mathbf { w } ^ { T } \\mathbf { I } \\mathbf { w } + C . } \\end{array}", + "type": "interline_equation", + "image_path": "3e92a4b74763a03469816778e023f02f170c348be759a20be4a6ae37b7713ea9.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 209, + 262, + 402, + 278 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 286, + 260, + 298 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 261, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 261, + 299 + ], + "score": 1.0, + "content": "Compare Eq. 13 with Eq. 12, we have", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 106, + 285, + 261, + 299 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 299, + 413, + 314 + ], + "lines": [ + { + "bbox": [ + 197, + 299, + 413, + 314 + ], + "spans": [ + { + "bbox": [ + 197, + 299, + 413, + 314 + ], + "score": 0.87, + "content": "( \\mathbf { H } + \\delta \\lambda \\mathbf { I } ) { \\hat { \\mathbf { w } } } ^ { * } = \\mathbf { H } \\mathbf { w } ^ { * } \\Rightarrow { \\hat { \\mathbf { w } } } ^ { * } = ( \\mathbf { H } + \\delta \\lambda \\mathbf { I } ) ^ { - 1 } \\mathbf { H } \\mathbf { w } ^ { * } .", + "type": "interline_equation", + "image_path": "5e5c2915d39bb703757f0810d89c822d24d94c22f481c25d5e2a659ee95f70b9.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 197, + 299, + 413, + 314 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 328, + 211, + 341 + ], + "lines": [ + { + "bbox": [ + 106, + 327, + 213, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 213, + 344 + ], + "score": 1.0, + "content": "C PROOF OF EQ. 7", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 106, + 327, + 213, + 344 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 350, + 450, + 380 + ], + "lines": [ + { + "bbox": [ + 159, + 350, + 450, + 380 + ], + "spans": [ + { + "bbox": [ + 159, + 350, + 450, + 380 + ], + "score": 0.93, + "content": "\\hat { \\mathbf { H } } = \\left\\{ \\begin{array} { c c } { h _ { 1 1 } + \\delta \\lambda } & { h _ { 1 2 } } \\\\ { h _ { 1 2 } } & { h _ { 2 2 } + \\delta \\lambda } \\end{array} \\right\\} \\Rightarrow \\hat { \\mathbf { H } } ^ { - 1 } = \\frac { 1 } { \\vert \\hat { \\mathbf { H } } \\vert } \\left\\{ \\begin{array} { c c } { h _ { 2 2 } + \\delta \\lambda } & { - h _ { 1 2 } } \\\\ { - h _ { 1 2 } } & { h _ { 1 1 } + \\delta \\lambda } \\end{array} \\right\\}", + "type": "interline_equation", + "image_path": "ea9dc30c0730797c1c511ebc3edb22e5cea7c1db75ba0e57f8f864a49be11f03.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 159, + 350, + 450, + 360.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 159, + 360.0, + 450, + 370.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 159, + 370.0, + 450, + 380.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 387, + 233, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 386, + 233, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 150, + 402 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + }, + { + "bbox": [ + 150, + 387, + 233, + 399 + ], + "score": 0.85, + "content": "\\hat { \\mathbf { w } } ^ { * } = \\hat { \\mathbf { H } } ^ { - 1 } \\mathbf { H } \\mathbf { w } ^ { * } \\Rightarrow", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 386, + 233, + 402 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 402, + 451, + 456 + ], + "lines": [ + { + "bbox": [ + 158, + 402, + 451, + 456 + ], + "spans": [ + { + "bbox": [ + 158, + 402, + 451, + 456 + ], + "score": 0.93, + "content": "\\begin{array} { r l r } & { } & { \\left\\{ \\hat { w } _ { 1 } ^ { * } \\right\\} = \\hat { \\bf H } ^ { - 1 } { \\bf H } \\left\\{ w _ { 1 } ^ { * } \\right\\} = \\frac { 1 } { \\left| \\hat { \\bf H } \\right| } \\left\\{ \\begin{array} { c c } { h _ { 2 2 } + \\delta \\lambda } & { - h _ { 1 2 } } \\\\ { - h _ { 1 1 } } & { h _ { 1 1 } + \\delta \\lambda } \\end{array} \\right\\} \\left\\{ \\begin{array} { c c } { h _ { 1 1 } } & { h _ { 1 2 } } \\\\ { h _ { 1 2 } } & { h _ { 2 2 } } \\end{array} \\right\\} \\left\\{ \\begin{array} { c } { w _ { 1 } ^ { * } } \\\\ { w _ { 2 } ^ { * } } \\end{array} \\right\\} } \\\\ & { } & { = \\frac { 1 } { \\left| \\hat { \\bf H } \\right| } \\left\\{ \\begin{array} { c c } { \\left( h _ { 1 1 } h _ { 2 2 } + h _ { 1 1 } \\delta \\lambda - h _ { 1 2 } ^ { 2 } \\right) w _ { 1 } ^ { * } + \\delta \\lambda h _ { 1 2 } w _ { 2 } ^ { * } } \\\\ { \\left( h _ { 1 1 } h _ { 2 2 } + h _ { 2 2 } \\delta \\lambda - h _ { 1 2 } ^ { 2 } \\right) w _ { 2 } ^ { * } + \\delta \\lambda h _ { 1 2 } w _ { 1 } ^ { * } } \\end{array} \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "51ddb2df0ebfaba7fefc2a8c3c7138da365237bf1ff5c5ce6187997a1c7278c3.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 158, + 402, + 451, + 420.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 158, + 420.0, + 451, + 438.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 158, + 438.0, + 451, + 456.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 466, + 212, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 213, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 164, + 482 + ], + "score": 1.0, + "content": "D GREG-", + "type": "text" + }, + { + "bbox": [ + 164, + 466, + 213, + 479 + ], + "score": 0.31, + "content": "1 + \\mathrm { O B D }", + "type": "inline_equation" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 491, + 505, + 546 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "score": 1.0, + "content": "In Sec. 4.1, we show when pruning the same weights, GReg-1 is significantly better than the one-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 357, + 514 + ], + "score": 1.0, + "content": "shot counterpart, where the pruned weights are selected by the", + "type": "text" + }, + { + "bbox": [ + 357, + 502, + 370, + 513 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "-norm criterion. Here we conduct", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "the same comparison just with a different pruning criterion introduced in OBD LeCun et al. (1990).", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "OBD is also an one-shot pruning method, using a Hessian-based criterion which is believed to be", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 535, + 228, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 189, + 547 + ], + "score": 1.0, + "content": "more advanced than", + "type": "text" + }, + { + "bbox": [ + 189, + 535, + 201, + 546 + ], + "score": 0.89, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 535, + 228, + 547 + ], + "score": 1.0, + "content": "-norm.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 491, + 505, + 547 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 551, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 565 + ], + "score": 1.0, + "content": "Results are shown in Tab. 7. As seen, using this more advanced importance criterion, our pruning", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 563, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 505, + 575 + ], + "score": 1.0, + "content": "scheme based on growing regularization is still consistently better than the one-shot counterpart.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "Besides, it is also verified here that a better pruning schedule can bring more accuracy gain when", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 584, + 195, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 195, + 598 + ], + "score": 1.0, + "content": "the speedup is larger.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 550, + 505, + 598 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 611, + 324, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 324, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 324, + 626 + ], + "score": 1.0, + "content": "E FILTER L1-NORM CHANGE OF VGG19", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 636, + 504, + 669 + ], + "lines": [ + { + "bbox": [ + 104, + 635, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 104, + 635, + 248, + 649 + ], + "score": 1.0, + "content": "In Fig. 1 (Row 2), we plot the filter", + "type": "text" + }, + { + "bbox": [ + 248, + 637, + 260, + 647 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 635, + 505, + 649 + ], + "score": 1.0, + "content": "-norm change over time for ResNet50 on ImageNet. Here we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 646, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 660 + ], + "score": 1.0, + "content": "plot the case of VGG19 on CIFAR100 to show the weight separation phenomenon under growing", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 659, + 386, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 386, + 670 + ], + "score": 1.0, + "content": "regularization is a general one across different datasets and networks.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 635, + 505, + 670 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 686, + 383, + 698 + ], + "lines": [ + { + "bbox": [ + 104, + 684, + 384, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 684, + 384, + 700 + ], + "score": 1.0, + "content": "F HYPER-PARAMETERS AND SENSITIVITY ANALYSIS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 502, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 501, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 401, + 722 + ], + "score": 1.0, + "content": "There are five introduced values in our methods: regularization ceiling", + "type": "text" + }, + { + "bbox": [ + 401, + 712, + 407, + 720 + ], + "score": 0.68, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 709, + 491, + 722 + ], + "score": 1.0, + "content": ", ceiling for picking", + "type": "text" + }, + { + "bbox": [ + 492, + 710, + 501, + 720 + ], + "score": 0.82, + "content": "\\tau ^ { \\prime }", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 139, + 734 + ], + "score": 1.0, + "content": "interval", + "type": "text" + }, + { + "bbox": [ + 139, + 721, + 171, + 732 + ], + "score": 0.92, + "content": "K _ { u } , K _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 719, + 221, + 734 + ], + "score": 1.0, + "content": ", granularity", + "type": "text" + }, + { + "bbox": [ + 221, + 721, + 233, + 730 + ], + "score": 0.81, + "content": "\\delta \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 719, + 505, + 734 + ], + "score": 1.0, + "content": ". Their settings are summarized in Tab. 8. Among them, the ceilings", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 709, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 89, + 505, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 88, + 505, + 102 + ], + "spans": [ + { + "bbox": [ + 105, + 88, + 505, + 102 + ], + "score": 1.0, + "content": "Table 7: Comparison between pruning schedules: one-shot pruning vs. our proposed GReg-1 using", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 100, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 505, + 113 + ], + "score": 1.0, + "content": "the Hessian-based criterion introduced in OBD LeCun et al. (1990). Each setting is randomly run", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 111, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 505, + 124 + ], + "score": 1.0, + "content": "for 3 times, mean and std accuracies reported. We vary the global pruning ratio from 0.7 to 0.95", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 505, + 135 + ], + "score": 1.0, + "content": "so as to cover the major speedup spectrum of interest. Same as Tab. 1, the pruned weights here are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 133, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 505, + 145 + ], + "score": 1.0, + "content": "exactly the same for the two methods under each speedup ratio. The finetuning processes (number", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 399, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 399, + 157 + ], + "score": 1.0, + "content": "of epochs, LR schedules, etc.) are also the same to keep fair comparison.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "image", + "bbox": [ + 109, + 160, + 501, + 341 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 160, + 501, + 341 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 160, + 501, + 341 + ], + "spans": [ + { + "bbox": [ + 109, + 160, + 501, + 341 + ], + "score": 0.541, + "type": "image", + "image_path": "0f2d6935b0744fbb2ed80e72482e888c222e79ac5a2f589b70d6f4f3caae6562.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 109, + 160, + 501, + 220.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 109, + 220.33333333333334, + 501, + 280.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 109, + 280.6666666666667, + 501, + 341.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 161, + 349, + 448, + 361 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 162, + 348, + 448, + 363 + ], + "spans": [ + { + "bbox": [ + 162, + 348, + 273, + 363 + ], + "score": 1.0, + "content": "Figure 2: Normalized filter", + "type": "text" + }, + { + "bbox": [ + 273, + 350, + 285, + 361 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 348, + 448, + 363 + ], + "score": 1.0, + "content": "-norm over iterations for VGG19 layer3.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + } + ], + "index": 8.0 + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 218, + 393 + ], + "score": 1.0, + "content": "are set through validation:", + "type": "text" + }, + { + "bbox": [ + 218, + 381, + 248, + 390 + ], + "score": 0.9, + "content": "\\tau = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "is set to make sure the unimportant weights are pushed down", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 392, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 404 + ], + "score": 1.0, + "content": "enough (as stated in the main paper, normally after the regularization training, their magnitudes are", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 451, + 415 + ], + "score": 1.0, + "content": "too small to cause significant accuracy degradation if they are completely removed).", + "type": "text" + }, + { + "bbox": [ + 451, + 402, + 494, + 413 + ], + "score": 0.91, + "content": "\\tau ^ { \\prime } = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 240, + 426 + ], + "score": 1.0, + "content": "set generally for the same goal as", + "type": "text" + }, + { + "bbox": [ + 241, + 415, + 247, + 424 + ], + "score": 0.77, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 414, + 505, + 426 + ], + "score": 1.0, + "content": ", but since it is applied to all the weight (not just the unimportant", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 424, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 367, + 438 + ], + "score": 1.0, + "content": "ones), we only expect it to be moderately large (thus smaller than", + "type": "text" + }, + { + "bbox": [ + 367, + 426, + 374, + 434 + ], + "score": 0.67, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 424, + 505, + 438 + ], + "score": 1.0, + "content": ") so that the important and unim-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 348, + 448 + ], + "score": 1.0, + "content": "portant can be differentiated with a clear boundary. For the", + "type": "text" + }, + { + "bbox": [ + 348, + 435, + 360, + 446 + ], + "score": 0.8, + "content": "\\delta \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 435, + 506, + 448 + ], + "score": 1.0, + "content": ", we use a very small regularization", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 153, + 459 + ], + "score": 1.0, + "content": "granularity", + "type": "text" + }, + { + "bbox": [ + 153, + 446, + 165, + 456 + ], + "score": 0.8, + "content": "\\delta \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 446, + 506, + 459 + ], + "score": 1.0, + "content": ", which our theoretical analysis is based on. We set its value to 1e-4 for GReg-1 and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 368, + 470 + ], + "score": 1.0, + "content": "1e-5 for GReg-2 with reference to the original weight decay value", + "type": "text" + }, + { + "bbox": [ + 369, + 457, + 405, + 468 + ], + "score": 0.92, + "content": "5 \\times 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "(for CIFAR models) and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 466, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 128, + 479 + ], + "score": 0.9, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 466, + 505, + 482 + ], + "score": 1.0, + "content": "(for ImageNet models). Note that, these values come from our methods per se, not directly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "score": 1.0, + "content": "related to datasets and networks, thus are invariant to them. This is why we can employ the same set-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "score": 1.0, + "content": "ting of these three hyper-parameters in all our experiments, freeing practitioners from heavy tuning", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 502, + 305, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 305, + 513 + ], + "score": 1.0, + "content": "when dealing with different networks or datasets.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15.5 + }, + { + "type": "table", + "bbox": [ + 177, + 547, + 432, + 611 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 219, + 532, + 392, + 543 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 218, + 531, + 393, + 544 + ], + "spans": [ + { + "bbox": [ + 218, + 531, + 393, + 544 + ], + "score": 1.0, + "content": "Table 8: Hyper-parameters of our methods.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "table_body", + "bbox": [ + 177, + 547, + 432, + 611 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 177, + 547, + 432, + 611 + ], + "spans": [ + { + "bbox": [ + 177, + 547, + 432, + 611 + ], + "score": 0.977, + "html": "
NotationDefault value (CIFAR)Default value (ImageNet)
8入GReg-1: 1e-4, GReg-2: 1e-5
T1
T0.01
Ku10 iterations5 iterations
Ks5k iterations40k iterations
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Both are generally to let the network have enough time to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "converge to the new equilibrium. Generally, we prefer large update intervals, yet we also need", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 642, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 104, + 642, + 506, + 657 + ], + "score": 1.0, + "content": "to consider the time complexity: Too large of them will bring too many iterations, which may", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 234, + 667 + ], + "score": 1.0, + "content": "be unnecessary. Among them,", + "type": "text" + }, + { + "bbox": [ + 234, + 655, + 248, + 666 + ], + "score": 0.89, + "content": "K _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "is less important since it is to stabilize the large regularization", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 109, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 109, + 666, + 135, + 677 + ], + "score": 0.84, + "content": "\\mathit { \\Psi } _ { \\tau } = 1 \\mathit { \\Psi } _ { . }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "). We introduce it simply to make sure the training is fully converged. Therefore, the possibly", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 259, + 689 + ], + "score": 1.0, + "content": "more sensitive hyper-parameter is the", + "type": "text" + }, + { + "bbox": [ + 260, + 677, + 275, + 688 + ], + "score": 0.89, + "content": "K _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "(set to 5 for ImageNet and 10 for CIFAR). Here we will", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 104, + 686, + 312, + 702 + ], + "score": 1.0, + "content": "show the performance is insensitive to the varying", + "type": "text" + }, + { + "bbox": [ + 312, + 688, + 326, + 699 + ], + "score": 0.87, + "content": "K _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 686, + 506, + 702 + ], + "score": 1.0, + "content": ". As shown in Tab. 9, the peak performance", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 698, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 181, + 712 + ], + "score": 1.0, + "content": "appears at around", + "type": "text" + }, + { + "bbox": [ + 182, + 699, + 223, + 710 + ], + "score": 0.91, + "content": "K _ { u } = 1 5", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 698, + 300, + 712 + ], + "score": 1.0, + "content": "for ResNet56 and", + "type": "text" + }, + { + "bbox": [ + 300, + 699, + 342, + 710 + ], + "score": 0.91, + "content": "K _ { u } = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 698, + 507, + 712 + ], + "score": 1.0, + "content": "for VGG19. We simply adopt 10 for a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "uniform setting in our paper. 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(1990). Each setting is randomly run", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 111, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 505, + 124 + ], + "score": 1.0, + "content": "for 3 times, mean and std accuracies reported. We vary the global pruning ratio from 0.7 to 0.95", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 505, + 135 + ], + "score": 1.0, + "content": "so as to cover the major speedup spectrum of interest. Same as Tab. 1, the pruned weights here are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 133, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 505, + 145 + ], + "score": 1.0, + "content": "exactly the same for the two methods under each speedup ratio. The finetuning processes (number", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 399, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 399, + 157 + ], + "score": 1.0, + "content": "of epochs, LR schedules, etc.) are also the same to keep fair comparison.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 88, + 505, + 157 + ] + }, + { + "type": "image", + "bbox": [ + 109, + 160, + 501, + 341 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 160, + 501, + 341 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 160, + 501, + 341 + ], + "spans": [ + { + "bbox": [ + 109, + 160, + 501, + 341 + ], + "score": 0.541, + "type": "image", + "image_path": "0f2d6935b0744fbb2ed80e72482e888c222e79ac5a2f589b70d6f4f3caae6562.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 109, + 160, + 501, + 220.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 109, + 220.33333333333334, + 501, + 280.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 109, + 280.6666666666667, + 501, + 341.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 161, + 349, + 448, + 361 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 162, + 348, + 448, + 363 + ], + "spans": [ + { + "bbox": [ + 162, + 348, + 273, + 363 + ], + "score": 1.0, + "content": "Figure 2: Normalized filter", + "type": "text" + }, + { + "bbox": [ + 273, + 350, + 285, + 361 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 348, + 448, + 363 + ], + "score": 1.0, + "content": "-norm over iterations for VGG19 layer3.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + } + ], + "index": 8.0 + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 218, + 393 + ], + "score": 1.0, + "content": "are set through validation:", + "type": "text" + }, + { + "bbox": [ + 218, + 381, + 248, + 390 + ], + "score": 0.9, + "content": "\\tau = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "is set to make sure the unimportant weights are pushed down", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 392, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 404 + ], + "score": 1.0, + "content": "enough (as stated in the main paper, normally after the regularization training, their magnitudes are", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 451, + 415 + ], + "score": 1.0, + "content": "too small to cause significant accuracy degradation if they are completely removed).", + "type": "text" + }, + { + "bbox": [ + 451, + 402, + 494, + 413 + ], + "score": 0.91, + "content": "\\tau ^ { \\prime } = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 240, + 426 + ], + "score": 1.0, + "content": "set generally for the same goal as", + "type": "text" + }, + { + "bbox": [ + 241, + 415, + 247, + 424 + ], + "score": 0.77, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 414, + 505, + 426 + ], + "score": 1.0, + "content": ", but since it is applied to all the weight (not just the unimportant", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 424, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 367, + 438 + ], + "score": 1.0, + "content": "ones), we only expect it to be moderately large (thus smaller than", + "type": "text" + }, + { + "bbox": [ + 367, + 426, + 374, + 434 + ], + "score": 0.67, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 424, + 505, + 438 + ], + "score": 1.0, + "content": ") so that the important and unim-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 348, + 448 + ], + "score": 1.0, + "content": "portant can be differentiated with a clear boundary. For the", + "type": "text" + }, + { + "bbox": [ + 348, + 435, + 360, + 446 + ], + "score": 0.8, + "content": "\\delta \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 435, + 506, + 448 + ], + "score": 1.0, + "content": ", we use a very small regularization", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 153, + 459 + ], + "score": 1.0, + "content": "granularity", + "type": "text" + }, + { + "bbox": [ + 153, + 446, + 165, + 456 + ], + "score": 0.8, + "content": "\\delta \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 446, + 506, + 459 + ], + "score": 1.0, + "content": ", which our theoretical analysis is based on. We set its value to 1e-4 for GReg-1 and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 368, + 470 + ], + "score": 1.0, + "content": "1e-5 for GReg-2 with reference to the original weight decay value", + "type": "text" + }, + { + "bbox": [ + 369, + 457, + 405, + 468 + ], + "score": 0.92, + "content": "5 \\times 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "(for CIFAR models) and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 466, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 128, + 479 + ], + "score": 0.9, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 466, + 505, + 482 + ], + "score": 1.0, + "content": "(for ImageNet models). Note that, these values come from our methods per se, not directly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "score": 1.0, + "content": "related to datasets and networks, thus are invariant to them. This is why we can employ the same set-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 504 + ], + "score": 1.0, + "content": "ting of these three hyper-parameters in all our experiments, freeing practitioners from heavy tuning", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 502, + 305, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 305, + 513 + ], + "score": 1.0, + "content": "when dealing with different networks or datasets.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 380, + 506, + 513 + ] + }, + { + "type": "table", + "bbox": [ + 177, + 547, + 432, + 611 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 219, + 532, + 392, + 543 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 218, + 531, + 393, + 544 + ], + "spans": [ + { + "bbox": [ + 218, + 531, + 393, + 544 + ], + "score": 1.0, + "content": "Table 8: Hyper-parameters of our methods.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "table_body", + "bbox": [ + 177, + 547, + 432, + 611 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 177, + 547, + 432, + 611 + ], + "spans": [ + { + "bbox": [ + 177, + 547, + 432, + 611 + ], + "score": 0.977, + "html": "
NotationDefault value (CIFAR)Default value (ImageNet)
8入GReg-1: 1e-4, GReg-2: 1e-5
T1
T0.01
Ku10 iterations5 iterations
Ks5k iterations40k iterations
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Both are generally to let the network have enough time to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "converge to the new equilibrium. Generally, we prefer large update intervals, yet we also need", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 642, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 104, + 642, + 506, + 657 + ], + "score": 1.0, + "content": "to consider the time complexity: Too large of them will bring too many iterations, which may", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 234, + 667 + ], + "score": 1.0, + "content": "be unnecessary. Among them,", + "type": "text" + }, + { + "bbox": [ + 234, + 655, + 248, + 666 + ], + "score": 0.89, + "content": "K _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "is less important since it is to stabilize the large regularization", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 109, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 109, + 666, + 135, + 677 + ], + "score": 0.84, + "content": "\\mathit { \\Psi } _ { \\tau } = 1 \\mathit { \\Psi } _ { . }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "). We introduce it simply to make sure the training is fully converged. Therefore, the possibly", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 259, + 689 + ], + "score": 1.0, + "content": "more sensitive hyper-parameter is the", + "type": "text" + }, + { + "bbox": [ + 260, + 677, + 275, + 688 + ], + "score": 0.89, + "content": "K _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "(set to 5 for ImageNet and 10 for CIFAR). Here we will", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 104, + 686, + 312, + 702 + ], + "score": 1.0, + "content": "show the performance is insensitive to the varying", + "type": "text" + }, + { + "bbox": [ + 312, + 688, + 326, + 699 + ], + "score": 0.87, + "content": "K _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 686, + 506, + 702 + ], + "score": 1.0, + "content": ". As shown in Tab. 9, the peak performance", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 698, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 181, + 712 + ], + "score": 1.0, + "content": "appears at around", + "type": "text" + }, + { + "bbox": [ + 182, + 699, + 223, + 710 + ], + "score": 0.91, + "content": "K _ { u } = 1 5", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 698, + 300, + 712 + ], + "score": 1.0, + "content": "for ResNet56 and", + "type": "text" + }, + { + "bbox": [ + 300, + 699, + 342, + 710 + ], + "score": 0.91, + "content": "K _ { u } = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 698, + 507, + 712 + ], + "score": 1.0, + "content": "for VGG19. We simply adopt 10 for a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "uniform setting in our paper. We did not heavily tune these hyper-parameters, yet as seen, they work", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "pretty well across different networks and datasets. 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Pruning ratio", + "type": "text" + }, + { + "bbox": [ + 285, + 135, + 305, + 146 + ], + "score": 0.88, + "content": "9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 135, + 370, + 148 + ], + "score": 1.0, + "content": "(ResNet56) and", + "type": "text" + }, + { + "bbox": [ + 371, + 135, + 390, + 146 + ], + "score": 0.88, + "content": "7 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 135, + 505, + 148 + ], + "score": 1.0, + "content": "(VGG19) are explored here.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 147, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 505, + 158 + ], + "score": 1.0, + "content": "Experiments are randomly run for 3 times with mean accuracy and standard deviation reported. The", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 158, + 403, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 403, + 169 + ], + "score": 1.0, + "content": "best is highlighted with bold and the worst is highlighted with blue color.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "table_body", + "bbox": [ + 107, + 172, + 505, + 208 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 172, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 107, + 172, + 505, + 208 + ], + "score": 0.976, + "html": "
Ku15101520L1+one-shot
Acc. (%,ResNet56)89.40±0.0489.38±0.1389.49±0.2389.69±0.0589.62±0.1387.34±0.21
Acc. (%,VGG19)67.22±0.3367.32±0.2467.35±0.1567.06±0.4066.93±0.2266.05±0.04
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This question is important because it will show if the gain from a better pruning schedule", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 251, + 313 + ], + "score": 1.0, + "content": "is only a bonus concurrent with the", + "type": "text" + }, + { + "bbox": [ + 251, + 300, + 263, + 311 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "criterion or a really universal phenomenon. Since there are", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "literally so many weight importance criteria, we cannot ablate them one by one. Nevertheless, given", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 104, + 322, + 263, + 334 + ], + "score": 1.0, + "content": "a pre-trained model and a pruning ratio", + "type": "text" + }, + { + "bbox": [ + 264, + 324, + 270, + 332 + ], + "score": 0.7, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 322, + 505, + 334 + ], + "score": 1.0, + "content": ", no matter what criterion, its role is to select a filter subset.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 332, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 309, + 347 + ], + "score": 1.0, + "content": "For example, if there are 100 filters in a layer and", + "type": "text" + }, + { + "bbox": [ + 309, + 334, + 343, + 344 + ], + "score": 0.89, + "content": "r = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 333, + 434, + 347 + ], + "score": 1.0, + "content": ", then they are at most", + "type": "text" + }, + { + "bbox": [ + 435, + 332, + 456, + 347 + ], + "score": 0.74, + "content": "\\binom { 1 0 0 } { 5 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 333, + 505, + 347 + ], + "score": 1.0, + "content": "importance", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "criteria in theory for this layer. We can simply randomly pick a subset of filters (which corresponds", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 354, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 370 + ], + "score": 1.0, + "content": "to certain criterion, albeit unknown) and compare the one-shot way with regularization-based way", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "on the subset. Based on this idea, we conduct five random runs on the ResNet56 and VGG19 to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 282, + 390 + ], + "score": 1.0, + "content": "explore this. The pruning ratio is chosen as", + "type": "text" + }, + { + "bbox": [ + 282, + 378, + 302, + 388 + ], + "score": 0.89, + "content": "9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 378, + 376, + 390 + ], + "score": 1.0, + "content": "for ResNet56 and", + "type": "text" + }, + { + "bbox": [ + 376, + 378, + 396, + 388 + ], + "score": 0.89, + "content": "7 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 378, + 506, + 390 + ], + "score": 1.0, + "content": "for VGG19 because under", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "this ratio the compression (or acceleration) ratio is about 10 times, neither too large nor too small", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 399, + 373, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 373, + 412 + ], + "score": 1.0, + "content": "(where the network can heal itself regardless of pruning methods).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 106, + 416, + 505, + 494 + ], + "lines": [ + { + "bbox": [ + 105, + 415, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 430 + ], + "score": 1.0, + "content": "The results are shown in Tab. 10. Here is a sanity check: Compared with Tab. 1, the mean accuracy", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 421, + 441 + ], + "score": 1.0, + "content": "of pruning randomly picked filters should be less than pruning those picked by", + "type": "text" + }, + { + "bbox": [ + 421, + 428, + 434, + 439 + ], + "score": 0.87, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "-norm, confirmed", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 437, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 119, + 452 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 120, + 439, + 152, + 450 + ], + "score": 0.88, + "content": "8 6 . 8 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 437, + 168, + 452 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 169, + 439, + 201, + 449 + ], + "score": 0.88, + "content": "8 7 . 3 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 437, + 278, + 452 + ], + "score": 1.0, + "content": "for ResNet56 and", + "type": "text" + }, + { + "bbox": [ + 278, + 439, + 310, + 449 + ], + "score": 0.87, + "content": "6 5 . 0 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 437, + 327, + 452 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 327, + 439, + 359, + 449 + ], + "score": 0.88, + "content": "6 6 . 0 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 437, + 506, + 452 + ], + "score": 1.0, + "content": "for VGG19. As seen, in each run,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 449, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 463 + ], + "score": 1.0, + "content": "the regularization-based way also significantly surpasses its one-shot counterpart. 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Pruning ratio is", + "type": "text" + }, + { + "bbox": [ + 469, + 514, + 489, + 524 + ], + "score": 0.86, + "content": "90 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 165, + 537 + ], + "score": 1.0, + "content": "ResNet56 and", + "type": "text" + }, + { + "bbox": [ + 166, + 524, + 186, + 535 + ], + "score": 0.86, + "content": "70 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "for VGG19. 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ResNet56 + CIFAR10Run #1Run #2Run #3Run #4Run #5Mean±std
Acc. (%, one-shot)Acc. (%, GReg-1, ours)87.5789.2687.0088.9886.2788.7886.7589.4286.6788.9686.85±0.4389.08±0.23
VGG19 + CIFAR100Run #1Run #2Run #3Run #4Run #5Mean±std
Acc. (%,one-shot)Acc. (%, GReg-1, ours)64.5666.6365.0666.5765.0766.8065.0566.8065.4867.1665.04±0.2966.79±0.21
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Ku15101520L1+one-shot
Acc. (%,ResNet56)89.40±0.0489.38±0.1389.49±0.2389.69±0.0589.62±0.1387.34±0.21
Acc. (%,VGG19)67.22±0.3367.32±0.2467.35±0.1567.06±0.4066.93±0.2266.05±0.04
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This question is important because it will show if the gain from a better pruning schedule", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 251, + 313 + ], + "score": 1.0, + "content": "is only a bonus concurrent with the", + "type": "text" + }, + { + "bbox": [ + 251, + 300, + 263, + 311 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "criterion or a really universal phenomenon. Since there are", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "literally so many weight importance criteria, we cannot ablate them one by one. Nevertheless, given", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 104, + 322, + 263, + 334 + ], + "score": 1.0, + "content": "a pre-trained model and a pruning ratio", + "type": "text" + }, + { + "bbox": [ + 264, + 324, + 270, + 332 + ], + "score": 0.7, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 322, + 505, + 334 + ], + "score": 1.0, + "content": ", no matter what criterion, its role is to select a filter subset.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 332, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 309, + 347 + ], + "score": 1.0, + "content": "For example, if there are 100 filters in a layer and", + "type": "text" + }, + { + "bbox": [ + 309, + 334, + 343, + 344 + ], + "score": 0.89, + "content": "r = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 333, + 434, + 347 + ], + "score": 1.0, + "content": ", then they are at most", + "type": "text" + }, + { + "bbox": [ + 435, + 332, + 456, + 347 + ], + "score": 0.74, + "content": "\\binom { 1 0 0 } { 5 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 333, + 505, + 347 + ], + "score": 1.0, + "content": "importance", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "criteria in theory for this layer. We can simply randomly pick a subset of filters (which corresponds", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 354, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 370 + ], + "score": 1.0, + "content": "to certain criterion, albeit unknown) and compare the one-shot way with regularization-based way", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "on the subset. Based on this idea, we conduct five random runs on the ResNet56 and VGG19 to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 282, + 390 + ], + "score": 1.0, + "content": "explore this. The pruning ratio is chosen as", + "type": "text" + }, + { + "bbox": [ + 282, + 378, + 302, + 388 + ], + "score": 0.89, + "content": "9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 378, + 376, + 390 + ], + "score": 1.0, + "content": "for ResNet56 and", + "type": "text" + }, + { + "bbox": [ + 376, + 378, + 396, + 388 + ], + "score": 0.89, + "content": "7 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 378, + 506, + 390 + ], + "score": 1.0, + "content": "for VGG19 because under", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "this ratio the compression (or acceleration) ratio is about 10 times, neither too large nor too small", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 399, + 373, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 373, + 412 + ], + "score": 1.0, + "content": "(where the network can heal itself regardless of pruning methods).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 256, + 506, + 412 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 416, + 505, + 494 + ], + "lines": [ + { + "bbox": [ + 105, + 415, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 430 + ], + "score": 1.0, + "content": "The results are shown in Tab. 10. Here is a sanity check: Compared with Tab. 1, the mean accuracy", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 421, + 441 + ], + "score": 1.0, + "content": "of pruning randomly picked filters should be less than pruning those picked by", + "type": "text" + }, + { + "bbox": [ + 421, + 428, + 434, + 439 + ], + "score": 0.87, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "-norm, confirmed", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 437, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 119, + 452 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 120, + 439, + 152, + 450 + ], + "score": 0.88, + "content": "8 6 . 8 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 437, + 168, + 452 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 169, + 439, + 201, + 449 + ], + "score": 0.88, + "content": "8 7 . 3 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 437, + 278, + 452 + ], + "score": 1.0, + "content": "for ResNet56 and", + "type": "text" + }, + { + "bbox": [ + 278, + 439, + 310, + 449 + ], + "score": 0.87, + "content": "6 5 . 0 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 437, + 327, + 452 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 327, + 439, + 359, + 449 + ], + "score": 0.88, + "content": "6 6 . 0 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 437, + 506, + 452 + ], + "score": 1.0, + "content": "for VGG19. As seen, in each run,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 449, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 463 + ], + "score": 1.0, + "content": "the regularization-based way also significantly surpasses its one-shot counterpart. Although five", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 459, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 104, + 459, + 506, + 474 + ], + "score": 1.0, + "content": "random runs are still too few given the exploding potential combinations, yet as shown by the ac-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "curacy standard deviations, the results are stable and thus qualified to support our finding that the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 483, + 408, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 408, + 495 + ], + "score": 1.0, + "content": "regularization-based pruning schedule is better to the one-shot counterpart.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 415, + 506, + 495 + ] + }, + { + "type": "table", + "bbox": [ + 106, + 551, + 508, + 623 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 513, + 506, + 546 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 469, + 525 + ], + "score": 1.0, + "content": "Table 10: Comparison between pruning schedules: one-shot vs. GReg-1. Pruning ratio is", + "type": "text" + }, + { + "bbox": [ + 469, + 514, + 489, + 524 + ], + "score": 0.86, + "content": "90 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 165, + 537 + ], + "score": 1.0, + "content": "ResNet56 and", + "type": "text" + }, + { + "bbox": [ + 166, + 524, + 186, + 535 + ], + "score": 0.86, + "content": "70 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "for VGG19. In each run, the weights to prune are picked randomly before the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 536, + 167, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 167, + 548 + ], + "score": 1.0, + "content": "training starts.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "table_body", + "bbox": [ + 106, + 551, + 508, + 623 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 551, + 508, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 508, + 623 + ], + "score": 0.979, + "html": "
ResNet56 + CIFAR10Run #1Run #2Run #3Run #4Run #5Mean±std
Acc. (%, one-shot)Acc. (%, GReg-1, ours)87.5789.2687.0088.9886.2788.7886.7589.4286.6788.9686.85±0.4389.08±0.23
VGG19 + CIFAR100Run #1Run #2Run #3Run #4Run #5Mean±std
Acc. (%,one-shot)Acc. (%, GReg-1, ours)64.5666.6365.0666.5765.0766.8065.0566.8065.4867.1665.04±0.2966.79±0.21
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However, generalization across different environments is known to be hard. A natural solution would be to keep training after deployment in the new environment, but this cannot be done if the new environment offers no reward signal. Our work explores the use of self-supervision to allow the policy to continue training after deployment without using any rewards. While previous methods explicitly anticipate changes in the new environment, we assume no prior knowledge of those changes yet still obtain significant improvements. Empirical evaluations are performed on diverse simulation environments from DeepMind Control suite and ViZDoom, as well as real robotic manipulation tasks in continuously changing environments, taking observations from an uncalibrated camera. Our method improves generalization in 31 out of 36 environments across various tasks and outperforms domain randomization on a majority of environments. + +# 1 INTRODUCTION + +Deep reinforcement learning (RL) has achieved considerable success when combined with convolutional neural networks for deriving actions from image pixels (Mnih et al., 2013; Levine et al., 2016; Nair et al., 2018; Yan et al., 2020; Andrychowicz et al., 2020). However, one significant challenge for real-world deployment of vision-based RL remains: a policy trained in one environment might not generalize to other new environments not seen during training. Already hard for RL alone, the challenge is exacerbated when a policy faces high-dimensional visual inputs. + +A well explored class of solutions is to learn robust policies that are simply invariant to changes in the environment (Rajeswaran et al., 2016; Tobin et al., 2017; Sadeghi & Levine, 2016; Pinto et al., 2017b; Lee et al., 2019). For example, domain randomization (Tobin et al., 2017; Peng et al., 2018; Pinto et al., 2017a; Yang et al., 2019) applies data augmentation in a simulated environment to train a single robust policy, with the hope that the augmented environment covers enough factors of variation in the test environment. However, this hope may be difficult to realize when the test environment is truly unknown. With too much randomization, training a policy that can simultaneously fit numerous augmented environments requires much larger model and sample complexity. With too little randomization, the actual changes in the test environment might not be covered, and domain randomization may do more harm than good since the randomized factors are now irrelevant. Both phenomena have been observed in our experiments. In all cases, this class of solutions requires human experts to anticipate the changes before the test environment is seen. This cannot scale as more test environments are added with more diverse changes. + +Instead of learning a robust policy invariant to all possible environmental changes, we argue that it is better for a policy to keep learning during deployment and adapt to its actual new environment. A naive way to implement this in RL is to fine-tune the policy in the new environment using rewards as supervision (Rusu et al., 2016; Kalashnikov et al., 2018; Julian et al., 2020). However, while it is relatively easy to craft a dense reward function during training (Gu et al., 2017; Pinto & Gupta, 2016), during deployment it is often impractical and may require substantial engineering efforts. + +In this paper, we tackle an alternative problem setting in vision-based RL: adapting a pre-trained policy to an unknown environment without any reward. We do this by introducing self-supervision to obtain “free” training signal during deployment. Standard self-supervised learning employs auxiliary tasks designed to automatically create training labels using only the input data (see Section 2 for details). Inspired by this, our policy is jointly trained with two objectives: a standard RL objective and, additionally, a self-supervised objective applied on an intermediate representation of the policy network. During training, both objectives are active, maximizing expected reward and simultaneously constraining the intermediate representation through self-supervision. During testing / deployment, only the self-supervised objective (on the raw observational data) remains active, forcing the intermediate representation to adapt to the new environment. + +We perform experiments both in simulation and with a real robot. In simulation, we evaluate on two sets of environments: DeepMind Control suite (Tassa et al., 2018) and the CRLMaze ViZDoom (Lomonaco et al., 2019; Wydmuch et al., 2018) navigation task. We evaluate generalization by testing in new environments with visual changes unknown during training. Our method improves generalization in 19 out of 22 test environments across various tasks in DeepMind Control suite, and in all considered test environments on CRLMaze. Besides simulations, we also perform Sim2Real transfer on both reaching and pushing tasks with a Kinova Gen3 robot. After training in simulation, we successfully transfer and adapt policies to 6 different environments, including continuously changing disco lights, on a real robot operating solely from an uncalibrated camera. In both simulation and real experiments, our approach outperforms domain randomization in most environments. + +# 2 RELATED WORK + +Self-supervised learning is a powerful way to learn visual representations from unlabeled data (Vincent et al., 2008; Doersch et al., 2015; Wang & Gupta, 2015; Zhang et al., 2016; Pathak et al., 2016; Noroozi & Favaro, 2016; Zhang et al., 2017; Gidaris et al., 2018). Researchers have proposed to use auxiliary data prediction tasks, such as undoing rotation (Gidaris et al., 2018), solving a jigsaw puzzle (Noroozi & Favaro, 2016), tracking (Wang et al., 2019), etc. to provide supervision in lieu of labels. In RL, the idea of learning visual representations and action at the same time has been investigated (Lange & Riedmiller, 2010; Jaderberg et al., 2016; Pathak et al., 2017; Ha & Schmidhuber, 2018; Yarats et al., 2019; Srinivas et al., 2020; Laskin et al., 2020; Yan et al., 2020). For example, Srinivas et al. (2020) use self-supervised contrastive learning techniques (Chen et al., 2020; Henaff ´ et al., 2019; Wu et al., 2018; He et al., 2020) to improve sample efficiency in RL by jointly training the self-supervised objective and RL objective. However, this has not been shown to generalize to unseen environments. Other works have applied self-supervision for better generalization across environments (Pathak et al., 2017; Ebert et al., 2018; Sekar et al., 2020). For example, Pathak et al. (2017) use a self-supervised prediction task to provide dense rewards for exploration in novel environments. While results on environment exploration from scratch are encouraging, how to transfer a trained policy (with extrinsic reward) to a novel environment remains unclear. Hence, these methods are not directly applicable to the proposed problem in our paper. + +Generalization across different distributions is a central challenge in machine learning. In domain adaptation, target domain data is assumed to be accessible (Geirhos et al., 2018; Tzeng et al., 2017; Ganin et al., 2016; Gong et al., 2012; Long et al., 2016; Sun et al., 2019; Julian et al., 2020). For example, Tzeng et al. (2017) use adversarial learning to align the feature representations in both the source and target domain during training. Similarly, the setting of domain generalization (Ghifary et al., 2015; Li et al., 2018; Matsuura & Harada, 2019) assumes that all domains are sampled from the same meta distribution, but the same challenge remains and now becomes generalization across meta-distributions. Our work focuses instead on the setting of generalizing to truly unseen changes in the environment which cannot be anticipated at training time. + +There have been several recent benchmarks in our setting for image recognition (Hendrycks & Dietterich, 2018; Recht et al., 2018; 2019; Shankar et al., 2019). For example, in Hendrycks & Dietterich (2018), a classifier trained on regular images is tested on corrupted images, with corruption types unknown during training; the method of Hendrycks et al. (2019) is proposed to improve robustness on this benchmark. Following similar spirit, in the context of RL, domain randomization (Tobin et al., 2017; Pinto et al., 2017a; Peng et al., 2018; Ramos et al., 2019; Yang et al., 2019; James et al., 2019) helps a policy trained in simulation to generalize to real robots. For example, Tobin et al. (2017); Sadeghi & Levine (2016) propose to render the simulation environment with random textures and train the policy on top. The learned policy is shown to generalize to real robot manipulation tasks. Instead of deploying a fixed policy, we train and adapt the policy to the new environment with observational data that is naturally revealed during deployment. + +![](images/57b2c63b42fbd5db91f946cce8e82fe594fcd240b2336a571da5151324e65062.jpg) +Figure 1. Left: Training before deployment. Observations are sampled from a replay buffer for off-policy methods and are collected during roll-outs for on-policy methods. We optimize the RL and self-supervised objectives jointly. Right: Policy adaptation during deployment. Observations are collected from the test environment online, and we optimize only the self-supervised objective. + +Test-time adaptation for deep learning is starting to be used in computer vision (Shocher et al., 2017; 2018; Bau et al., 2019; Mullapudi et al., 2019; Sun et al., 2020; Wortsman et al., 2018). For example, Shocher et al. (2018) shows that image super-resolution can be learned at test time (from scratch) simply by trying to upsample a downsampled version of the input image. Bau et al. (2019) show that adapting the prior of a generative adversarial network to the statistics of the test image improves photo manipulation tasks. Our work is closely related to the test-time training method of Sun et al. (2020), which performs joint optimization of image recognition and self-supervised learning with rotation prediction (Gidaris et al., 2018), then uses the self-supervised objective to adapt the representation of individual images during testing. Instead of image recognition, we perform test-time adaptation for RL with visual inputs in an online fashion. As the agent interacts with an environment, we keep obtaining new observational data in a stream for training the visual representations. + +# 3 METHOD + +In this section, we describe our proposed Policy Adaptation during Deployment (PAD) approach. It can be implemented on top of any policy network and standard RL algorithm (both on-policy and off-policy) that can be described by minimizing some RL objective $J ( \theta )$ w.r.t. the collection of parameters $\theta$ using stochastic gradient descent. + +# 3.1 NETWORK ARCHITECTURE + +We design the network architecture to allow the policy and the self-supervised prediction to share features. For the collection of parameters $\theta$ of a given policy network $\pi$ , we split it sequentially into $\theta = \left( \theta _ { e } , \theta _ { a } \right)$ , where $\theta _ { e }$ collects the parameters of the feature extractor, and $\theta _ { a }$ is the head that outputs a distribution over actions. We define networks $\pi _ { e }$ with parameters $\theta _ { e }$ and $\pi _ { a }$ with parameters $\theta _ { a }$ such that $\pi ( \mathbf { s } ; \theta ) = \pi _ { a } ( \pi _ { e } ( \mathbf { s } ) )$ , where s represents an image observation. Intuitively, one can think of $\pi _ { e }$ as a feature extractor, and $\pi _ { a }$ as a controller based on these features. The goal of our method is to update $\pi _ { e }$ at test-time using gradients from a self-supervised task, such that $\pi _ { e }$ (and consequently $\pi _ { \theta }$ ) can generalize. Let $\pi _ { s }$ with parameters $\theta _ { s }$ be the self-supervised prediction head and its collection of parameters, and the input to $\pi _ { s }$ be the output of $\pi _ { e }$ (as illustrated in Figure 1). In this work, the self-supervised task is inverse dynamics prediction for control, and rotation prediction for navigation. + +# 3.2 INVERSE DYNAMICS PREDICTION AND ROTATION PREDICTION + +At each time step, we always observe a transition sequence in the form of $\left( \mathbf { s } _ { t } , \mathbf { a } _ { t } , \mathbf { s } _ { t + 1 } \right)$ , during both training and testing. Naturally, self-supervision can be derived from taking parts of the sequence and predicting the rest. An inverse dynamics model takes the states before and after transition, and predicts the action in between. In this work, the inverse dynamics model $\pi _ { s }$ operates on the feature space extracted by $\pi _ { e }$ . We can write the inverse dynamics prediction objective formally as + +$$ +L ( \theta _ { s } , \theta _ { e } ) = \ell \big ( \mathbf { a } _ { t } , \pi _ { s } ( \pi _ { e } ( \mathbf { s } _ { t } ) , \pi _ { e } ( \mathbf { s } _ { t + 1 } ) ) \big ) . +$$ + +For continuous actions, $\ell$ is the mean squared error between the ground truth and the model output. For discrete actions, the output is a soft-max distribution over the action space, and $\ell$ is the crossentropy loss. Empirically, we find this self-supervised task to be most effective with continuous actions, possibly because inverse dynamics prediction in a small space of discrete actions is not as challenging. Note that we predict the inverse dynamics instead of the forward dynamics, because when operating in feature space, the latter can produce trivial solutions such as the constant zero feature for every state2. If we instead performed prediction with forward dynamics in pixel space, the task would be extremely challenging given the large uncertainty in pixel prediction. + +As an alternative self-supervised task, we use rotation prediction (Gidaris et al., 2018). We rotate an image by one of 0, 90, 180 and 270 degrees as input to the network, and cast this as a four-way classification problem to determine which one of these four ways the image has been rotated. This task is shown to be effective for learning representations for object configuration and scene structure, which is beneficial for visual recognition (Hendrycks et al., 2019; Doersch & Zisserman, 2017). + +# 3.3 TRAINING AND TESTING + +Before deployment of the policy, because we have signals from both the reward and self-supervised auxiliary task, we can train with both in the fashion of multi-task learning. This corresponds to the following optimization problem during training $\begin{array} { r } { \operatorname* { m i n } _ { \theta _ { a } , \theta _ { s } , \theta _ { e } } J ( \theta _ { a } , \theta _ { e } ) + \alpha L ( \theta _ { s } , \theta _ { e } ) } \end{array}$ , where $\alpha > 0$ is a trade-off hyperparameter. During deployment, we cannot optimize $J$ anymore since the reward is unavailable, but we can still optimize $L$ to update both $\theta _ { s }$ and $\theta _ { e }$ . Empirically, we find only negligible difference with keeping $\theta _ { s }$ fixed at test-time, so we update both since the gradients have to be computed regardless; we ablate this decision in appendix C. As we obtain new images from the stream of visual inputs in the environment, $\theta$ keeps being updated until the episode ends. This corresponds to, for each iteration $t = 1 . . . T$ : + +$$ +\begin{array} { r l } & { \quad \mathbf s _ { t } \sim p ( \mathbf s _ { t } | \mathbf a _ { t - 1 } , \mathbf s _ { t - 1 } ) } \\ & { \quad \theta _ { s } ( t ) = \theta _ { s } ( t - 1 ) - \nabla _ { \theta _ { s } } L ( \mathbf s _ { t } ; \theta _ { s } ( t - 1 ) , \theta _ { e } ( t - 1 ) ) } \\ & { \quad \theta _ { e } ( t ) = \theta _ { e } ( t - 1 ) - \nabla _ { \theta _ { e } } L ( \mathbf s _ { t } ; \theta _ { s } ( t - 1 ) , \theta _ { e } ( t - 1 ) ) } \\ & { \quad \mathbf a _ { t } = \pi ( \mathbf s _ { t } ; \theta ( t ) ) \mathrm { ~ w i t h ~ } \theta ( t ) = ( \theta _ { e } ( t ) , \theta _ { a } ) , } \end{array} +$$ + +where $\theta _ { s } ( 0 ) = \theta _ { s } , \theta _ { e } ( 0 ) = \theta _ { e }$ , $\mathbf { s } _ { 0 }$ is the initial condition given by the environment, $\mathbf { a } _ { 0 } = \pi _ { \theta } ( \mathbf { s } _ { 0 } )$ , $p$ is the unknown environment transition, and $L$ is the self-supervised objective as previously introduced. + +# 4 EXPERIMENTS + +In this work, we investigate how well an agent trained in one environment (denoted the training environment) generalizes to unseen and diverse test environments. During evaluation, agents have no access to reward signals and are expected to generalize without trials nor prior knowledge about the test environments. In simulation, we evaluate our method (PAD) and baselines extensively on continuous control tasks from DeepMind Control (DMControl) suite (Tassa et al., 2018) as well as the CRLMaze (Lomonaco et al., 2019) navigation task, and experiment with both stationary (colors, objects, textures, lighting) and non-stationary (videos) environment changes. We further show that PAD transfers from simulation to a real robot and successfully adapts to environmental differences during deployment in two robotic manipulation tasks. Samples from DMControl and CRLMaze environments are shown in Figure 2, and samples from the robot experiments are shown in Figure 4. Implementation is available at https://nicklashansen.github.io/PAD/. + +Network details. For DMControl and the robotic manipulation tasks we implement PAD on top of Soft Actor-Critic (SAC) (Haarnoja et al., 2018), and adopt both network architecture and hyperparameters from Yarats et al. (2019), with minor modifications: the feature extractor $\pi _ { e }$ has 8 convolutional layers shared between the RL head $\pi _ { a }$ and self-supervised head $\pi _ { s }$ , and we split the network into architecturally identical heads following $\pi _ { e }$ . Each head consists of 3 convolutional layers followed by 4 fully connected layers. For CRLMaze, we use Advantage Actor-Critic (A2C) as base algorithm (Mnih et al., 2016) and apply the same architecture as for the other experiments, but implement $\pi _ { e }$ with only 6 convolutional layers. Observations are stacks of $k$ colored frames $k = 3$ on DMControl and CRLMaze; $k = 1$ in robotic manipulation) of size $1 0 0 \times 1 0 0$ and time-consistent random crop is applied as in Srinivas et al. (2020). During deployment, we optimize the self-supervised objective online w.r.t. $\theta _ { e } , \theta _ { s }$ for one gradient step per time iteration. See appendix F for implementation details. + +![](images/bc62cee880a17c8b0b2832092fcc074bd1b01dcd8fb62086e20850e09ae21b15.jpg) +Figure 2. Left: Training environments of DMControl (top) and CRLMaze (bottom). Right: Test environments of DMControl (top) and CRLMaze (bottom). Changes to DMControl include randomized colors, video backgrounds, and distractors; changes to CRLMaze include textures and lighting. + +Table 1. Episodic return in test environments with randomized colors, mean and std. dev. for 10 seeds. Best method on each task is in bold and blue compares $\mathrm { S A C + I D M }$ with and without PAD. + +10x episode length + +
Random colorsSAC+DR+IDM+IDM (PAD)+IDM+IDM (PAD)
Walker, walk414±74594±104406±29468±473830±5475505±592
Walker, stand719±74715±96743±37797±467832±2098566±121
Cartpole, swingup592±50647±48585±73630±636528±5397093±592
Cartpole,balance857±60867±37835±40848±297746±5267670±293
Ball in cup,catch411±183470±252471±75563±50
Finger, spin626±163465±314757±62803±727249±6427496±655
Finger, turn_easy270±43167±26283±51304±461
Cheetah, run154±41145±29121±38159±281117±5301208±487
Reacher, easy163±45105±37201±32214±441788±4412152±506
+ +# 4.1 DEEPMIND CONTROL + +DeepMind Control (DMControl) (Tassa et al., 2018) is a collection of continuous control tasks where agents only observe raw pixels. Generalization benchmarks on DMControl represent diverse real-world tasks for motor control, and contain distracting surroundings not correlated with the reward signals. + +Experimental setup. We experiment with 9 tasks from DMControl and measure generalization to four types of test environments: (i) randomized colors; (ii) natural videos as background; (iii) distracting objects placed in the scene; and (iv) the unmodified training environment. For each test environment, we evaluate methods across 10 seeds and 100 random initializations. If a given test environment is not applicable to certain tasks, e.g. if a task has no background for the video background setting, they are excluded. Tasks are selected on the basis of diversity, as well as the success of vision-based RL in prior work (Yarats et al., 2019; Srinivas et al., 2020; Laskin et al., 2020; Kostrikov et al., 2020). We implement PAD on top of SAC and use an Inverse Dynamics Model (IDM) for self-supervision, as we find that learning a model of the dynamics works well for motor control. For completeness, we ablate the choice of self-supervision. Learning curves are provided in appendix B. + +![](images/1d57abcc32301ede5d2b8455a03059fd30b9cd73bdc63abef544626395d0fffc.jpg) +Figure 3. Relative improvement in instantaneous reward over time for PAD on the random color env. + +We compare our method to the following baselines: (i) SAC with no changes (denoted $S A C$ ); (ii) SAC trained with domain randomization on a fixed set of 100 colors (denoted $+ D R$ ); and (iii) SAC trained jointly with an IDM but without PAD (denoted $+ I D M )$ ). Our method using an IDM with PAD is denoted by $+ I D M \left( P A D \right)$ . For domain randomization, colors are sampled from the same distribution as in evaluation, but with lower variance, as we find that training directly on the test distribution does not converge. + +Random perturbation of color. Robustness to subtle changes such as color is essential to realworld deployment of RL policies. We evaluate generalization on a fixed set of 100 colors of foreground, background and the agent itself, and report the results in Table 1 (first 4 columns). We find PAD to improve generalization in all tasks considered, outperforming SAC trained with domain randomization in 6 out of 9 tasks. Surprisingly, despite a substantial overlap between training and test domains of domain randomization, it generalizes no better than vanilla SAC on a majority of tasks. + +Long-term stability. We find the relative improvement of PAD to improve over time, as shown in Figure 3. To examine the long-term stability of PAD, we further evaluate on $1 0 \mathrm { x }$ episode lengths and summarize the results in the last two columns in Table 1 (goal-oriented tasks excluded). While we do not explicitly prevent the embedding from drifting away from the RL task, we find empirically that PAD does not degrade the performance of the policy, even over long horizons, and when PAD does not improve, we find it to hurt minimally. We conjecture this is because we are not learning a new task, but simply continue to optimize the same (self-supervised) objective as during joint training, where both two tasks are compatible. In this setting, PAD still improves generalization in 6 out of 7 tasks, and thus naturally extends beyond episodic deployment. For completeness, we also evaluate methods in the environment in which they were trained, and report the results in appendix A. We find that, while PAD improves generalization to novel environments, performance is virtually unchanged on the training environment. We conjecture this is because the self-supervised task is already fully learned and any continued training on the same data distribution thus has little impact. + +Non-stationary environments. To investigate whether PAD can adapt in non-stationary environments, we evaluate generalization to diverse video backgrounds (refer to Figure 2). We find PAD to outperform all baselines on 7 out of 8 tasks, as shown in Table 2, by as much as $104 \%$ over domain randomization on Finger, spin. Domain randomization generalizes comparably worse to videos, which we conjecture is not because the environments are non-stationary, but rather because the image statistics of videos are not covered by its training domain of randomized colors. In fact, domain randomization is outperformed by the vanilla SAC in most tasks with video back + +Table 2. Episodic return in test environments with video backgrounds (top) and distracting objects (bottom), mean and std. dev. for 10 seeds. Best method on each task is in bold and blue compares $\mathrm { S A C + I D M }$ with and without PAD. + +
Video backgroundsSAC+DR+IDM+IDM (PAD)
Walker, walk616±80655±55694±85717±79
Walker, stand899±53869±60902±51935±20
Cartpole, swingup375±90485±67487±90521±76
Cartpole, balance693±109766±92691±76687±58
Ball in cup, catch393±175271±189362±69436±55
Finger, spin447±102338±207605±61691±80
Finger, turn_easy355±108223±91355±110362±101
Cheetah, run194±30150±34164±42206±34
Distracting objectsSAC+DR+IDM+IDM (PAD)
Cartpole, swingup815±60809±24776±58771±64
Cartpole,balance969±20938±35964±26960±29
Ball in cup, catch177±111331±189482±128545±173
Finger, spin652±184564±288836±62867±72
Finger, turn_easy302±68165±12326±101347±48
+ +grounds, which is in line with the findings of Packer et al. (2018). + +Scene content. We hypothesize that: (i) an agent trained with an IDM is comparably less distracted by scene content since objects uncorrelated to actions yield no predictive power; and (ii) that PAD can adapt to unexpected objects in the scene. We test these hypotheses by measuring robustness to colored shapes at a variety of positions in both the foreground and background of the scene (no physical interaction). Results are summarized in Table 2. PAD outperforms all baselines in 3 out of 5 tasks, with a relative improvement of $20 \%$ over SAC on Ball in cup, catch. In the two cartpole tasks in which PAD does not improve, all methods are already relatively unaffected by the distractors. + +Choice of self-supervised task. We investigate how much the choice of self-supervised task contributes to the overall success of our method, and consider the following ablations: (i) replacing inverse dynamics with the rotation prediction task described in Section 3.2; and (ii) replacing it with the recently proposed CURL (Srinivas et al., 2020) contrastive learning algorithm for RL. As shown in Table 3, PAD improves generalization of CURL in a majority of tasks on the randomized color benchmark, and in 4 out of 9 tasks using rotation prediction. However, inverse dynamics as auxiliary task produces more consistent results and offers better generalization overall. We argue that learning an IDM produces better representations for motor control since it connects observations directly to actions, whereas CURL and rotation prediction operates purely on observations. In general, we find the improvement of PAD to be bigger in tasks that benefit significantly from visual information (see appendix A), and conjecture that selecting a self-supervised task that learns features useful to the RL task is crucial to the success of PAD, which we discuss further in Section 4.2. + +Table 3. Ablations on the randomized color domain of DMC. All methods use SAC. CURL represents RL with a contrastive learning task (Srinivas et al., 2020) and Rot represents the rotation prediction (Gidaris et al., 2018). Offline PAD is here denoted O-PAD for brevity, whereas the default usage of PAD is in an online setting. Best method is in bold and blue compares $+ \mathrm { I D M }$ w/ and w/o PAD. + +
Random colorsCURLCURL (PAD)RotRot (PAD)IDMIDM (O-PAD)IDM (PAD)
Walker, walk445±99495±70335±7330±30406±29441±16468±47
Walker, stand662±54753±49673±4653±27743±37727±21797±46
Cartpole, swingup454±110413±67493±52477±38585±73578±69630±63
Cartpole,balance782±13763±5710±72734±81835±40796±37848±29
Ball in cup, catch231±92332±78291±54314±60471±75490±16563±50
Finger, spin691±12588±22695±36689±20757±62767±43803±72
Finger, turn_easy202±32186±2283±68230±53283±51321±10304±46
Cheetah, run202±22211±20127±3135±12121±38112±35159±28
Reacher, easy325±32378±6299±29120±7201±32241±24214±44
+ +Table 4. Episodic return of PAD and baselines in CRLMaze environments. PAD improves generalization in all considered environments and outperforms both A2C and domain randomization by a large margin. All methods use A2C. We report mean and std. error of 10 seeds. Best method in each environment is in bold and blue compares rotation prediction with and without PAD. + +
CRLMazeRandomA2C+DR+IDM+IDM (PAD)+Rot+Rot (PAD)
Walls-870±30-380±145-260±137-302±150-428±135-206±166-74±116
Floor-868±23-320±167-438±59-47±198-530±106-294±123-209±94
Ceiling-872±30-171±175-400±74166±215-508±104128±196281±83
Lights-900±29-30±213-310±106239±270-460±114-84±53312±104
+ +Offline versus online learning. Observations that arrive sequentially are highly correlated, and we thus hypothesize that our method benefits significantly from learning online. To test this hypothesis, we run an offline variant of our method in which network updates are forgotten after each step. In this setting, our method can only adapt to single observations and does not benefit from learning over time. Results are shown in Table 3. We find that our method benefits substantially from online learning, but learning offline still improves generalization on select tasks. + +# 4.2 CRLMAZE + +CRLMaze (Lomonaco et al., 2019) is a time-constrained, discrete-action 3D navigation task for ViZDoom (Wydmuch et al., 2018), in which an agent is to navigate a maze and collect objects. There is a positive reward associated with green columns, and a negative reward for lanterns as well as for living. Readers are referred to the respective papers for details on the task and environment. + +Experimental setup. We train agents on a single environment and measure generalization to environments with novel textures for walls, floor, and ceiling, as well as lighting, as shown in Figure 2. We implement PAD on top of A2C (Mnih et al., 2016) and use rotation prediction (see Section 3.2) as self-supervised task. Learning to navigate novel scenes requires a generalized scene understanding, and we find that rotation prediction facilitates that more so than an IDM. We compare to the following baselines: (i) a random agent (denoted Random); (ii) A2C with no changes (denoted $A 2 C$ ); (iii) A2C trained with domain randomization (denoted $+ D R$ ); (iv) A2C with an IDM as auxiliary task (denoted $+ I D M )$ ; and (v) A2C with rotation prediction as auxiliary task (denoted $+ R o t )$ . We denote Rot with PAD as $+ R o t \ ( P A D )$ . Domain randomization uses 56 combinations of diverse textures, partially overlapping with the test distribution, and we find it necessary to train domain randomization for twice as many episodes in order to converge. We closely follow the evaluation procedure of (Lomonaco et al., 2019) and evaluate methods across 20 starting positions and 10 random seeds. + +Results. We report performance on the CRLMaze environments in Table 4. PAD improves generalization in all considered test environments, outperforming both A2C and domain randomization by a large margin. Domain randomization performs consistently across all environments but is less successful overall. We further examine the importance of selecting appropriate auxiliary tasks by a simple ablation: replacing rotation prediction with an IDM for the navigation task. We conjecture that, while an auxiliary task can enforce structure in the learned representations, its features (and consequently gradients) need to be sufficiently correlated with the primary RL task for PAD to be + +![](images/7a75facacf5c2ac9f0591865730aa213a149b2aae09f6d5a1610fc3b0f899d74.jpg) +(a) Simulation. (b) Default transfer. (c) Table cloth. (d) Disco lights. + +Figure 4. Samples from the push robotic manipulation task. The task is to push the yellow cube to the location of the red disc. Agents are trained in setting (a) and evaluated in settings (b-d). + +successful during deployment. While PAD with rotation prediction improves generalization across all test environments considered, IDM does not, which suggests that rotation prediction is more suitable for tasks that require scene understanding, whereas IDM is useful for tasks that require motor control. We leave it to future work to automate the process of selecting appropriate auxiliary tasks. + +# 4.3 ROBOTIC MANIPULATION TASKS + +We deploy our method and baselines on a real Kinova Gen3 robot and evaluate on two manipulation tasks: (i) reach, a task in which the robot reaches for a goal marked by a red disc; and (ii) push, a task in which the robot pushes a cube to the location of the red disc. Both tasks use an XY action space, where the Z position of the actuator is fixed. Agents operate purely from pixel observations with no access to state information. During deployment, we make no effort to calibrate camera, lighting, or + +Table 5. Success rate of PAD and baselines on a real robotic arm. Best method in each environment is in bold and blue compares $+ \mathrm { I D M }$ with and without PAD. + +
Real robotSAC+DR+IDM+IDM (PAD)
Reach (default)100%100%100%100%
Reach (cloth)48%80%56%80%
Reach (disco)72%76%88%92%
Push (default)88%88%92%100%
Push (cloth)60%64%64%88%
Push (disco)60%68%72%84%
+ +physical properties such as dimensions, mass, and friction, and policies are expected to generalize with no prior knowledge of the test environment. Samples from the push task are shown in Figure 4, and samples from reach are shown in appendix E. + +Experimental setup. We implement PAD on top of SAC (Haarnoja et al., 2018) and apply the same experimental setup as in Section 4.1 using an Inverse Dynamics Model (IDM) for self-supervision, but without frame-stacking (i.e. $k = 1$ ). Agents are trained in simulation with dense rewards and randomized initial configurations of arm, goal, and box, and we measure generalization to 3 novel environments in the real-world: (i) default environment with pixel observations that roughly mimic the simulation; (ii) a patterned table cloth that + +Table 6. Success rate of PAD and baselines for the push task on a simulated robotic arm in test environments with changes to dynamics. Changes include object mass, size, and friction, arm mount position, and end effector velocity. Best method in each environment is in bold and blue compares $+ \mathrm { I D M }$ with and without PAD. + +
Simulated robotSAC+DR+IDM+IDM (PAD)
Push (object)66%64%72%82%
Push (mount)68%58%86%84%
Push (velocity)70%68%70%78%
Push (all)56%50%48%76%
+ +distracts visually and greatly increases friction; and (iii) disco, an environment with non-stationary visual disco light distractions. Notably, all 3 environments also feature subtle differences in dynamics compared to the training environment, such as object dimensions, mass, friction, and uncalibrated actions. In each setting, we evaluate the success rate across 25 test runs spanning across 5 pre-defined goal locations throughout the table. The goal locations vary between the two tasks, and the robot is reset after each run. We perform comparison against direct transfer and domain randomization baselines as in Section 4.1. We further evaluate generalization to changes in dynamics by considering a variant of the simulated environment in which object mass, size, and friction, arm mount position, and end effector velocity is modified. We consider each setting both individually and jointly, and evaluate success rate across 50 unique configurations with the robot reset after each run. + +Results. We report transfer results in Table 5. While all methods transfer successfully to reach (default), we observe PAD to improve generalization in all settings in which the baselines show sub-optimal performance. We find PAD to be especially powerful for the push task that involves dynamics, improving by as much as $24 \%$ in push (cloth). While domain randomization proves highly effective in reach (cloth), we observe no significant benefit in the other settings, which suggests that PAD can be more suitable in challenging tasks like push. To isolate the effect of dynamics, we further evaluate generalization to a number of simulated changes in dynamics on the push task. Results are shown in Table 6. We find PAD to improve generalization to changes in the physical properties of the object and end effector, whereas both $S A C { + } I D M$ and PAD are relatively unaffected by changes to the mount position. Consistent with the real robot results in Section 5, PAD is found to be most effective when changes in dynamics are non-trivial, improving by as much as $28 \%$ in the push (all) setting, where all 3 environmental changes are considered jointly. These results suggest that PAD can be a simple, yet effective method for generalization to diverse, unseen environments that vary in both visuals and dynamics. + +# 5 CONCLUSION + +While previous work addresses generalization in RL by learning policies that are invariant to any environment changes that can be anticipated, we formulate an alternative problem setting in visionbased RL: can we instead adapt a pretrained-policy to new environments without any reward. We propose Policy Adaptation during Deployment, a self-supervised framework for online adaptation at test-time, and show empirically that our method improves generalization of policies to diverse simulated and real-world environmental changes across a variety of tasks. We find our approach benefits greatly from learning online, and we systematically evaluate how the choice of self-supervised task impacts performance. While the current framework relies on prior knowledge on selecting selfsupervised tasks for policy adaptation, we see our work as the initial step in addressing the problem of adapting vision-based policies to unknown environments. We ultimately envision embodied agents in the future to be learning all the time, with the flexibility to learn both with and without rewards, before and during deployment. + +Acknowledgements. This work was supported, in part, by grants from DARPA, NSF 1730158 CI-New: Cognitive Hardware and Software Ecosystem Community Infrastructure (CHASE-CI), NSF ACI-1541349 CC\*DNI Pacific Research Platform, and gifts from Qualcomm and TuSimple. This work was also funded, in part, by grants from Berkeley DeepDrive, SAP and European Research Council (ERC) from the European Union Horizon 2020 Programme under grant agreement no. 741930 (CLOTHILDE). We would like to thank Fenglu Hong and Joey Hejna for helpful discussions. + +# REFERENCES + +OpenAI: Marcin Andrychowicz, Bowen Baker, Maciek Chociej, Rafal Jozefowicz, Bob McGrew, Jakub Pachocki, Arthur Petron, Matthias Plappert, Glenn Powell, Alex Ray, et al. Learning dexterous in-hand manipulation. The International Journal of Robotics Research, 39(1):3–20, 2020. 1 +David Bau, Hendrik Strobelt, William Peebles, Jonas Wulff, Bolei Zhou, Jun-Yan Zhu, and Antonio Torralba. Semantic photo manipulation with a generative image prior. ACM Trans. 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In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1058–1067, 2017. 2 + +# A PERFORMANCE ON THE TRAINING ENVIRONMENT + +Historically, agents have commonly been trained and evaluated in the same environment when benchmarking RL algorithms exclusively in simulation. Although such an evaluation procedure does not consider generalization, it is still a useful metric for comparison of sample efficiency and stability of algorithms. For completeness, we also evaluate our method and baselines in this setting on both DMControl and CRLMaze. DMControl results are reported in Table 7 and results on the CRLMaze environment are shown in Table 8. In this setting, we also compare to an additional baseline on DMControl: a blind SAC agent that operates purely on its previous actions. The performance of a blind agent indicates to which degree a given task benefits from visual information. We find that, while PAD improves generalization to novel environments, performance is virtually unchanged when evaluated on the same environment as in training. We conjecture that this is because the algorithm already is adapted to the training environment and any continued training on the same data distribution thus has little influence. We further emphasize that, even when evaluated on the training environment, PAD still outperforms baselines on most tasks. For example, we observe a $15 \%$ relative improvement over SAC on the Finger, spin task. We hypothesize that this gain in performance is because the selfsupervised objective improves learning by constraining the intermediate representation of policies. A blind agent is no better than random on this particular task, which would suggest that agents benefit substantially from visual information in Finger, spin. Therefore, learning a good intermediate representation of that information is highly beneficial to the RL objective, which we find PAD to facilitate through its self-supervised learning framework. Likewise, the SAC baseline only achieves a $51 \%$ improvement over the blind agent on Cartpole, balance, which indicates that extracting visual information from observations is not as crucial on this task. Consequently, both PAD and baselines achieve similar performance on this task. + +Table 7. Episodic return on the training environment for each of the 9 tasks considered in DMControl, mean and std. dev. for 10 seeds. Best method on each task is in bold and blue compares $+ \mathrm { I D M }$ with and without PAD. It is shown that PAD hurts minimally when the environment is unchanged. + +
Training env.BlindSAC+DR+IDM+IDM (PAD)
Walker, walk235±17847±71756±71911±24895±28
Walker, stand388±10959±11928±36966±8956±20
Cartpole, swingup132±41850±28807±36849±30845±34
Cartpole,balance646±131978±22971±30982±20979±21
Ball in cup, catch150±96725±355469±339919±118910±129
Finger, spin3±2809±138686±295928±45927±45
Finger, turn_easy172±27462±146243±124462±152455±160
Cheetah, run264±75387±74195±46384±88380±91
Reacher, easy107±11264±11392±45390±126365±114
+ +Table 8. Episodic return of PAD and baselines in the CRLMaze training environment. All methods use A2C. We report mean and std. error of 10 seeds. Best method is in bold and blue compares rotation prediction with and without PAD. + +
CRLMazeRandomA2C+DR+IDM+IDM (PAD)+Rot+Rot (PAD)
Training env.-868±34371±198-355±93585±246-416±135729±148681±99
+ +# B LEARNING CURVES ON DEEPMIND CONTROL + +All methods are trained until convergence (500,000 frames) on DMControl. While we do not consider the sample efficiency of our method and baselines in this study, we report learning curves for SAC, $\mathrm { S A C + I D M }$ and SAC trained with domain randomization on three tasks in Figure 5 for completeness. SAC trained with and without an IDM are similar in terms of sample efficiency and final performance, whereas domain randomization consistently displays worse sample efficiency, larger variation between seeds, and converges to sub-optimal performance in two out of the three tasks shown. + +![](images/9988caa8b9a55fc3afee7ef0f724429fd455ed6f365e57a9ccd860a250478099.jpg) +Figure 5. Learning curves for SAC, SAC trained with domain randomization (denoted $S A C ( D R )$ here), and $\mathrm { S A C + I D M }$ on three tasks from the DeepMind Control suite (DMControl). Episodic return is averaged across 10 seeds and the $9 5 \%$ confidence intervals are visualized as shaded regions. SAC and $\mathrm { S A C + I D M }$ exhibit similar sample efficiency and final performance, whereas domain randomization consistently displays worse sample efficiency, larger variation between seeds, and converges to sub-optimal performance in two out of the three tasks shown. + +# C KEEPING $\pi _ { s }$ FIXED DURING POLICY ADAPTATION + +We now consider a variant of PAD where the self-supervised task head $\pi _ { s }$ is fixed at test-time such that the self-supervised objective $L$ is optimized only wrt $\pi _ { e }$ , as discussed in Section 3.3. We measure generalization to test environments with randomized colors and report the results in Table 9 for three tasks from the DeepMind Control suite. We empirically find the difference between updating $\pi _ { s }$ and keeping it fixed negligible, and we choose to update $\pi _ { s }$ by default since its gradients are computed by back-propagation regardless. + +Table 9. Episodic return in test environments with randomized colors, mean and std. dev. for 10 seeds. All methods use SAC. IDM (PAD, fixed $\pi _ { s . }$ ) considers a variant of PAD where $\pi _ { s }$ is fixed at test-time, whereas $I D M \left( P A D \right)$ denotes the default usage of PAD in which both $\pi _ { e }$ and $\pi _ { s }$ are optimized at test-time using the self-supervised objective. + +
Random colorsIDMIDM (PAD, fixed π s)IDM (PAD)
Walker, walk406±29452±38468±47
Walker, stand743±37802±41797±46
Cartpole, swingup585±73623±57630±63
+ +# D COMPARISON TO ADAPTATION WITH REWARDS + +While our method does not require data collected prior to deployment and does not assume access to a reward signal, we additionally compare our method to a na¨ıve fine-tuning approach using transitions and rewards collected from the target environment prior to deployment. To fine-tune the pre-trained policy using rewards, we collect datasets consisting of 1, 10, and 100 episodes in each target environment using the learned policy while keeping its parameters fixed, and then subsequently fine-tune both $\pi _ { e }$ and $\pi _ { a }$ on the collected data, following the same training procedure as during the training phase. This fine-tuning approach is analogous to Julian et al. (2020) but does not use data from the original environment during adaptation. Results are shown in Table 10. We find that na¨ıvely fine-tuning the policy using data collected prior to deployment can improve generalization but requires comparably more data than PAD, as well as access to a reward signal in the target environment. This finding suggests that PAD may be a more suitable method for settings where data from the target environment is scarce and not easily accessible prior to deployment. + +# E ADDITIONAL ROBOTIC MANIPULATION SAMPLES + +Figure 6 provides samples from the training and test environments for the reach robotic manipulation task. Agents are trained in simulation and deployed on a real robot. Samples from the push task are shown in Figure 4. + +Table 10. Episodic return in test environments with randomized colors, mean and std. dev. for 10 seeds. All methods use SAC trained with an inverse dynamics model (IDM) as auxiliary task. Our method is denoted IDM (PAD), and we compare to a na¨ıve fine-tuning approach that assumes access to transitions and rewards collected from 1, 10, and 100 episodes, respectively, from target environments prior to deployment. + +Fine-tuning w/ rewards + +
Random colorsIDMIDM (PAD)1 episode10 episodes100 episodes
Walker, walk406±29468±47395±78489±104561±62
Walker, stand743±37797±46661±65728±44784±31
Cartpole, swingup585±73630±63538±53605±51650±58
+ +![](images/bb8251dc060f2862ca990a7216bf7fd39204605009caf6d0e9e8d640e1c1e5a1.jpg) +Figure 6. Samples from the reach robotic manipulation task. The task is to move the robot gripper to the location of the red disc. Agents are trained in setting (a) and evaluated in settings (b-d) on a real robot, taking observations from an uncalibrated camera. + +# F IMPLEMENTATION DETAILS + +In this section, we elaborate on implementation details for our experiments on DeepMind Control (DMControl) suite (Tassa et al., 2018) and CRLMaze (Lomonaco et al., 2019) for ViZDoom (Wydmuch et al., 2018). Our implementation for the robotic manipulation experiments closely follows that of DMControl. Code is available at https://nicklashansen.github.io/PAD/. + +![](images/9e0be3b5d460fc2992f725ba0dd7adde3d5e70bc06c843e908bf8d4d2266f9a5.jpg) +Figure 7. Network architecture for the DMControl, CRLMaze, and robotic manipulation experiments. $\pi ^ { s }$ and $\pi ^ { a }$ uses a shared feature extractor $\pi ^ { e }$ . Observations are stacks of $1 0 0 \times 1 0 0$ colored frames. Implementation of policy and value function depends on the learning algorithm. + +Architecture. Our network architecture is illustrated in Figure 7. Observations are stacked frames $k = 3 ,$ ) rendered at $1 0 0 \times 1 0 0$ and cropped to $8 4 \times 8 4$ , i.e. inputs to the network are of dimensions $9 \times 8 4 \times 8 4$ , where the first dimension indicates the channel numbers and the following ones represent spatial dimensions. The same crop is applied to all frames in a stack. The shared feature extractor $\pi ^ { e }$ consists of 8 (DMControl, robotic manipulation) or 6 (CRLMaze) convolutional layers and outputs features of size $3 2 \times 2 1 \times 2 1$ in DMControl and robotic manipulation, and size $3 2 \times 2 5 \times 2 5$ in CRLMaze. The output from $\pi ^ { e }$ is used as input to both the self-supervised head $\pi ^ { s }$ and RL head $\pi ^ { a }$ , both of which consist of 3 convolutional layers followed by 3 fully-connected layers. All convolutional layers use 32 filters and all fully connected layers use a hidden size of 1024, as in Yarats et al. (2019). + +Table 11. Hyperparameters used for the DMControl (Tassa et al., 2018) tasks. + +
HyperparameterValue
Frame rendering3 ×100×100
Frame after crop3×84×84
Stacked frames3
Action repeat2 (finger)
8 (cartpole) 4(otherwise)
Discount factor y0.99
Episode length1,000
Learning algorithmSoft Actor-Critic
Self-supervised taskInverse Dynamics Model
Number of training steps500,000
Replay buffer size500,000
Optimizer(πe,πä,π)Adam (β=0.9,β=0.999)
Optimizer (α)Adam(β=0.5,β=0.999)
Learning rate (πe,π,π$)3e-4 (cheetah)
Learning rate (α)le-3 (otherwise) 1e-4
Batch size128
Batch size (test-time)32
πe,π update freq.2
πe,π update freq. (test-time)1
+ +Table 12. Hyperparameters used for the CRLMaze (Lomonaco et al., 2019) navigation task. + +
HyperparameterValue
Frame rendering3×100×100
Frame after crop3×84×84
Stacked frames3
Action repeat4
Discount factor y0.99
Episode length1,000
Learning algorithmAdvantage Actor-Critic
Self-supervised taskRotation Prediction
Number of training episodes1,000 (dom. rand.) 500 (otherwise)
Number of processes20
OptimizerAdam (β=0.9,β=0.999)
Learning rate1e-4
Learning rate (test-time)1e-5
Batch size20
32
Batch size (test-time) π,πloss coefficient0.5
1
πe,πloss coefficient (test-time)1
πe,π update freq. πe,π update freq.(test-time)1
+ +Learning algorithm. We use Soft Actor-Critic (SAC) (Haarnoja et al., 2018) for DMControl and robotic manipulation, and Advantage Actor-Critic (A2C) for CRLMaze. Network outputs depend on the task and learning algorithm. As the action spaces of both DMControl and robotic manipulation are continuous, the policy learned by SAC outputs the mean and variance of a Gaussian distribution over actions. CRLMaze has a discrete action space and the policy learned by A2C thus learns a soft-max distribution over actions. For details on the critics learned by SAC and A2C, the reader is referred to Haarnoja et al. (2018) and Mnih et al. (2016), respectively. + +Hyperparameters. When applicable, we adopt our hyperparameters from Yarats et al. (2019) (DMControl, robotic manipulation) and Lomonaco et al. (2019) (CRLMaze). For the robotic manipulation experiments, our implementation closely follows that of DMControl, only differing by number of frames in an observation. We use a frame stack of $k = 3$ frames for DMControl and CRLMaze, and only $k = 1$ frame for robotic manipulation. For completeness, we detail all hyperparameters used for the DMControl and CRLMaze environments in Table 11 and Table 12. + +Data augmentation. Random cropping is a commonly used data augmentation used in computer vision systems (Krizhevsky et al., 2012; Szegedy et al., 2015) but has only recently gained interest as a stochastic regularization technique in the RL literature (Srinivas et al., 2020; Kostrikov et al., 2020; Laskin et al., 2020). We adopt the random crop proposed in Srinivas et al. (2020): crop rendered observations of size $1 0 0 \times 1 0 0$ to $8 4 \times 8 4$ , applying the same crop to all frames in a stacked observation. This has the added benefits of regularization while still preserving spatio-temporal patterns between frames. When learning an inverse dynamics model, we apply the same crop to all frames of a given observation but apply two different crops to the consecutive observations $\left( \mathbf { s } _ { t } , \mathbf { s } _ { t + 1 } \right)$ used to predict action $\mathbf { a } _ { t }$ . + +Policy Adaptation during Deployment. We evaluate our method and baselines by episodic return of an agent trained in a single environment and tested in a collection of test environments, each with distinct changes from the training environment. We assume no reward signal at test-time and agents are expected to generalize without pre-training or resetting in the new environment. Therefore, we make updates to the policy using a self-supervised objective, and we train using observations from the environment in an online manner without memory, i.e. we make one update per step using the most-recent observation. + +Empirically, we find that: (i) the random crop data augmentation used during training helps regularize learning at test-time; and (ii) our algorithm benefits from learning from a batch of randomly cropped observations rather than single observations, even when all observations in the batch are augmented copies of the most-recent observation. As such, we apply both of these techniques when performing Policy Adaptation during Deployment and use a batch size of 32. 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Our method improves general-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 362, + 469, + 375 + ], + "spans": [ + { + "bbox": [ + 141, + 362, + 469, + 375 + ], + "score": 1.0, + "content": "ization in 31 out of 36 environments across various tasks and outperforms domain", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 374, + 328, + 386 + ], + "spans": [ + { + "bbox": [ + 141, + 374, + 328, + 386 + ], + "score": 1.0, + "content": "randomization on a majority of environments.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 108, + 406, + 206, + 418 + ], + "lines": [ + { + "bbox": [ + 105, + 404, + 208, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 208, + 421 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 427, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 506, + 438 + ], + "score": 1.0, + "content": "Deep reinforcement learning (RL) has achieved considerable success when combined with convolu-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "tional neural networks for deriving actions from image pixels (Mnih et al., 2013; Levine et al., 2016;", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 447, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 461 + ], + "score": 1.0, + "content": "Nair et al., 2018; Yan et al., 2020; Andrychowicz et al., 2020). However, one significant challenge", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "for real-world deployment of vision-based RL remains: a policy trained in one environment might", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 471, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 482 + ], + "score": 1.0, + "content": "not generalize to other new environments not seen during training. Already hard for RL alone, the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 482, + 415, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 415, + 494 + ], + "score": 1.0, + "content": "challenge is exacerbated when a policy faces high-dimensional visual inputs.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 499, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "A well explored class of solutions is to learn robust policies that are simply invariant to changes in", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "score": 1.0, + "content": "the environment (Rajeswaran et al., 2016; Tobin et al., 2017; Sadeghi & Levine, 2016; Pinto et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "2017b; Lee et al., 2019). 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However, this hope may be difficult to realize when the test environment", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "is truly unknown. With too much randomization, training a policy that can simultaneously fit", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 574, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 104, + 574, + 506, + 588 + ], + "score": 1.0, + "content": "numerous augmented environments requires much larger model and sample complexity. 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This cannot scale as", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 630, + 354, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 354, + 643 + ], + "score": 1.0, + "content": "more test environments are added with more diverse changes.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 647, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 647, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 647, + 505, + 659 + ], + "score": 1.0, + "content": "Instead of learning a robust policy invariant to all possible environmental changes, we argue that it is", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 659, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 506, + 670 + ], + "score": 1.0, + "content": "better for a policy to keep learning during deployment and adapt to its actual new environment. 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Our method improves general-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 362, + 469, + 375 + ], + "spans": [ + { + "bbox": [ + 141, + 362, + 469, + 375 + ], + "score": 1.0, + "content": "ization in 31 out of 36 environments across various tasks and outperforms domain", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 374, + 328, + 386 + ], + "spans": [ + { + "bbox": [ + 141, + 374, + 328, + 386 + ], + "score": 1.0, + "content": "randomization on a majority of environments.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 13.5, + "bbox_fs": [ + 141, + 231, + 471, + 386 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 406, + 206, + 418 + ], + "lines": [ + { + "bbox": [ + 105, + 404, + 208, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 208, + 421 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 427, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 506, + 438 + ], + "score": 1.0, + "content": "Deep reinforcement learning (RL) has achieved considerable success when combined with convolu-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "tional neural networks for deriving actions from image pixels (Mnih et al., 2013; Levine et al., 2016;", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 447, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 461 + ], + "score": 1.0, + "content": "Nair et al., 2018; Yan et al., 2020; Andrychowicz et al., 2020). However, one significant challenge", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "for real-world deployment of vision-based RL remains: a policy trained in one environment might", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 471, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 482 + ], + "score": 1.0, + "content": "not generalize to other new environments not seen during training. 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For example, domain randomization (Tobin et al., 2017; Peng et al., 2018;", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 529, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 545 + ], + "score": 1.0, + "content": "Pinto et al., 2017a; Yang et al., 2019) applies data augmentation in a simulated environment to train a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "single robust policy, with the hope that the augmented environment covers enough factors of variation", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 554, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 505, + 565 + ], + "score": 1.0, + "content": "in the test environment. However, this hope may be difficult to realize when the test environment", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "is truly unknown. With too much randomization, training a policy that can simultaneously fit", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 574, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 104, + 574, + 506, + 588 + ], + "score": 1.0, + "content": "numerous augmented environments requires much larger model and sample complexity. With too", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 587, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 598 + ], + "score": 1.0, + "content": "little randomization, the actual changes in the test environment might not be covered, and domain", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "randomization may do more harm than good since the randomized factors are now irrelevant. Both", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 608, + 504, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 504, + 620 + ], + "score": 1.0, + "content": "phenomena have been observed in our experiments. In all cases, this class of solutions requires", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "score": 1.0, + "content": "human experts to anticipate the changes before the test environment is seen. This cannot scale as", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 630, + 354, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 354, + 643 + ], + "score": 1.0, + "content": "more test environments are added with more diverse changes.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 498, + 506, + 643 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 647, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 647, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 647, + 505, + 659 + ], + "score": 1.0, + "content": "Instead of learning a robust policy invariant to all possible environmental changes, we argue that it is", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 659, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 506, + 670 + ], + "score": 1.0, + "content": "better for a policy to keep learning during deployment and adapt to its actual new environment. A", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 669, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 505, + 681 + ], + "score": 1.0, + "content": "naive way to implement this in RL is to fine-tune the policy in the new environment using rewards", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 679, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 506, + 691 + ], + "score": 1.0, + "content": "as supervision (Rusu et al., 2016; Kalashnikov et al., 2018; Julian et al., 2020). However, while it", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 690, + 507, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 507, + 703 + ], + "score": 1.0, + "content": "is relatively easy to craft a dense reward function during training (Gu et al., 2017; Pinto & Gupta,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 701, + 493, + 714 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 493, + 714 + ], + "score": 1.0, + "content": "2016), during deployment it is often impractical and may require substantial engineering efforts.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 647, + 507, + 714 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "In this paper, we tackle an alternative problem setting in vision-based RL: adapting a pre-trained", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "policy to an unknown environment without any reward. We do this by introducing self-supervision", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "to obtain “free” training signal during deployment. Standard self-supervised learning employs", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "auxiliary tasks designed to automatically create training labels using only the input data (see Section", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "2 for details). Inspired by this, our policy is jointly trained with two objectives: a standard RL", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "objective and, additionally, a self-supervised objective applied on an intermediate representation of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "score": 1.0, + "content": "the policy network. During training, both objectives are active, maximizing expected reward and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "simultaneously constraining the intermediate representation through self-supervision. During testing /", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 168, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 506, + 185 + ], + "score": 1.0, + "content": "deployment, only the self-supervised objective (on the raw observational data) remains active, forcing", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 182, + 365, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 365, + 193 + ], + "score": 1.0, + "content": "the intermediate representation to adapt to the new environment.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "We perform experiments both in simulation and with a real robot. In simulation, we evaluate", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "on two sets of environments: DeepMind Control suite (Tassa et al., 2018) and the CRLMaze", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "ViZDoom (Lomonaco et al., 2019; Wydmuch et al., 2018) navigation task. We evaluate generalization", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "by testing in new environments with visual changes unknown during training. Our method improves", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "generalization in 19 out of 22 test environments across various tasks in DeepMind Control suite, and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 264 + ], + "score": 1.0, + "content": "in all considered test environments on CRLMaze. Besides simulations, we also perform Sim2Real", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "transfer on both reaching and pushing tasks with a Kinova Gen3 robot. After training in simulation, we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 273, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 289 + ], + "score": 1.0, + "content": "successfully transfer and adapt policies to 6 different environments, including continuously changing", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "disco lights, on a real robot operating solely from an uncalibrated camera. In both simulation and real", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 450, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 450, + 309 + ], + "score": 1.0, + "content": "experiments, our approach outperforms domain randomization in most environments.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 108, + 317, + 211, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 213, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 213, + 332 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 506, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 336, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 506, + 348 + ], + "score": 1.0, + "content": "Self-supervised learning is a powerful way to learn visual representations from unlabeled data (Vin-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 347, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 506, + 358 + ], + "score": 1.0, + "content": "cent et al., 2008; Doersch et al., 2015; Wang & Gupta, 2015; Zhang et al., 2016; Pathak et al., 2016;", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "Noroozi & Favaro, 2016; Zhang et al., 2017; Gidaris et al., 2018). Researchers have proposed to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 367, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 382 + ], + "score": 1.0, + "content": "use auxiliary data prediction tasks, such as undoing rotation (Gidaris et al., 2018), solving a jigsaw", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "puzzle (Noroozi & Favaro, 2016), tracking (Wang et al., 2019), etc. to provide supervision in lieu of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 390, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 506, + 402 + ], + "score": 1.0, + "content": "labels. In RL, the idea of learning visual representations and action at the same time has been investi-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "gated (Lange & Riedmiller, 2010; Jaderberg et al., 2016; Pathak et al., 2017; Ha & Schmidhuber,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 411, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 425 + ], + "score": 1.0, + "content": "2018; Yarats et al., 2019; Srinivas et al., 2020; Laskin et al., 2020; Yan et al., 2020). For example,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 424, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 506, + 435 + ], + "score": 1.0, + "content": "Srinivas et al. (2020) use self-supervised contrastive learning techniques (Chen et al., 2020; Henaff ´", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 433, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 448 + ], + "score": 1.0, + "content": "et al., 2019; Wu et al., 2018; He et al., 2020) to improve sample efficiency in RL by jointly training", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 446, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 457 + ], + "score": 1.0, + "content": "the self-supervised objective and RL objective. However, this has not been shown to generalize to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 457, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 468 + ], + "score": 1.0, + "content": "unseen environments. Other works have applied self-supervision for better generalization across envi-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 467, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 479 + ], + "score": 1.0, + "content": "ronments (Pathak et al., 2017; Ebert et al., 2018; Sekar et al., 2020). For example, Pathak et al. (2017)", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "use a self-supervised prediction task to provide dense rewards for exploration in novel environments.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "While results on environment exploration from scratch are encouraging, how to transfer a trained", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 501, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 512 + ], + "score": 1.0, + "content": "policy (with extrinsic reward) to a novel environment remains unclear. Hence, these methods are not", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 510, + 335, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 335, + 524 + ], + "score": 1.0, + "content": "directly applicable to the proposed problem in our paper.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "Generalization across different distributions is a central challenge in machine learning. In domain", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 539, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 506, + 551 + ], + "score": 1.0, + "content": "adaptation, target domain data is assumed to be accessible (Geirhos et al., 2018; Tzeng et al., 2017;", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 549, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 506, + 562 + ], + "score": 1.0, + "content": "Ganin et al., 2016; Gong et al., 2012; Long et al., 2016; Sun et al., 2019; Julian et al., 2020). For", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "example, Tzeng et al. (2017) use adversarial learning to align the feature representations in both the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "source and target domain during training. Similarly, the setting of domain generalization (Ghifary", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 583, + 504, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 504, + 595 + ], + "score": 1.0, + "content": "et al., 2015; Li et al., 2018; Matsuura & Harada, 2019) assumes that all domains are sampled from", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "the same meta distribution, but the same challenge remains and now becomes generalization across", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "meta-distributions. Our work focuses instead on the setting of generalizing to truly unseen changes in", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 615, + 353, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 353, + 628 + ], + "score": 1.0, + "content": "the environment which cannot be anticipated at training time.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 633, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "There have been several recent benchmarks in our setting for image recognition (Hendrycks &", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "Dietterich, 2018; Recht et al., 2018; 2019; Shankar et al., 2019). For example, in Hendrycks", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "& Dietterich (2018), a classifier trained on regular images is tested on corrupted images, with", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "corruption types unknown during training; the method of Hendrycks et al. (2019) is proposed to", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "improve robustness on this benchmark. Following similar spirit, in the context of RL, domain", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 686, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 701 + ], + "score": 1.0, + "content": "randomization (Tobin et al., 2017; Pinto et al., 2017a; Peng et al., 2018; Ramos et al., 2019; Yang", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "et al., 2019; James et al., 2019) helps a policy trained in simulation to generalize to real robots. For", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "example, Tobin et al. (2017); Sadeghi & Levine (2016) propose to render the simulation environment", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "score": 1.0, + "content": "with random textures and train the policy on top. The learned policy is shown to generalize to real", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 51 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "In this paper, we tackle an alternative problem setting in vision-based RL: adapting a pre-trained", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "policy to an unknown environment without any reward. We do this by introducing self-supervision", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "to obtain “free” training signal during deployment. Standard self-supervised learning employs", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "auxiliary tasks designed to automatically create training labels using only the input data (see Section", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "2 for details). Inspired by this, our policy is jointly trained with two objectives: a standard RL", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "objective and, additionally, a self-supervised objective applied on an intermediate representation of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "score": 1.0, + "content": "the policy network. During training, both objectives are active, maximizing expected reward and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "simultaneously constraining the intermediate representation through self-supervision. During testing /", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 168, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 506, + 185 + ], + "score": 1.0, + "content": "deployment, only the self-supervised objective (on the raw observational data) remains active, forcing", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 182, + 365, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 365, + 193 + ], + "score": 1.0, + "content": "the intermediate representation to adapt to the new environment.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 82, + 506, + 193 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "We perform experiments both in simulation and with a real robot. In simulation, we evaluate", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "on two sets of environments: DeepMind Control suite (Tassa et al., 2018) and the CRLMaze", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "ViZDoom (Lomonaco et al., 2019; Wydmuch et al., 2018) navigation task. We evaluate generalization", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "by testing in new environments with visual changes unknown during training. Our method improves", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "generalization in 19 out of 22 test environments across various tasks in DeepMind Control suite, and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 264 + ], + "score": 1.0, + "content": "in all considered test environments on CRLMaze. Besides simulations, we also perform Sim2Real", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "transfer on both reaching and pushing tasks with a Kinova Gen3 robot. After training in simulation, we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 273, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 289 + ], + "score": 1.0, + "content": "successfully transfer and adapt policies to 6 different environments, including continuously changing", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "disco lights, on a real robot operating solely from an uncalibrated camera. In both simulation and real", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 450, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 450, + 309 + ], + "score": 1.0, + "content": "experiments, our approach outperforms domain randomization in most environments.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 197, + 506, + 309 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 317, + 211, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 213, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 213, + 332 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 506, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 336, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 506, + 348 + ], + "score": 1.0, + "content": "Self-supervised learning is a powerful way to learn visual representations from unlabeled data (Vin-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 347, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 506, + 358 + ], + "score": 1.0, + "content": "cent et al., 2008; Doersch et al., 2015; Wang & Gupta, 2015; Zhang et al., 2016; Pathak et al., 2016;", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "Noroozi & Favaro, 2016; Zhang et al., 2017; Gidaris et al., 2018). Researchers have proposed to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 367, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 382 + ], + "score": 1.0, + "content": "use auxiliary data prediction tasks, such as undoing rotation (Gidaris et al., 2018), solving a jigsaw", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "puzzle (Noroozi & Favaro, 2016), tracking (Wang et al., 2019), etc. to provide supervision in lieu of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 390, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 506, + 402 + ], + "score": 1.0, + "content": "labels. In RL, the idea of learning visual representations and action at the same time has been investi-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "gated (Lange & Riedmiller, 2010; Jaderberg et al., 2016; Pathak et al., 2017; Ha & Schmidhuber,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 411, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 425 + ], + "score": 1.0, + "content": "2018; Yarats et al., 2019; Srinivas et al., 2020; Laskin et al., 2020; Yan et al., 2020). For example,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 424, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 506, + 435 + ], + "score": 1.0, + "content": "Srinivas et al. (2020) use self-supervised contrastive learning techniques (Chen et al., 2020; Henaff ´", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 433, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 448 + ], + "score": 1.0, + "content": "et al., 2019; Wu et al., 2018; He et al., 2020) to improve sample efficiency in RL by jointly training", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 446, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 457 + ], + "score": 1.0, + "content": "the self-supervised objective and RL objective. However, this has not been shown to generalize to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 457, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 468 + ], + "score": 1.0, + "content": "unseen environments. Other works have applied self-supervision for better generalization across envi-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 467, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 479 + ], + "score": 1.0, + "content": "ronments (Pathak et al., 2017; Ebert et al., 2018; Sekar et al., 2020). For example, Pathak et al. (2017)", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "use a self-supervised prediction task to provide dense rewards for exploration in novel environments.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "While results on environment exploration from scratch are encouraging, how to transfer a trained", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 501, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 512 + ], + "score": 1.0, + "content": "policy (with extrinsic reward) to a novel environment remains unclear. Hence, these methods are not", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 510, + 335, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 335, + 524 + ], + "score": 1.0, + "content": "directly applicable to the proposed problem in our paper.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 336, + 506, + 524 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "Generalization across different distributions is a central challenge in machine learning. In domain", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 539, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 506, + 551 + ], + "score": 1.0, + "content": "adaptation, target domain data is assumed to be accessible (Geirhos et al., 2018; Tzeng et al., 2017;", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 549, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 506, + 562 + ], + "score": 1.0, + "content": "Ganin et al., 2016; Gong et al., 2012; Long et al., 2016; Sun et al., 2019; Julian et al., 2020). For", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "example, Tzeng et al. (2017) use adversarial learning to align the feature representations in both the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "source and target domain during training. Similarly, the setting of domain generalization (Ghifary", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 583, + 504, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 504, + 595 + ], + "score": 1.0, + "content": "et al., 2015; Li et al., 2018; Matsuura & Harada, 2019) assumes that all domains are sampled from", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "the same meta distribution, but the same challenge remains and now becomes generalization across", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "meta-distributions. Our work focuses instead on the setting of generalizing to truly unseen changes in", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 615, + 353, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 353, + 628 + ], + "score": 1.0, + "content": "the environment which cannot be anticipated at training time.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 528, + 506, + 628 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 633, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "There have been several recent benchmarks in our setting for image recognition (Hendrycks &", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "Dietterich, 2018; Recht et al., 2018; 2019; Shankar et al., 2019). For example, in Hendrycks", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "& Dietterich (2018), a classifier trained on regular images is tested on corrupted images, with", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "corruption types unknown during training; the method of Hendrycks et al. (2019) is proposed to", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "improve robustness on this benchmark. Following similar spirit, in the context of RL, domain", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 686, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 701 + ], + "score": 1.0, + "content": "randomization (Tobin et al., 2017; Pinto et al., 2017a; Peng et al., 2018; Ramos et al., 2019; Yang", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "et al., 2019; James et al., 2019) helps a policy trained in simulation to generalize to real robots. For", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "example, Tobin et al. 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Left: Training before deployment. Observations are sampled from a replay buffer for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 200, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 506, + 213 + ], + "score": 1.0, + "content": "off-policy methods and are collected during roll-outs for on-policy methods. We optimize the RL and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 212, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 223 + ], + "score": 1.0, + "content": "self-supervised objectives jointly. Right: Policy adaptation during deployment. Observations are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 222, + 487, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 487, + 235 + ], + "score": 1.0, + "content": "collected from the test environment online, and we optimize only the self-supervised objective.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 106, + 244, + 504, + 266 + ], + "lines": [ + { + "bbox": [ + 106, + 243, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 256 + ], + "score": 1.0, + "content": "robot manipulation tasks. Instead of deploying a fixed policy, we train and adapt the policy to the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 254, + 454, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 454, + 267 + ], + "score": 1.0, + "content": "new environment with observational data that is naturally revealed during deployment.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 271, + 505, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 283 + ], + "score": 1.0, + "content": "Test-time adaptation for deep learning is starting to be used in computer vision (Shocher et al.,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "score": 1.0, + "content": "2017; 2018; Bau et al., 2019; Mullapudi et al., 2019; Sun et al., 2020; Wortsman et al., 2018). For", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "score": 1.0, + "content": "example, Shocher et al. (2018) shows that image super-resolution can be learned at test time (from", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "scratch) simply by trying to upsample a downsampled version of the input image. Bau et al. (2019)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 314, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 506, + 329 + ], + "score": 1.0, + "content": "show that adapting the prior of a generative adversarial network to the statistics of the test image", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 327, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 338 + ], + "score": 1.0, + "content": "improves photo manipulation tasks. Our work is closely related to the test-time training method of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 336, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 351 + ], + "score": 1.0, + "content": "Sun et al. (2020), which performs joint optimization of image recognition and self-supervised learning", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 349, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 360 + ], + "score": 1.0, + "content": "with rotation prediction (Gidaris et al., 2018), then uses the self-supervised objective to adapt the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 358, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 104, + 358, + 506, + 372 + ], + "score": 1.0, + "content": "representation of individual images during testing. Instead of image recognition, we perform test-time", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 371, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 506, + 382 + ], + "score": 1.0, + "content": "adaptation for RL with visual inputs in an online fashion. As the agent interacts with an environment,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 382, + 475, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 475, + 393 + ], + "score": 1.0, + "content": "we keep obtaining new observational data in a stream for training the visual representations.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 107, + 404, + 173, + 417 + ], + "lines": [ + { + "bbox": [ + 104, + 402, + 175, + 420 + ], + "spans": [ + { + "bbox": [ + 104, + 402, + 175, + 420 + ], + "score": 1.0, + "content": "3 METHOD", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 422, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 422, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 435 + ], + "score": 1.0, + "content": "In this section, we describe our proposed Policy Adaptation during Deployment (PAD) approach.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "It can be implemented on top of any policy network and standard RL algorithm (both on-policy", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 393, + 456 + ], + "score": 1.0, + "content": "and off-policy) that can be described by minimizing some RL objective", + "type": "text" + }, + { + "bbox": [ + 394, + 444, + 414, + 456 + ], + "score": 0.92, + "content": "J ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "w.r.t. the collection of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 455, + 295, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 153, + 467 + ], + "score": 1.0, + "content": "parameters", + "type": "text" + }, + { + "bbox": [ + 153, + 455, + 159, + 465 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 456, + 295, + 467 + ], + "score": 1.0, + "content": "using stochastic gradient descent.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 108, + 479, + 247, + 490 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 248, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 248, + 491 + ], + "score": 1.0, + "content": "3.1 NETWORK ARCHITECTURE", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 496, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "We design the network architecture to allow the policy and the self-supervised prediction to share", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 506, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 272, + 520 + ], + "score": 1.0, + "content": "features. 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Left: Training before deployment. Observations are sampled from a replay buffer for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 200, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 506, + 213 + ], + "score": 1.0, + "content": "off-policy methods and are collected during roll-outs for on-policy methods. We optimize the RL and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 212, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 223 + ], + "score": 1.0, + "content": "self-supervised objectives jointly. Right: Policy adaptation during deployment. 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For", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "score": 1.0, + "content": "example, Shocher et al. (2018) shows that image super-resolution can be learned at test time (from", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "scratch) simply by trying to upsample a downsampled version of the input image. Bau et al. 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We can write the inverse dynamics prediction objective formally as", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 635, + 506, + 694 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 693, + 392, + 708 + ], + "lines": [ + { + "bbox": [ + 219, + 693, + 392, + 708 + ], + "spans": [ + { + "bbox": [ + 219, + 693, + 392, + 708 + ], + "score": 0.93, + "content": "L ( \\theta _ { s } , \\theta _ { e } ) = \\ell \\big ( \\mathbf { a } _ { t } , \\pi _ { s } ( \\pi _ { e } ( \\mathbf { s } _ { t } ) , \\pi _ { e } ( \\mathbf { s } _ { t + 1 } ) ) \\big ) .", + "type": "interline_equation", + "image_path": "7468b6ceb4f801113991097451372b16a2bf35641902476ec43690677aa145b5.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 219, + 693, + 392, + 708 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 508, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 203, + 722 + ], + "score": 1.0, + "content": "For continuous actions,", + "type": "text" + }, + { + "bbox": [ + 203, + 710, + 209, + 720 + ], + "score": 0.79, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "is the mean squared error between the ground truth and the model output.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 507, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 448, + 733 + ], + "score": 1.0, + "content": "For discrete actions, the output is a soft-max distribution over the action space, and", + "type": "text" + }, + { + "bbox": [ + 448, + 721, + 454, + 730 + ], + "score": 0.78, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 720, + 507, + 733 + ], + "score": 1.0, + "content": "is the cross-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "entropy loss. Empirically, we find this self-supervised task to be most effective with continuous", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "actions, possibly because inverse dynamics prediction in a small space of discrete actions is not as", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "challenging. Note that we predict the inverse dynamics instead of the forward dynamics, because", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "when operating in feature space, the latter can produce trivial solutions such as the constant zero", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "feature for every state2. If we instead performed prediction with forward dynamics in pixel space, the", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 441, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 441, + 149 + ], + "score": 1.0, + "content": "task would be extremely challenging given the large uncertainty in pixel prediction.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 709, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "entropy loss. Empirically, we find this self-supervised task to be most effective with continuous", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "actions, possibly because inverse dynamics prediction in a small space of discrete actions is not as", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "challenging. Note that we predict the inverse dynamics instead of the forward dynamics, because", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "when operating in feature space, the latter can produce trivial solutions such as the constant zero", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "feature for every state2. If we instead performed prediction with forward dynamics in pixel space, the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 441, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 441, + 149 + ], + "score": 1.0, + "content": "task would be extremely challenging given the large uncertainty in pixel prediction.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "As an alternative self-supervised task, we use rotation prediction (Gidaris et al., 2018). We rotate", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "an image by one of 0, 90, 180 and 270 degrees as input to the network, and cast this as a four-way", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "classification problem to determine which one of these four ways the image has been rotated. This", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 507, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 507, + 200 + ], + "score": 1.0, + "content": "task is shown to be effective for learning representations for object configuration and scene structure,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 490, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 490, + 210 + ], + "score": 1.0, + "content": "which is beneficial for visual recognition (Hendrycks et al., 2019; Doersch & Zisserman, 2017).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 108, + 222, + 237, + 234 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 238, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 238, + 235 + ], + "score": 1.0, + "content": "3.3 TRAINING AND TESTING", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 239, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 505, + 252 + ], + "score": 1.0, + "content": "Before deployment of the policy, because we have signals from both the reward and self-supervised", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 250, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 505, + 262 + ], + "score": 1.0, + "content": "auxiliary task, we can train with both in the fashion of multi-task learning. This corresponds to the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 260, + 507, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 298, + 276 + ], + "score": 1.0, + "content": "following optimization problem during training", + "type": "text" + }, + { + "bbox": [ + 298, + 261, + 438, + 273 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\operatorname* { m i n } _ { \\theta _ { a } , \\theta _ { s } , \\theta _ { e } } J ( \\theta _ { a } , \\theta _ { e } ) + \\alpha L ( \\theta _ { s } , \\theta _ { e } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 260, + 469, + 276 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 469, + 262, + 495, + 272 + ], + "score": 0.9, + "content": "\\alpha > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 260, + 507, + 276 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 388, + 285 + ], + "score": 1.0, + "content": "a trade-off hyperparameter. 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Empirically, we find only", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 246, + 307 + ], + "score": 1.0, + "content": "negligible difference with keeping", + "type": "text" + }, + { + "bbox": [ + 247, + 295, + 257, + 305 + ], + "score": 0.87, + "content": "\\theta _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "fixed at test-time, so we update both since the gradients have", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 306, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 505, + 317 + ], + "score": 1.0, + "content": "to be computed regardless; we ablate this decision in appendix C. As we obtain new images from", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 316, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 297, + 329 + ], + "score": 1.0, + "content": "the stream of visual inputs in the environment,", + "type": "text" + }, + { + "bbox": [ + 298, + 317, + 304, + 326 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 316, + 506, + 329 + ], + "score": 1.0, + "content": "keeps being updated until the episode ends. This", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 327, + 282, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 240, + 339 + ], + "score": 1.0, + "content": "corresponds to, for each iteration", + "type": "text" + }, + { + "bbox": [ + 240, + 327, + 279, + 337 + ], + "score": 0.89, + "content": "t = 1 . . . T", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 327, + 282, + 339 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 340, + 410, + 397 + ], + "lines": [ + { + "bbox": [ + 200, + 340, + 410, + 397 + ], + "spans": [ + { + "bbox": [ + 200, + 340, + 410, + 397 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\quad \\mathbf s _ { t } \\sim p ( \\mathbf s _ { t } | \\mathbf a _ { t - 1 } , \\mathbf s _ { t - 1 } ) } \\\\ & { \\quad \\theta _ { s } ( t ) = \\theta _ { s } ( t - 1 ) - \\nabla _ { \\theta _ { s } } L ( \\mathbf s _ { t } ; \\theta _ { s } ( t - 1 ) , \\theta _ { e } ( t - 1 ) ) } \\\\ & { \\quad \\theta _ { e } ( t ) = \\theta _ { e } ( t - 1 ) - \\nabla _ { \\theta _ { e } } L ( \\mathbf s _ { t } ; \\theta _ { s } ( t - 1 ) , \\theta _ { e } ( t - 1 ) ) } \\\\ & { \\quad \\mathbf a _ { t } = \\pi ( \\mathbf s _ { t } ; \\theta ( t ) ) \\mathrm { ~ w i t h ~ } \\theta ( t ) = ( \\theta _ { e } ( t ) , \\theta _ { a } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "09a2d43c370c6c00533a9ad2763599d0bd064ebf1e85936b3031644a0c616642.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 200, + 340, + 410, + 354.25 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 200, + 354.25, + 410, + 368.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 200, + 368.5, + 410, + 382.75 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 200, + 382.75, + 410, + 397.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 398, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 132, + 411 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 398, + 225, + 410 + ], + "score": 0.88, + "content": "\\theta _ { s } ( 0 ) = \\theta _ { s } , \\theta _ { e } ( 0 ) = \\theta _ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 396, + 229, + 411 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 229, + 399, + 240, + 410 + ], + "score": 0.8, + "content": "\\mathbf { s } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 396, + 433, + 411 + ], + "score": 1.0, + "content": "is the initial condition given by the environment,", + "type": "text" + }, + { + "bbox": [ + 434, + 398, + 485, + 410 + ], + "score": 0.87, + "content": "\\mathbf { a } _ { 0 } = \\pi _ { \\theta } ( \\mathbf { s } _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 396, + 488, + 411 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 488, + 400, + 495, + 410 + ], + "score": 0.48, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 396, + 506, + 411 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 271, + 421 + ], + "score": 1.0, + "content": "the unknown environment transition, and", + "type": "text" + }, + { + "bbox": [ + 271, + 410, + 279, + 419 + ], + "score": 0.83, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 408, + 506, + 421 + ], + "score": 1.0, + "content": "is the self-supervised objective as previously introduced.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 108, + 443, + 200, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 201, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 201, + 457 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 105, + 459, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 474 + ], + "score": 1.0, + "content": "In this work, we investigate how well an agent trained in one environment (denoted the training", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "environment) generalizes to unseen and diverse test environments. During evaluation, agents have", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "no access to reward signals and are expected to generalize without trials nor prior knowledge about", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "the test environments. In simulation, we evaluate our method (PAD) and baselines extensively on", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 504, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 517 + ], + "score": 1.0, + "content": "continuous control tasks from DeepMind Control (DMControl) suite (Tassa et al., 2018) as well as", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "the CRLMaze (Lomonaco et al., 2019) navigation task, and experiment with both stationary (colors,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 527, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 538 + ], + "score": 1.0, + "content": "objects, textures, lighting) and non-stationary (videos) environment changes. We further show that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "PAD transfers from simulation to a real robot and successfully adapts to environmental differences", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "during deployment in two robotic manipulation tasks. Samples from DMControl and CRLMaze", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 559, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 506, + 572 + ], + "score": 1.0, + "content": "environments are shown in Figure 2, and samples from the robot experiments are shown in Figure 4.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 570, + 449, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 449, + 582 + ], + "score": 1.0, + "content": "Implementation is available at https://nicklashansen.github.io/PAD/.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 586, + 505, + 708 + ], + "lines": [ + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "score": 1.0, + "content": "Network details. For DMControl and the robotic manipulation tasks we implement PAD on top of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 597, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 506, + 610 + ], + "score": 1.0, + "content": "Soft Actor-Critic (SAC) (Haarnoja et al., 2018), and adopt both network architecture and hyperparam-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 609, + 504, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 414, + 620 + ], + "score": 1.0, + "content": "eters from Yarats et al. 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Each head consists of 3 convolutional layers followed", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "score": 1.0, + "content": "by 4 fully connected layers. For CRLMaze, we use Advantage Actor-Critic (A2C) as base algorithm", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 651, + 504, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 492, + 666 + ], + "score": 1.0, + "content": "(Mnih et al., 2016) and apply the same architecture as for the other experiments, but implement", + "type": "text" + }, + { + "bbox": [ + 493, + 654, + 504, + 664 + ], + "score": 0.83, + "content": "\\pi _ { e }", + "type": "inline_equation" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 663, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 345, + 675 + ], + "score": 1.0, + "content": "with only 6 convolutional layers. 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(2020). During deployment, we optimize the self-supervised objective", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 695, + 507, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 155, + 710 + ], + "score": 1.0, + "content": "online w.r.t.", + "type": "text" + }, + { + "bbox": [ + 155, + 697, + 179, + 708 + ], + "score": 0.9, + "content": "\\theta _ { e } , \\theta _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 695, + 507, + 710 + ], + "score": 1.0, + "content": "for one gradient step per time iteration. See appendix F for implementation details.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 44 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 106, + 712, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "2A forward dynamics model operating in feature space can trivially achieve a loss of 0 by learning to map", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "every state to a constant vector, e.g. 0. An inverse dynamics model, however, does not have such trivial solutions.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 105, + 82, + 505, + 149 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "As an alternative self-supervised task, we use rotation prediction (Gidaris et al., 2018). 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This", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 507, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 507, + 200 + ], + "score": 1.0, + "content": "task is shown to be effective for learning representations for object configuration and scene structure,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 490, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 490, + 210 + ], + "score": 1.0, + "content": "which is beneficial for visual recognition (Hendrycks et al., 2019; Doersch & Zisserman, 2017).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 154, + 507, + 210 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 222, + 237, + 234 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 238, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 238, + 235 + ], + "score": 1.0, + "content": "3.3 TRAINING AND TESTING", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 239, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 505, + 252 + ], + "score": 1.0, + "content": "Before deployment of the policy, because we have signals from both the reward and self-supervised", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 250, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 505, + 262 + ], + "score": 1.0, + "content": "auxiliary task, we can train with both in the fashion of multi-task learning. 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During evaluation, agents have", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "score": 1.0, + "content": "no access to reward signals and are expected to generalize without trials nor prior knowledge about", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "the test environments. In simulation, we evaluate our method (PAD) and baselines extensively on", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 504, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 517 + ], + "score": 1.0, + "content": "continuous control tasks from DeepMind Control (DMControl) suite (Tassa et al., 2018) as well as", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "the CRLMaze (Lomonaco et al., 2019) navigation task, and experiment with both stationary (colors,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 527, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 538 + ], + "score": 1.0, + "content": "objects, textures, lighting) and non-stationary (videos) environment changes. We further show that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 549 + ], + "score": 1.0, + "content": "PAD transfers from simulation to a real robot and successfully adapts to environmental differences", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "during deployment in two robotic manipulation tasks. Samples from DMControl and CRLMaze", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 559, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 506, + 572 + ], + "score": 1.0, + "content": "environments are shown in Figure 2, and samples from the robot experiments are shown in Figure 4.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 570, + 449, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 449, + 582 + ], + "score": 1.0, + "content": "Implementation is available at https://nicklashansen.github.io/PAD/.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 459, + 506, + 582 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 586, + 505, + 708 + ], + "lines": [ + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "score": 1.0, + "content": "Network details. For DMControl and the robotic manipulation tasks we implement PAD on top of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 597, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 506, + 610 + ], + "score": 1.0, + "content": "Soft Actor-Critic (SAC) (Haarnoja et al., 2018), and adopt both network architecture and hyperparam-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 609, + 504, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 414, + 620 + ], + "score": 1.0, + "content": "eters from Yarats et al. (2019), with minor modifications: the feature extractor", + "type": "text" + }, + { + "bbox": [ + 414, + 610, + 425, + 620 + ], + "score": 0.87, + "content": "\\pi _ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 609, + 504, + 620 + ], + "score": 1.0, + "content": "has 8 convolutional", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 620, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 251, + 632 + ], + "score": 1.0, + "content": "layers shared between the RL head", + "type": "text" + }, + { + "bbox": [ + 252, + 621, + 263, + 631 + ], + "score": 0.87, + "content": "\\pi _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 620, + 368, + 632 + ], + "score": 1.0, + "content": "and self-supervised head", + "type": "text" + }, + { + "bbox": [ + 368, + 621, + 379, + 631 + ], + "score": 0.86, + "content": "\\pi _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 620, + 505, + 632 + ], + "score": 1.0, + "content": ", and we split the network into", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 271, + 643 + ], + "score": 1.0, + "content": "architecturally identical heads following", + "type": "text" + }, + { + "bbox": [ + 271, + 632, + 282, + 642 + ], + "score": 0.85, + "content": "\\pi _ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 631, + 505, + 643 + ], + "score": 1.0, + "content": ". Each head consists of 3 convolutional layers followed", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "score": 1.0, + "content": "by 4 fully connected layers. For CRLMaze, we use Advantage Actor-Critic (A2C) as base algorithm", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 651, + 504, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 492, + 666 + ], + "score": 1.0, + "content": "(Mnih et al., 2016) and apply the same architecture as for the other experiments, but implement", + "type": "text" + }, + { + "bbox": [ + 493, + 654, + 504, + 664 + ], + "score": 0.83, + "content": "\\pi _ { e }", + "type": "inline_equation" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 663, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 345, + 675 + ], + "score": 1.0, + "content": "with only 6 convolutional layers. Observations are stacks of", + "type": "text" + }, + { + "bbox": [ + 345, + 664, + 352, + 673 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 663, + 417, + 675 + ], + "score": 1.0, + "content": "colored frames", + "type": "text" + }, + { + "bbox": [ + 418, + 664, + 443, + 674 + ], + "score": 0.88, + "content": "k = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 663, + 505, + 675 + ], + "score": 1.0, + "content": "on DMControl", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 675, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 169, + 687 + ], + "score": 1.0, + "content": "and CRLMaze;", + "type": "text" + }, + { + "bbox": [ + 169, + 675, + 194, + 685 + ], + "score": 0.9, + "content": "k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 675, + 321, + 687 + ], + "score": 1.0, + "content": "in robotic manipulation) of size", + "type": "text" + }, + { + "bbox": [ + 321, + 675, + 364, + 685 + ], + "score": 0.91, + "content": "1 0 0 \\times 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 675, + 505, + 687 + ], + "score": 1.0, + "content": "and time-consistent random crop is", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 685, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 506, + 699 + ], + "score": 1.0, + "content": "applied as in Srinivas et al. (2020). During deployment, we optimize the self-supervised objective", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 695, + 507, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 155, + 710 + ], + "score": 1.0, + "content": "online w.r.t.", + "type": "text" + }, + { + "bbox": [ + 155, + 697, + 179, + 708 + ], + "score": 0.9, + "content": "\\theta _ { e } , \\theta _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 695, + 507, + 710 + ], + "score": 1.0, + "content": "for one gradient step per time iteration. 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Left: Training environments of DMControl (top) and CRLMaze (bottom). Right: Test en-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 182, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 505, + 194 + ], + "score": 1.0, + "content": "vironments of DMControl (top) and CRLMaze (bottom). Changes to DMControl include randomized", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 192, + 487, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 487, + 207 + ], + "score": 1.0, + "content": "colors, video backgrounds, and distractors; changes to CRLMaze include textures and lighting.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 211, + 505, + 234 + ], + "lines": [ + { + "bbox": [ + 105, + 211, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 505, + 224 + ], + "score": 1.0, + "content": "Table 1. 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Random colorsSAC+DR+IDM+IDM (PAD)+IDM+IDM (PAD)
Walker, walk414±74594±104406±29468±473830±5475505±592
Walker, stand719±74715±96743±37797±467832±2098566±121
Cartpole, swingup592±50647±48585±73630±636528±5397093±592
Cartpole,balance857±60867±37835±40848±297746±5267670±293
Ball in cup,catch411±183470±252471±75563±50
Finger, spin626±163465±314757±62803±727249±6427496±655
Finger, turn_easy270±43167±26283±51304±461
Cheetah, run154±41145±29121±38159±281117±5301208±487
Reacher, easy163±45105±37201±32214±441788±4412152±506
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Generalization benchmarks on DMControl represent diverse", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 413, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 424 + ], + "score": 1.0, + "content": "real-world tasks for motor control, and contain distracting surroundings not correlated with the reward", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 423, + 140, + 437 + ], + "spans": [ + { + "bbox": [ + 104, + 423, + 140, + 437 + ], + "score": 1.0, + "content": "signals.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 441, + 357, + 615 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 357, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 357, + 452 + ], + "score": 1.0, + "content": "Experimental setup. We experiment with 9 tasks from DM-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 451, + 358, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 358, + 464 + ], + "score": 1.0, + "content": "Control and measure generalization to four types of test environ-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 462, + 357, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 357, + 475 + ], + "score": 1.0, + "content": "ments: (i) randomized colors; (ii) natural videos as background;", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 473, + 358, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 358, + 486 + ], + "score": 1.0, + "content": "(iii) distracting objects placed in the scene; and (iv) the un-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 485, + 356, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 356, + 496 + ], + "score": 1.0, + "content": "modified training environment. For each test environment, we", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 496, + 357, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 357, + 507 + ], + "score": 1.0, + "content": "evaluate methods across 10 seeds and 100 random initializa-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 506, + 356, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 356, + 518 + ], + "score": 1.0, + "content": "tions. If a given test environment is not applicable to certain", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 518, + 357, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 357, + 529 + ], + "score": 1.0, + "content": "tasks, e.g. if a task has no background for the video background", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 528, + 358, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 358, + 539 + ], + "score": 1.0, + "content": "setting, they are excluded. Tasks are selected on the basis of di-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 540, + 357, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 357, + 551 + ], + "score": 1.0, + "content": "versity, as well as the success of vision-based RL in prior work", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 550, + 357, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 357, + 561 + ], + "score": 1.0, + "content": "(Yarats et al., 2019; Srinivas et al., 2020; Laskin et al., 2020;", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 561, + 356, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 356, + 573 + ], + "score": 1.0, + "content": "Kostrikov et al., 2020). We implement PAD on top of SAC and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 572, + 357, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 357, + 584 + ], + "score": 1.0, + "content": "use an Inverse Dynamics Model (IDM) for self-supervision,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 583, + 356, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 356, + 595 + ], + "score": 1.0, + "content": "as we find that learning a model of the dynamics works well", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 594, + 357, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 357, + 605 + ], + "score": 1.0, + "content": "for motor control. For completeness, we ablate the choice of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 605, + 357, + 616 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 357, + 616 + ], + "score": 1.0, + "content": "self-supervision. 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Relative improvement in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 363, + 587, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 363, + 587, + 505, + 598 + ], + "score": 1.0, + "content": "instantaneous reward over time for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 363, + 597, + 487, + 609 + ], + "spans": [ + { + "bbox": [ + 363, + 597, + 487, + 609 + ], + "score": 1.0, + "content": "PAD on the random color env.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + } + ], + "index": 34.75 + }, + { + "type": "text", + "bbox": [ + 106, + 616, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 444, + 628 + ], + "score": 1.0, + "content": "We compare our method to the following baselines: (i) SAC with no changes (denoted", + "type": "text" + }, + { + "bbox": [ + 444, + 616, + 463, + 627 + ], + "score": 0.41, + "content": "S A C", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "); (ii) SAC", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 392, + 639 + ], + "score": 1.0, + "content": "trained with domain randomization on a fixed set of 100 colors (denoted", + "type": "text" + }, + { + "bbox": [ + 392, + 627, + 414, + 637 + ], + "score": 0.86, + "content": "+ D R", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "); and (iii) SAC trained", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 304, + 650 + ], + "score": 1.0, + "content": "jointly with an IDM but without PAD (denoted", + "type": "text" + }, + { + "bbox": [ + 304, + 639, + 333, + 649 + ], + "score": 0.85, + "content": "+ I D M )", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "). 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Left: Training environments of DMControl (top) and CRLMaze (bottom). Right: Test en-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 182, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 505, + 194 + ], + "score": 1.0, + "content": "vironments of DMControl (top) and CRLMaze (bottom). Changes to DMControl include randomized", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 192, + 487, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 487, + 207 + ], + "score": 1.0, + "content": "colors, video backgrounds, and distractors; changes to CRLMaze include textures and lighting.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 211, + 505, + 234 + ], + "lines": [ + { + "bbox": [ + 105, + 211, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 505, + 224 + ], + "score": 1.0, + "content": "Table 1. 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Random colorsSAC+DR+IDM+IDM (PAD)+IDM+IDM (PAD)
Walker, walk414±74594±104406±29468±473830±5475505±592
Walker, stand719±74715±96743±37797±467832±2098566±121
Cartpole, swingup592±50647±48585±73630±636528±5397093±592
Cartpole,balance857±60867±37835±40848±297746±5267670±293
Ball in cup,catch411±183470±252471±75563±50
Finger, spin626±163465±314757±62803±727249±6427496±655
Finger, turn_easy270±43167±26283±51304±461
Cheetah, run154±41145±29121±38159±281117±5301208±487
Reacher, easy163±45105±37201±32214±441788±4412152±506
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Tasks are selected on the basis of di-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 540, + 357, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 357, + 551 + ], + "score": 1.0, + "content": "versity, as well as the success of vision-based RL in prior work", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 550, + 357, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 357, + 561 + ], + "score": 1.0, + "content": "(Yarats et al., 2019; Srinivas et al., 2020; Laskin et al., 2020;", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 561, + 356, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 356, + 573 + ], + "score": 1.0, + "content": "Kostrikov et al., 2020). 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Relative improvement in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 363, + 587, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 363, + 587, + 505, + 598 + ], + "score": 1.0, + "content": "instantaneous reward over time for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 363, + 597, + 487, + 609 + ], + "spans": [ + { + "bbox": [ + 363, + 597, + 487, + 609 + ], + "score": 1.0, + "content": "PAD on the random color env.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + } + ], + "index": 34.75 + }, + { + "type": "text", + "bbox": [ + 106, + 616, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 444, + 628 + ], + "score": 1.0, + "content": "We compare our method to the following baselines: (i) SAC with no changes (denoted", + "type": "text" + }, + { + "bbox": [ + 444, + 616, + 463, + 627 + ], + "score": 0.41, + "content": "S A C", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "); (ii) SAC", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 392, + 639 + ], + "score": 1.0, + "content": "trained with domain randomization on a fixed set of 100 colors (denoted", + "type": "text" + }, + { + "bbox": [ + 392, + 627, + 414, + 637 + ], + "score": 0.86, + "content": "+ D R", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "); and (iii) SAC trained", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 304, + 650 + ], + "score": 1.0, + "content": "jointly with an IDM but without PAD (denoted", + "type": "text" + }, + { + "bbox": [ + 304, + 639, + 333, + 649 + ], + "score": 0.85, + "content": "+ I D M )", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "). Our method using an IDM with PAD is", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 153, + 661 + ], + "score": 1.0, + "content": "denoted by", + "type": "text" + }, + { + "bbox": [ + 153, + 649, + 207, + 660 + ], + "score": 0.76, + "content": "+ I D M \\left( P A D \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 649, + 505, + 661 + ], + "score": 1.0, + "content": ". For domain randomization, colors are sampled from the same distribution", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "as in evaluation, but with lower variance, as we find that training directly on the test distribution does", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 671, + 162, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 162, + 685 + ], + "score": 1.0, + "content": "not converge.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 615, + 506, + 685 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 699 + ], + "score": 1.0, + "content": "Random perturbation of color. Robustness to subtle changes such as color is essential to real-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "world deployment of RL policies. We evaluate generalization on a fixed set of 100 colors of", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "foreground, background and the agent itself, and report the results in Table 1 (first 4 columns). We", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "find PAD to improve generalization in all tasks considered, outperforming SAC trained with domain", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "randomization in 6 out of 9 tasks. Surprisingly, despite a substantial overlap between training and test", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 504, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 504, + 106 + ], + "score": 1.0, + "content": "domains of domain randomization, it generalizes no better than vanilla SAC on a majority of tasks.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 687, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "randomization in 6 out of 9 tasks. Surprisingly, despite a substantial overlap between training and test", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 504, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 504, + 106 + ], + "score": 1.0, + "content": "domains of domain randomization, it generalizes no better than vanilla SAC on a majority of tasks.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "score": 1.0, + "content": "Long-term stability. We find the relative improvement of PAD to improve over time, as shown in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 408, + 133 + ], + "score": 1.0, + "content": "Figure 3. To examine the long-term stability of PAD, we further evaluate on", + "type": "text" + }, + { + "bbox": [ + 408, + 122, + 424, + 132 + ], + "score": 0.33, + "content": "1 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "episode lengths and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "summarize the results in the last two columns in Table 1 (goal-oriented tasks excluded). While we do", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "not explicitly prevent the embedding from drifting away from the RL task, we find empirically that", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "PAD does not degrade the performance of the policy, even over long horizons, and when PAD does", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "not improve, we find it to hurt minimally. We conjecture this is because we are not learning a new", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 506, + 190 + ], + "score": 1.0, + "content": "task, but simply continue to optimize the same (self-supervised) objective as during joint training,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 104, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "where both two tasks are compatible. In this setting, PAD still improves generalization in 6 out of 7", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 212 + ], + "score": 1.0, + "content": "tasks, and thus naturally extends beyond episodic deployment. For completeness, we also evaluate", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "methods in the environment in which they were trained, and report the results in appendix A. We find", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "that, while PAD improves generalization to novel environments, performance is virtually unchanged", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "on the training environment. We conjecture this is because the self-supervised task is already fully", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 243, + 452, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 452, + 254 + ], + "score": 1.0, + "content": "learned and any continued training on the same data distribution thus has little impact.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 259, + 237, + 500 + ], + "lines": [ + { + "bbox": [ + 106, + 259, + 239, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 239, + 271 + ], + "score": 1.0, + "content": "Non-stationary environments.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 269, + 237, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 237, + 281 + ], + "score": 1.0, + "content": "To investigate whether PAD can", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 239, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 239, + 293 + ], + "score": 1.0, + "content": "adapt in non-stationary envi-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 239, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 239, + 303 + ], + "score": 1.0, + "content": "ronments, we evaluate general-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 302, + 239, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 239, + 314 + ], + "score": 1.0, + "content": "ization to diverse video back-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 238, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 238, + 325 + ], + "score": 1.0, + "content": "grounds (refer to Figure 2). We", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 324, + 239, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 239, + 336 + ], + "score": 1.0, + "content": "find PAD to outperform all base-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 335, + 238, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 238, + 347 + ], + "score": 1.0, + "content": "lines on 7 out of 8 tasks, as", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 345, + 238, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 238, + 359 + ], + "score": 1.0, + "content": "shown in Table 2, by as much", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 357, + 239, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 118, + 369 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 357, + 144, + 368 + ], + "score": 0.85, + "content": "104 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 357, + 239, + 369 + ], + "score": 1.0, + "content": "over domain random-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 368, + 238, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 238, + 380 + ], + "score": 1.0, + "content": "ization on Finger, spin. Domain", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 380, + 239, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 239, + 391 + ], + "score": 1.0, + "content": "randomization generalizes com-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 390, + 238, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 238, + 401 + ], + "score": 1.0, + "content": "parably worse to videos, which", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 402, + 237, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 237, + 412 + ], + "score": 1.0, + "content": "we conjecture is not because the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 411, + 239, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 239, + 425 + ], + "score": 1.0, + "content": "environments are non-stationary,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 422, + 238, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 238, + 436 + ], + "score": 1.0, + "content": "but rather because the image", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 434, + 239, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 239, + 445 + ], + "score": 1.0, + "content": "statistics of videos are not cov-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 446, + 239, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 239, + 457 + ], + "score": 1.0, + "content": "ered by its training domain of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 456, + 239, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 239, + 467 + ], + "score": 1.0, + "content": "randomized colors. In fact, do-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 467, + 239, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 239, + 479 + ], + "score": 1.0, + "content": "main randomization is outper-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 478, + 238, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 238, + 489 + ], + "score": 1.0, + "content": "formed by the vanilla SAC in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 489, + 238, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 238, + 500 + ], + "score": 1.0, + "content": "most tasks with video back-", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 25.5 + }, + { + "type": "table", + "bbox": [ + 245, + 309, + 509, + 486 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 245, + 259, + 504, + 303 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 244, + 259, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 244, + 259, + 506, + 271 + ], + "score": 1.0, + "content": "Table 2. Episodic return in test environments with video back-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 244, + 270, + 507, + 282 + ], + "spans": [ + { + "bbox": [ + 244, + 270, + 507, + 282 + ], + "score": 1.0, + "content": "grounds (top) and distracting objects (bottom), mean and std.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 244, + 281, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 244, + 281, + 506, + 293 + ], + "score": 1.0, + "content": "dev. for 10 seeds. Best method on each task is in bold and blue", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 244, + 293, + 426, + 304 + ], + "spans": [ + { + "bbox": [ + 244, + 293, + 285, + 304 + ], + "score": 1.0, + "content": "compares", + "type": "text" + }, + { + "bbox": [ + 286, + 293, + 331, + 303 + ], + "score": 0.69, + "content": "\\mathrm { S A C + I D M }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 293, + 426, + 304 + ], + "score": 1.0, + "content": "with and without PAD.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5 + }, + { + "type": "table_body", + "bbox": [ + 245, + 309, + 509, + 486 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 245, + 309, + 509, + 486 + ], + "spans": [ + { + "bbox": [ + 245, + 309, + 509, + 486 + ], + "score": 0.981, + "html": "
Video backgroundsSAC+DR+IDM+IDM (PAD)
Walker, walk616±80655±55694±85717±79
Walker, stand899±53869±60902±51935±20
Cartpole, swingup375±90485±67487±90521±76
Cartpole, balance693±109766±92691±76687±58
Ball in cup, catch393±175271±189362±69436±55
Finger, spin447±102338±207605±61691±80
Finger, turn_easy355±108223±91355±110362±101
Cheetah, run194±30150±34164±42206±34
Distracting objectsSAC+DR+IDM+IDM (PAD)
Cartpole, swingup815±60809±24776±58771±64
Cartpole,balance969±20938±35964±26960±29
Ball in cup, catch177±111331±189482±128545±173
Finger, spin652±184564±288836±62867±72
Finger, turn_easy302±68165±12326±101347±48
", + "type": "table", + "image_path": "4bc66c9ffdfda4de28170ce557a14b812efeda8a00f84a592bc63390067c438f.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 245, + 309, + 509, + 368.0 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 245, + 368.0, + 509, + 427.0 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 245, + 427.0, + 509, + 486.0 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 108, + 500, + 369, + 511 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 499, + 371, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 371, + 513 + ], + "score": 1.0, + "content": "grounds, which is in line with the findings of Packer et al. (2018).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 516, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "Scene content. We hypothesize that: (i) an agent trained with an IDM is comparably less distracted", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "by scene content since objects uncorrelated to actions yield no predictive power; and (ii) that PAD", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "can adapt to unexpected objects in the scene. We test these hypotheses by measuring robustness", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "score": 1.0, + "content": "to colored shapes at a variety of positions in both the foreground and background of the scene (no", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "score": 1.0, + "content": "physical interaction). Results are summarized in Table 2. PAD outperforms all baselines in 3 out of 5", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 572, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 253, + 583 + ], + "score": 1.0, + "content": "tasks, with a relative improvement of", + "type": "text" + }, + { + "bbox": [ + 253, + 572, + 279, + 582 + ], + "score": 0.87, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 572, + 505, + 583 + ], + "score": 1.0, + "content": "over SAC on Ball in cup, catch. In the two cartpole tasks", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 582, + 492, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 492, + 596 + ], + "score": 1.0, + "content": "in which PAD does not improve, all methods are already relatively unaffected by the distractors.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 48 + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 506, + 612 + ], + "score": 1.0, + "content": "Choice of self-supervised task. We investigate how much the choice of self-supervised task con-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "score": 1.0, + "content": "tributes to the overall success of our method, and consider the following ablations: (i) replacing", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "inverse dynamics with the rotation prediction task described in Section 3.2; and (ii) replacing it with", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "the recently proposed CURL (Srinivas et al., 2020) contrastive learning algorithm for RL. As shown", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "in Table 3, PAD improves generalization of CURL in a majority of tasks on the randomized color", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "benchmark, and in 4 out of 9 tasks using rotation prediction. However, inverse dynamics as auxiliary", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 664, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 679 + ], + "score": 1.0, + "content": "task produces more consistent results and offers better generalization overall. We argue that learning", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 676, + 506, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 688 + ], + "score": 1.0, + "content": "an IDM produces better representations for motor control since it connects observations directly to", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "actions, whereas CURL and rotation prediction operates purely on observations. In general, we find", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "the improvement of PAD to be bigger in tasks that benefit significantly from visual information (see", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 710, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 721 + ], + "score": 1.0, + "content": "appendix A), and conjecture that selecting a self-supervised task that learns features useful to the RL", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 721, + 417, + 731 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 417, + 731 + ], + "score": 1.0, + "content": "task is crucial to the success of PAD, which we discuss further in Section 4.2.", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 57.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 505, + 106 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "score": 1.0, + "content": "Long-term stability. We find the relative improvement of PAD to improve over time, as shown in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 408, + 133 + ], + "score": 1.0, + "content": "Figure 3. To examine the long-term stability of PAD, we further evaluate on", + "type": "text" + }, + { + "bbox": [ + 408, + 122, + 424, + 132 + ], + "score": 0.33, + "content": "1 0 \\mathrm { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "episode lengths and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "summarize the results in the last two columns in Table 1 (goal-oriented tasks excluded). While we do", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "not explicitly prevent the embedding from drifting away from the RL task, we find empirically that", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "PAD does not degrade the performance of the policy, even over long horizons, and when PAD does", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "not improve, we find it to hurt minimally. We conjecture this is because we are not learning a new", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 506, + 190 + ], + "score": 1.0, + "content": "task, but simply continue to optimize the same (self-supervised) objective as during joint training,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 104, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "where both two tasks are compatible. In this setting, PAD still improves generalization in 6 out of 7", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 212 + ], + "score": 1.0, + "content": "tasks, and thus naturally extends beyond episodic deployment. For completeness, we also evaluate", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "methods in the environment in which they were trained, and report the results in appendix A. We find", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "that, while PAD improves generalization to novel environments, performance is virtually unchanged", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "on the training environment. We conjecture this is because the self-supervised task is already fully", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 243, + 452, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 452, + 254 + ], + "score": 1.0, + "content": "learned and any continued training on the same data distribution thus has little impact.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 8, + "bbox_fs": [ + 104, + 111, + 506, + 254 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 259, + 237, + 500 + ], + "lines": [ + { + "bbox": [ + 106, + 259, + 239, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 239, + 271 + ], + "score": 1.0, + "content": "Non-stationary environments.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 269, + 237, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 237, + 281 + ], + "score": 1.0, + "content": "To investigate whether PAD can", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 239, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 239, + 293 + ], + "score": 1.0, + "content": "adapt in non-stationary envi-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 239, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 239, + 303 + ], + "score": 1.0, + "content": "ronments, we evaluate general-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 302, + 239, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 239, + 314 + ], + "score": 1.0, + "content": "ization to diverse video back-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 238, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 238, + 325 + ], + "score": 1.0, + "content": "grounds (refer to Figure 2). We", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 324, + 239, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 239, + 336 + ], + "score": 1.0, + "content": "find PAD to outperform all base-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 335, + 238, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 238, + 347 + ], + "score": 1.0, + "content": "lines on 7 out of 8 tasks, as", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 345, + 238, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 238, + 359 + ], + "score": 1.0, + "content": "shown in Table 2, by as much", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 357, + 239, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 118, + 369 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 357, + 144, + 368 + ], + "score": 0.85, + "content": "104 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 357, + 239, + 369 + ], + "score": 1.0, + "content": "over domain random-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 368, + 238, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 238, + 380 + ], + "score": 1.0, + "content": "ization on Finger, spin. Domain", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 380, + 239, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 239, + 391 + ], + "score": 1.0, + "content": "randomization generalizes com-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 390, + 238, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 238, + 401 + ], + "score": 1.0, + "content": "parably worse to videos, which", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 402, + 237, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 237, + 412 + ], + "score": 1.0, + "content": "we conjecture is not because the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 411, + 239, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 239, + 425 + ], + "score": 1.0, + "content": "environments are non-stationary,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 422, + 238, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 238, + 436 + ], + "score": 1.0, + "content": "but rather because the image", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 434, + 239, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 239, + 445 + ], + "score": 1.0, + "content": "statistics of videos are not cov-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 446, + 239, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 239, + 457 + ], + "score": 1.0, + "content": "ered by its training domain of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 456, + 239, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 239, + 467 + ], + "score": 1.0, + "content": "randomized colors. In fact, do-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 467, + 239, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 239, + 479 + ], + "score": 1.0, + "content": "main randomization is outper-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 478, + 238, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 238, + 489 + ], + "score": 1.0, + "content": "formed by the vanilla SAC in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 489, + 238, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 238, + 500 + ], + "score": 1.0, + "content": "most tasks with video back-", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 259, + 239, + 500 + ] + }, + { + "type": "table", + "bbox": [ + 245, + 309, + 509, + 486 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 245, + 259, + 504, + 303 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 244, + 259, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 244, + 259, + 506, + 271 + ], + "score": 1.0, + "content": "Table 2. Episodic return in test environments with video back-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 244, + 270, + 507, + 282 + ], + "spans": [ + { + "bbox": [ + 244, + 270, + 507, + 282 + ], + "score": 1.0, + "content": "grounds (top) and distracting objects (bottom), mean and std.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 244, + 281, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 244, + 281, + 506, + 293 + ], + "score": 1.0, + "content": "dev. for 10 seeds. Best method on each task is in bold and blue", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 244, + 293, + 426, + 304 + ], + "spans": [ + { + "bbox": [ + 244, + 293, + 285, + 304 + ], + "score": 1.0, + "content": "compares", + "type": "text" + }, + { + "bbox": [ + 286, + 293, + 331, + 303 + ], + "score": 0.69, + "content": "\\mathrm { S A C + I D M }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 293, + 426, + 304 + ], + "score": 1.0, + "content": "with and without PAD.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5 + }, + { + "type": "table_body", + "bbox": [ + 245, + 309, + 509, + 486 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 245, + 309, + 509, + 486 + ], + "spans": [ + { + "bbox": [ + 245, + 309, + 509, + 486 + ], + "score": 0.981, + "html": "
Video backgroundsSAC+DR+IDM+IDM (PAD)
Walker, walk616±80655±55694±85717±79
Walker, stand899±53869±60902±51935±20
Cartpole, swingup375±90485±67487±90521±76
Cartpole, balance693±109766±92691±76687±58
Ball in cup, catch393±175271±189362±69436±55
Finger, spin447±102338±207605±61691±80
Finger, turn_easy355±108223±91355±110362±101
Cheetah, run194±30150±34164±42206±34
Distracting objectsSAC+DR+IDM+IDM (PAD)
Cartpole, swingup815±60809±24776±58771±64
Cartpole,balance969±20938±35964±26960±29
Ball in cup, catch177±111331±189482±128545±173
Finger, spin652±184564±288836±62867±72
Finger, turn_easy302±68165±12326±101347±48
", + "type": "table", + "image_path": "4bc66c9ffdfda4de28170ce557a14b812efeda8a00f84a592bc63390067c438f.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 245, + 309, + 509, + 368.0 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 245, + 368.0, + 509, + 427.0 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 245, + 427.0, + 509, + 486.0 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 108, + 500, + 369, + 511 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 499, + 371, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 371, + 513 + ], + "score": 1.0, + "content": "grounds, which is in line with the findings of Packer et al. (2018).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 516, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "Scene content. We hypothesize that: (i) an agent trained with an IDM is comparably less distracted", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "by scene content since objects uncorrelated to actions yield no predictive power; and (ii) that PAD", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "can adapt to unexpected objects in the scene. We test these hypotheses by measuring robustness", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "score": 1.0, + "content": "to colored shapes at a variety of positions in both the foreground and background of the scene (no", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "score": 1.0, + "content": "physical interaction). Results are summarized in Table 2. PAD outperforms all baselines in 3 out of 5", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 572, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 253, + 583 + ], + "score": 1.0, + "content": "tasks, with a relative improvement of", + "type": "text" + }, + { + "bbox": [ + 253, + 572, + 279, + 582 + ], + "score": 0.87, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 572, + 505, + 583 + ], + "score": 1.0, + "content": "over SAC on Ball in cup, catch. In the two cartpole tasks", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 582, + 492, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 492, + 596 + ], + "score": 1.0, + "content": "in which PAD does not improve, all methods are already relatively unaffected by the distractors.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 517, + 506, + 596 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 506, + 612 + ], + "score": 1.0, + "content": "Choice of self-supervised task. We investigate how much the choice of self-supervised task con-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 624 + ], + "score": 1.0, + "content": "tributes to the overall success of our method, and consider the following ablations: (i) replacing", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "inverse dynamics with the rotation prediction task described in Section 3.2; and (ii) replacing it with", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "the recently proposed CURL (Srinivas et al., 2020) contrastive learning algorithm for RL. As shown", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "in Table 3, PAD improves generalization of CURL in a majority of tasks on the randomized color", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "benchmark, and in 4 out of 9 tasks using rotation prediction. However, inverse dynamics as auxiliary", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 664, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 679 + ], + "score": 1.0, + "content": "task produces more consistent results and offers better generalization overall. We argue that learning", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 676, + 506, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 688 + ], + "score": 1.0, + "content": "an IDM produces better representations for motor control since it connects observations directly to", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "actions, whereas CURL and rotation prediction operates purely on observations. In general, we find", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "the improvement of PAD to be bigger in tasks that benefit significantly from visual information (see", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 710, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 721 + ], + "score": 1.0, + "content": "appendix A), and conjecture that selecting a self-supervised task that learns features useful to the RL", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 721, + 417, + 731 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 417, + 731 + ], + "score": 1.0, + "content": "task is crucial to the success of PAD, which we discuss further in Section 4.2.", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 57.5, + "bbox_fs": [ + 105, + 600, + 506, + 731 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 130, + 507, + 237 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 506, + 125 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "score": 1.0, + "content": "Table 3. Ablations on the randomized color domain of DMC. All methods use SAC. CURL represents", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 506, + 103 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 506, + 103 + ], + "score": 1.0, + "content": "RL with a contrastive learning task (Srinivas et al., 2020) and Rot represents the rotation predic-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 101, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 101, + 506, + 115 + ], + "score": 1.0, + "content": "tion (Gidaris et al., 2018). Offline PAD is here denoted O-PAD for brevity, whereas the default usage", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 112, + 498, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 112, + 398, + 126 + ], + "score": 1.0, + "content": "of PAD is in an online setting. Best method is in bold and blue compares", + "type": "text" + }, + { + "bbox": [ + 399, + 113, + 426, + 124 + ], + "score": 0.86, + "content": "+ \\mathrm { I D M }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 112, + 498, + 126 + ], + "score": 1.0, + "content": "w/ and w/o PAD.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 130, + 507, + 237 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 130, + 507, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 130, + 507, + 237 + ], + "score": 0.983, + "html": "
Random colorsCURLCURL (PAD)RotRot (PAD)IDMIDM (O-PAD)IDM (PAD)
Walker, walk445±99495±70335±7330±30406±29441±16468±47
Walker, stand662±54753±49673±4653±27743±37727±21797±46
Cartpole, swingup454±110413±67493±52477±38585±73578±69630±63
Cartpole,balance782±13763±5710±72734±81835±40796±37848±29
Ball in cup, catch231±92332±78291±54314±60471±75490±16563±50
Finger, spin691±12588±22695±36689±20757±62767±43803±72
Finger, turn_easy202±32186±2283±68230±53283±51321±10304±46
Cheetah, run202±22211±20127±3135±12121±38112±35159±28
Reacher, easy325±32378±6299±29120±7201±32241±24214±44
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CRLMazeRandomA2C+DR+IDM+IDM (PAD)+Rot+Rot (PAD)
Walls-870±30-380±145-260±137-302±150-428±135-206±166-74±116
Floor-868±23-320±167-438±59-47±198-530±106-294±123-209±94
Ceiling-872±30-171±175-400±74166±215-508±104128±196281±83
Lights-900±29-30±213-310±106239±270-460±114-84±53312±104
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Observations that arrive sequentially are highly correlated, and we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 383, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 397 + ], + "score": 1.0, + "content": "thus hypothesize that our method benefits significantly from learning online. To test this hypothesis,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "we run an offline variant of our method in which network updates are forgotten after each step. In", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 404, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 419 + ], + "score": 1.0, + "content": "this setting, our method can only adapt to single observations and does not benefit from learning", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "over time. Results are shown in Table 3. We find that our method benefits substantially from online", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 428, + 401, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 401, + 440 + ], + "score": 1.0, + "content": "learning, but learning offline still improves generalization on select tasks.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 107, + 450, + 179, + 461 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 181, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 181, + 464 + ], + "score": 1.0, + "content": "4.2 CRLMAZE", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 511 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "CRLMaze (Lomonaco et al., 2019) is a time-constrained, discrete-action 3D navigation task for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "ViZDoom (Wydmuch et al., 2018), in which an agent is to navigate a maze and collect objects. There", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "is a positive reward associated with green columns, and a negative reward for lanterns as well as for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 500, + 474, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 474, + 513 + ], + "score": 1.0, + "content": "living. Readers are referred to the respective papers for details on the task and environment.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 517, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "Experimental setup. We train agents on a single environment and measure generalization to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "environments with novel textures for walls, floor, and ceiling, as well as lighting, as shown in Figure 2.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "We implement PAD on top of A2C (Mnih et al., 2016) and use rotation prediction (see Section 3.2) as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 549, + 507, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 507, + 563 + ], + "score": 1.0, + "content": "self-supervised task. Learning to navigate novel scenes requires a generalized scene understanding,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "and we find that rotation prediction facilitates that more so than an IDM. We compare to the following", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 459, + 585 + ], + "score": 1.0, + "content": "baselines: (i) a random agent (denoted Random); (ii) A2C with no changes (denoted", + "type": "text" + }, + { + "bbox": [ + 460, + 573, + 480, + 583 + ], + "score": 0.5, + "content": "A 2 C", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "); (iii)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 312, + 595 + ], + "score": 1.0, + "content": "A2C trained with domain randomization (denoted", + "type": "text" + }, + { + "bbox": [ + 313, + 583, + 335, + 594 + ], + "score": 0.87, + "content": "+ D R", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "); (iv) A2C with an IDM as auxiliary task", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 144, + 606 + ], + "score": 1.0, + "content": "(denoted", + "type": "text" + }, + { + "bbox": [ + 144, + 594, + 172, + 605 + ], + "score": 0.86, + "content": "+ I D M )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 594, + 431, + 606 + ], + "score": 1.0, + "content": "; and (v) A2C with rotation prediction as auxiliary task (denoted", + "type": "text" + }, + { + "bbox": [ + 431, + 595, + 454, + 605 + ], + "score": 0.83, + "content": "+ R o t )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 594, + 506, + 606 + ], + "score": 1.0, + "content": ". We denote", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 180, + 618 + ], + "score": 1.0, + "content": "Rot with PAD as", + "type": "text" + }, + { + "bbox": [ + 180, + 605, + 231, + 616 + ], + "score": 0.63, + "content": "+ R o t \\ ( P A D )", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 604, + 506, + 618 + ], + "score": 1.0, + "content": ". Domain randomization uses 56 combinations of diverse textures,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "partially overlapping with the test distribution, and we find it necessary to train domain randomization", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "for twice as many episodes in order to converge. We closely follow the evaluation procedure of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 638, + 492, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 492, + 650 + ], + "score": 1.0, + "content": "(Lomonaco et al., 2019) and evaluate methods across 20 starting positions and 10 random seeds.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "Results. We report performance on the CRLMaze environments in Table 4. PAD improves gener-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "alization in all considered test environments, outperforming both A2C and domain randomization", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "by a large margin. Domain randomization performs consistently across all environments but is less", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "successful overall. We further examine the importance of selecting appropriate auxiliary tasks by a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "simple ablation: replacing rotation prediction with an IDM for the navigation task. We conjecture", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "that, while an auxiliary task can enforce structure in the learned representations, its features (and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "consequently gradients) need to be sufficiently correlated with the primary RL task for PAD to be", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 130, + 507, + 237 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 506, + 125 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "score": 1.0, + "content": "Table 3. Ablations on the randomized color domain of DMC. All methods use SAC. CURL represents", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 506, + 103 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 506, + 103 + ], + "score": 1.0, + "content": "RL with a contrastive learning task (Srinivas et al., 2020) and Rot represents the rotation predic-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 101, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 101, + 506, + 115 + ], + "score": 1.0, + "content": "tion (Gidaris et al., 2018). Offline PAD is here denoted O-PAD for brevity, whereas the default usage", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 112, + 498, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 112, + 398, + 126 + ], + "score": 1.0, + "content": "of PAD is in an online setting. Best method is in bold and blue compares", + "type": "text" + }, + { + "bbox": [ + 399, + 113, + 426, + 124 + ], + "score": 0.86, + "content": "+ \\mathrm { I D M }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 112, + 498, + 126 + ], + "score": 1.0, + "content": "w/ and w/o PAD.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 130, + 507, + 237 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 130, + 507, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 130, + 507, + 237 + ], + "score": 0.983, + "html": "
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Walker, walk445±99495±70335±7330±30406±29441±16468±47
Walker, stand662±54753±49673±4653±27743±37727±21797±46
Cartpole, swingup454±110413±67493±52477±38585±73578±69630±63
Cartpole,balance782±13763±5710±72734±81835±40796±37848±29
Ball in cup, catch231±92332±78291±54314±60471±75490±16563±50
Finger, spin691±12588±22695±36689±20757±62767±43803±72
Finger, turn_easy202±32186±2283±68230±53283±51321±10304±46
Cheetah, run202±22211±20127±3135±12121±38112±35159±28
Reacher, easy325±32378±6299±29120±7201±32241±24214±44
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CRLMazeRandomA2C+DR+IDM+IDM (PAD)+Rot+Rot (PAD)
Walls-870±30-380±145-260±137-302±150-428±135-206±166-74±116
Floor-868±23-320±167-438±59-47±198-530±106-294±123-209±94
Ceiling-872±30-171±175-400±74166±215-508±104128±196281±83
Lights-900±29-30±213-310±106239±270-460±114-84±53312±104
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Observations that arrive sequentially are highly correlated, and we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 383, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 397 + ], + "score": 1.0, + "content": "thus hypothesize that our method benefits significantly from learning online. To test this hypothesis,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "we run an offline variant of our method in which network updates are forgotten after each step. In", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 404, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 419 + ], + "score": 1.0, + "content": "this setting, our method can only adapt to single observations and does not benefit from learning", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "over time. Results are shown in Table 3. We find that our method benefits substantially from online", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 428, + 401, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 401, + 440 + ], + "score": 1.0, + "content": "learning, but learning offline still improves generalization on select tasks.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 373, + 506, + 440 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 450, + 179, + 461 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 181, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 181, + 464 + ], + "score": 1.0, + "content": "4.2 CRLMAZE", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 511 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "CRLMaze (Lomonaco et al., 2019) is a time-constrained, discrete-action 3D navigation task for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "ViZDoom (Wydmuch et al., 2018), in which an agent is to navigate a maze and collect objects. There", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "is a positive reward associated with green columns, and a negative reward for lanterns as well as for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 500, + 474, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 474, + 513 + ], + "score": 1.0, + "content": "living. Readers are referred to the respective papers for details on the task and environment.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 466, + 506, + 513 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 517, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "Experimental setup. We train agents on a single environment and measure generalization to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "environments with novel textures for walls, floor, and ceiling, as well as lighting, as shown in Figure 2.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "We implement PAD on top of A2C (Mnih et al., 2016) and use rotation prediction (see Section 3.2) as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 549, + 507, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 507, + 563 + ], + "score": 1.0, + "content": "self-supervised task. 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We compare to the following", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 459, + 585 + ], + "score": 1.0, + "content": "baselines: (i) a random agent (denoted Random); (ii) A2C with no changes (denoted", + "type": "text" + }, + { + "bbox": [ + 460, + 573, + 480, + 583 + ], + "score": 0.5, + "content": "A 2 C", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "); (iii)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 312, + 595 + ], + "score": 1.0, + "content": "A2C trained with domain randomization (denoted", + "type": "text" + }, + { + "bbox": [ + 313, + 583, + 335, + 594 + ], + "score": 0.87, + "content": "+ D R", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "); (iv) A2C with an IDM as auxiliary task", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 144, + 606 + ], + "score": 1.0, + "content": "(denoted", + "type": "text" + }, + { + "bbox": [ + 144, + 594, + 172, + 605 + ], + "score": 0.86, + "content": "+ I D M )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 594, + 431, + 606 + ], + "score": 1.0, + "content": "; and (v) A2C with rotation prediction as auxiliary task (denoted", + "type": "text" + }, + { + "bbox": [ + 431, + 595, + 454, + 605 + ], + "score": 0.83, + "content": "+ R o t )", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 594, + 506, + 606 + ], + "score": 1.0, + "content": ". We denote", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 180, + 618 + ], + "score": 1.0, + "content": "Rot with PAD as", + "type": "text" + }, + { + "bbox": [ + 180, + 605, + 231, + 616 + ], + "score": 0.63, + "content": "+ R o t \\ ( P A D )", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 604, + 506, + 618 + ], + "score": 1.0, + "content": ". Domain randomization uses 56 combinations of diverse textures,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "partially overlapping with the test distribution, and we find it necessary to train domain randomization", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "for twice as many episodes in order to converge. We closely follow the evaluation procedure of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 638, + 492, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 492, + 650 + ], + "score": 1.0, + "content": "(Lomonaco et al., 2019) and evaluate methods across 20 starting positions and 10 random seeds.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 518, + 507, + 650 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "Results. We report performance on the CRLMaze environments in Table 4. PAD improves gener-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "alization in all considered test environments, outperforming both A2C and domain randomization", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "by a large margin. Domain randomization performs consistently across all environments but is less", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "successful overall. We further examine the importance of selecting appropriate auxiliary tasks by a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "simple ablation: replacing rotation prediction with an IDM for the navigation task. We conjecture", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "that, while an auxiliary task can enforce structure in the learned representations, its features (and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "consequently gradients) need to be sufficiently correlated with the primary RL task for PAD to be", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 655, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 76, + 502, + 154 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 76, + 502, + 154 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 76, + 502, + 154 + ], + "spans": [ + { + "bbox": [ + 109, + 76, + 502, + 154 + ], + "score": 0.901, + "type": "image", + "image_path": "7a75facacf5c2ac9f0591865730aa213a149b2aae09f6d5a1610fc3b0f899d74.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 76, + 502, + 102.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 102.0, + 502, + 128.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 128.0, + 502, + 154.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 111, + 157, + 456, + 168 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 156, + 455, + 170 + ], + "spans": [ + { + "bbox": [ + 117, + 156, + 334, + 170 + ], + "score": 1.0, + "content": "(a) Simulation. (b) Default transfer. (c) Table cloth.", + "type": "text" + }, + { + "bbox": [ + 392, + 156, + 455, + 170 + ], + "score": 1.0, + "content": "(d) Disco lights.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 106, + 171, + 504, + 193 + ], + "lines": [ + { + "bbox": [ + 106, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "Figure 4. Samples from the push robotic manipulation task. The task is to push the yellow cube to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 181, + 471, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 471, + 194 + ], + "score": 1.0, + "content": "the location of the red disc. Agents are trained in setting (a) and evaluated in settings (b-d).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "successful during deployment. While PAD with rotation prediction improves generalization across all", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "test environments considered, IDM does not, which suggests that rotation prediction is more suitable", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 226, + 507, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 507, + 240 + ], + "score": 1.0, + "content": "for tasks that require scene understanding, whereas IDM is useful for tasks that require motor control.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 237, + 468, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 468, + 250 + ], + "score": 1.0, + "content": "We leave it to future work to automate the process of selecting appropriate auxiliary tasks.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 108, + 264, + 268, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 270, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 270, + 276 + ], + "score": 1.0, + "content": "4.3 ROBOTIC MANIPULATION TASKS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 281, + 276, + 413 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 277, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 277, + 293 + ], + "score": 1.0, + "content": "We deploy our method and baselines on", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 291, + 277, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 277, + 303 + ], + "score": 1.0, + "content": "a real Kinova Gen3 robot and evaluate on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 303, + 277, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 277, + 314 + ], + "score": 1.0, + "content": "two manipulation tasks: (i) reach, a task in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 314, + 277, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 277, + 325 + ], + "score": 1.0, + "content": "which the robot reaches for a goal marked", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 325, + 277, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 277, + 336 + ], + "score": 1.0, + "content": "by a red disc; and (ii) push, a task in which", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 336, + 277, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 277, + 347 + ], + "score": 1.0, + "content": "the robot pushes a cube to the location of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 347, + 277, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 277, + 358 + ], + "score": 1.0, + "content": "the red disc. Both tasks use an XY action", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 358, + 278, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 278, + 370 + ], + "score": 1.0, + "content": "space, where the Z position of the actua-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 369, + 277, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 277, + 381 + ], + "score": 1.0, + "content": "tor is fixed. Agents operate purely from", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 380, + 277, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 277, + 392 + ], + "score": 1.0, + "content": "pixel observations with no access to state", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 391, + 277, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 277, + 403 + ], + "score": 1.0, + "content": "information. During deployment, we make", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 401, + 277, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 277, + 415 + ], + "score": 1.0, + "content": "no effort to calibrate camera, lighting, or", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16.5 + }, + { + "type": "table", + "bbox": [ + 284, + 322, + 502, + 406 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 284, + 282, + 504, + 315 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 284, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 284, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "Table 5. Success rate of PAD and baselines on a real", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 284, + 293, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 284, + 293, + 505, + 304 + ], + "score": 1.0, + "content": "robotic arm. Best method in each environment is in bold", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 284, + 304, + 484, + 315 + ], + "spans": [ + { + "bbox": [ + 284, + 304, + 361, + 315 + ], + "score": 1.0, + "content": "and blue compares", + "type": "text" + }, + { + "bbox": [ + 362, + 304, + 389, + 314 + ], + "score": 0.86, + "content": "+ \\mathrm { I D M }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 304, + 484, + 315 + ], + "score": 1.0, + "content": "with and without PAD.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "table_body", + "bbox": [ + 284, + 322, + 502, + 406 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 284, + 322, + 502, + 406 + ], + "spans": [ + { + "bbox": [ + 284, + 322, + 502, + 406 + ], + "score": 0.979, + "html": "
Real robotSAC+DR+IDM+IDM (PAD)
Reach (default)100%100%100%100%
Reach (cloth)48%80%56%80%
Reach (disco)72%76%88%92%
Push (default)88%88%92%100%
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Push (disco)60%68%72%84%
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Samples from the push task are shown in Figure 4,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 434, + 307, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 307, + 446 + ], + "score": 1.0, + "content": "and samples from reach are shown in appendix E.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 276, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 451, + 277, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 277, + 464 + ], + "score": 1.0, + "content": "Experimental setup. We implement PAD", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 462, + 277, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 277, + 474 + ], + "score": 1.0, + "content": "on top of SAC (Haarnoja et al., 2018) and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 473, + 277, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 277, + 486 + ], + "score": 1.0, + "content": "apply the same experimental setup as in", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 484, + 277, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 277, + 497 + ], + "score": 1.0, + "content": "Section 4.1 using an Inverse Dynamics", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 496, + 278, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 278, + 507 + ], + "score": 1.0, + "content": "Model (IDM) for self-supervision, but with-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 506, + 277, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 200, + 518 + ], + "score": 1.0, + "content": "out frame-stacking (i.e.", + "type": "text" + }, + { + "bbox": [ + 200, + 506, + 226, + 517 + ], + "score": 0.89, + "content": "k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 506, + 277, + 518 + ], + "score": 1.0, + "content": "). 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Simulated robotSAC+DR+IDM+IDM (PAD)
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Push (mount)68%58%86%84%
Push (velocity)70%68%70%78%
Push (all)56%50%48%76%
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The goal locations vary between the two tasks, and the robot", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "is reset after each run. We perform comparison against direct transfer and domain randomization", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 104, + 658, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 104, + 658, + 506, + 674 + ], + "score": 1.0, + "content": "baselines as in Section 4.1. We further evaluate generalization to changes in dynamics by considering", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 104, + 671, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 104, + 671, + 506, + 684 + ], + "score": 1.0, + "content": "a variant of the simulated environment in which object mass, size, and friction, arm mount position,", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "and end effector velocity is modified. We consider each setting both individually and jointly, and", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 106, + 693, + 459, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 459, + 705 + ], + "score": 1.0, + "content": "evaluate success rate across 50 unique configurations with the robot reset after each run.", + "type": "text" + } + ], + "index": 69 + } + ], + "index": 64.5 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Results. We report transfer results in Table 5. While all methods transfer successfully to reach", + "type": "text" + } + ], + "index": 70 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "(default), we observe PAD to improve generalization in all settings in which the baselines show", + "type": "text" + } + ], + "index": 71 + } + ], + "index": 70.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 76, + 502, + 154 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 76, + 502, + 154 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 76, + 502, + 154 + ], + "spans": [ + { + "bbox": [ + 109, + 76, + 502, + 154 + ], + "score": 0.901, + "type": "image", + "image_path": "7a75facacf5c2ac9f0591865730aa213a149b2aae09f6d5a1610fc3b0f899d74.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 76, + 502, + 102.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 102.0, + 502, + 128.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 128.0, + 502, + 154.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 111, + 157, + 456, + 168 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 156, + 455, + 170 + ], + "spans": [ + { + "bbox": [ + 117, + 156, + 334, + 170 + ], + "score": 1.0, + "content": "(a) Simulation. (b) Default transfer. (c) Table cloth.", + "type": "text" + }, + { + "bbox": [ + 392, + 156, + 455, + 170 + ], + "score": 1.0, + "content": "(d) Disco lights.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 106, + 171, + 504, + 193 + ], + "lines": [ + { + "bbox": [ + 106, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "Figure 4. Samples from the push robotic manipulation task. The task is to push the yellow cube to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 181, + 471, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 471, + 194 + ], + "score": 1.0, + "content": "the location of the red disc. Agents are trained in setting (a) and evaluated in settings (b-d).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 170, + 505, + 194 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "successful during deployment. While PAD with rotation prediction improves generalization across all", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "test environments considered, IDM does not, which suggests that rotation prediction is more suitable", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 226, + 507, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 507, + 240 + ], + "score": 1.0, + "content": "for tasks that require scene understanding, whereas IDM is useful for tasks that require motor control.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 237, + 468, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 468, + 250 + ], + "score": 1.0, + "content": "We leave it to future work to automate the process of selecting appropriate auxiliary tasks.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 205, + 507, + 250 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 264, + 268, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 270, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 270, + 276 + ], + "score": 1.0, + "content": "4.3 ROBOTIC MANIPULATION TASKS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 281, + 276, + 413 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 277, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 277, + 293 + ], + "score": 1.0, + "content": "We deploy our method and baselines on", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 291, + 277, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 277, + 303 + ], + "score": 1.0, + "content": "a real Kinova Gen3 robot and evaluate on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 303, + 277, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 277, + 314 + ], + "score": 1.0, + "content": "two manipulation tasks: (i) reach, a task in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 314, + 277, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 277, + 325 + ], + "score": 1.0, + "content": "which the robot reaches for a goal marked", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 325, + 277, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 277, + 336 + ], + "score": 1.0, + "content": "by a red disc; and (ii) push, a task in which", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 336, + 277, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 277, + 347 + ], + "score": 1.0, + "content": "the robot pushes a cube to the location of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 347, + 277, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 277, + 358 + ], + "score": 1.0, + "content": "the red disc. Both tasks use an XY action", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 358, + 278, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 278, + 370 + ], + "score": 1.0, + "content": "space, where the Z position of the actua-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 369, + 277, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 277, + 381 + ], + "score": 1.0, + "content": "tor is fixed. Agents operate purely from", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 380, + 277, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 277, + 392 + ], + "score": 1.0, + "content": "pixel observations with no access to state", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 391, + 277, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 277, + 403 + ], + "score": 1.0, + "content": "information. 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Real robotSAC+DR+IDM+IDM (PAD)
Reach (default)100%100%100%100%
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Samples from the push task are shown in Figure 4,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 434, + 307, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 307, + 446 + ], + "score": 1.0, + "content": "and samples from reach are shown in appendix E.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 412, + 507, + 446 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 276, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 451, + 277, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 277, + 464 + ], + "score": 1.0, + "content": "Experimental setup. We implement PAD", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 462, + 277, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 277, + 474 + ], + "score": 1.0, + "content": "on top of SAC (Haarnoja et al., 2018) and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 473, + 277, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 277, + 486 + ], + "score": 1.0, + "content": "apply the same experimental setup as in", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 484, + 277, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 277, + 497 + ], + "score": 1.0, + "content": "Section 4.1 using an Inverse Dynamics", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 496, + 278, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 278, + 507 + ], + "score": 1.0, + "content": "Model (IDM) for self-supervision, but with-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 506, + 277, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 200, + 518 + ], + "score": 1.0, + "content": "out frame-stacking (i.e.", + "type": "text" + }, + { + "bbox": [ + 200, + 506, + 226, + 517 + ], + "score": 0.89, + "content": "k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 506, + 277, + 518 + ], + "score": 1.0, + "content": "). 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Simulated robotSAC+DR+IDM+IDM (PAD)
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The goal locations vary between the two tasks, and the robot", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "is reset after each run. We perform comparison against direct transfer and domain randomization", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 104, + 658, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 104, + 658, + 506, + 674 + ], + "score": 1.0, + "content": "baselines as in Section 4.1. We further evaluate generalization to changes in dynamics by considering", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 104, + 671, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 104, + 671, + 506, + 684 + ], + "score": 1.0, + "content": "a variant of the simulated environment in which object mass, size, and friction, arm mount position,", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "and end effector velocity is modified. We consider each setting both individually and jointly, and", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 106, + 693, + 459, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 459, + 705 + ], + "score": 1.0, + "content": "evaluate success rate across 50 unique configurations with the robot reset after each run.", + "type": "text" + } + ], + "index": 69 + } + ], + "index": 64.5, + "bbox_fs": [ + 104, + 593, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Results. We report transfer results in Table 5. While all methods transfer successfully to reach", + "type": "text" + } + ], + "index": 70 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "(default), we observe PAD to improve generalization in all settings in which the baselines show", + "type": "text" + } + ], + "index": 71 + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "sub-optimal performance. We find PAD to be especially powerful for the push task that involves", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 248, + 106 + ], + "score": 1.0, + "content": "dynamics, improving by as much as", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 249, + 94, + 270, + 104 + ], + "score": 0.87, + "content": "24 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 270, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "in push (cloth). While domain randomization proves highly", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "effective in reach (cloth), we observe no significant benefit in the other settings, which suggests that", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "PAD can be more suitable in challenging tasks like push. To isolate the effect of dynamics, we further", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "evaluate generalization to a number of simulated changes in dynamics on the push task. Results are", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "shown in Table 6. We find PAD to improve generalization to changes in the physical properties of", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 273, + 161 + ], + "score": 1.0, + "content": "the object and end effector, whereas both", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 273, + 149, + 319, + 159 + ], + "score": 0.83, + "content": "S A C { + } I D M", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 319, + 147, + 505, + 161 + ], + "score": 1.0, + "content": "and PAD are relatively unaffected by changes", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 160, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 171 + ], + "score": 1.0, + "content": "to the mount position. Consistent with the real robot results in Section 5, PAD is found to be most", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 417, + 183 + ], + "score": 1.0, + "content": "effective when changes in dynamics are non-trivial, improving by as much as", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 417, + 171, + 438, + 181 + ], + "score": 0.87, + "content": "28 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 438, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "in the push (all)", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "score": 1.0, + "content": "setting, where all 3 environmental changes are considered jointly. These results suggest that PAD can", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "be a simple, yet effective method for generalization to diverse, unseen environments that vary in both", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 196, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 196, + 216 + ], + "score": 1.0, + "content": "visuals and dynamics.", + "type": "text", + "cross_page": true + } + ], + "index": 11 + } + ], + "index": 70.5, + "bbox_fs": [ + 105, + 709, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 214 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "sub-optimal performance. We find PAD to be especially powerful for the push task that involves", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 248, + 106 + ], + "score": 1.0, + "content": "dynamics, improving by as much as", + "type": "text" + }, + { + "bbox": [ + 249, + 94, + 270, + 104 + ], + "score": 0.87, + "content": "24 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "in push (cloth). While domain randomization proves highly", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "effective in reach (cloth), we observe no significant benefit in the other settings, which suggests that", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "PAD can be more suitable in challenging tasks like push. To isolate the effect of dynamics, we further", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "evaluate generalization to a number of simulated changes in dynamics on the push task. Results are", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "shown in Table 6. 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Consistent with the real robot results in Section 5, PAD is found to be most", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 417, + 183 + ], + "score": 1.0, + "content": "effective when changes in dynamics are non-trivial, improving by as much as", + "type": "text" + }, + { + "bbox": [ + 417, + 171, + 438, + 181 + ], + "score": 0.87, + "content": "28 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "in the push (all)", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "score": 1.0, + "content": "setting, where all 3 environmental changes are considered jointly. These results suggest that PAD can", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "be a simple, yet effective method for generalization to diverse, unseen environments that vary in both", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 196, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 196, + 216 + ], + "score": 1.0, + "content": "visuals and dynamics.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 108, + 231, + 195, + 244 + ], + "lines": [ + { + "bbox": [ + 104, + 230, + 197, + 247 + ], + "spans": [ + { + "bbox": [ + 104, + 230, + 197, + 247 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 252, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 267 + ], + "score": 1.0, + "content": "While previous work addresses generalization in RL by learning policies that are invariant to any", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "environment changes that can be anticipated, we formulate an alternative problem setting in vision-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "based RL: can we instead adapt a pretrained-policy to new environments without any reward. We", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "propose Policy Adaptation during Deployment, a self-supervised framework for online adaptation", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "at test-time, and show empirically that our method improves generalization of policies to diverse", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "simulated and real-world environmental changes across a variety of tasks. We find our approach", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 319, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 506, + 331 + ], + "score": 1.0, + "content": "benefits greatly from learning online, and we systematically evaluate how the choice of self-supervised", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 330, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 506, + 342 + ], + "score": 1.0, + "content": "task impacts performance. While the current framework relies on prior knowledge on selecting self-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "supervised tasks for policy adaptation, we see our work as the initial step in addressing the problem", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "of adapting vision-based policies to unknown environments. We ultimately envision embodied agents", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 362, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 374 + ], + "score": 1.0, + "content": "in the future to be learning all the time, with the flexibility to learn both with and without rewards,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 373, + 231, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 231, + 387 + ], + "score": 1.0, + "content": "before and during deployment.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 398, + 505, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 411 + ], + "score": 1.0, + "content": "Acknowledgements. This work was supported, in part, by grants from DARPA, NSF 1730158", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "CI-New: Cognitive Hardware and Software Ecosystem Community Infrastructure (CHASE-CI), NSF", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 419, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 433 + ], + "score": 1.0, + "content": "ACI-1541349 CC*DNI Pacific Research Platform, and gifts from Qualcomm and TuSimple. This", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "work was also funded, in part, by grants from Berkeley DeepDrive, SAP and European Research", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 441, + 507, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 507, + 455 + ], + "score": 1.0, + "content": "Council (ERC) from the European Union Horizon 2020 Programme under grant agreement no.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 451, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 466 + ], + "score": 1.0, + "content": "741930 (CLOTHILDE). We would like to thank Fenglu Hong and Joey Hejna for helpful discus-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 464, + 133, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 133, + 476 + ], + "score": 1.0, + "content": "sions.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 108, + 492, + 175, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 492, + 177, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 177, + 506 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 105, + 510, + 507, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 506, + 525 + ], + "score": 1.0, + "content": "OpenAI: Marcin Andrychowicz, Bowen Baker, Maciek Chociej, Rafal Jozefowicz, Bob McGrew,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 114, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 114, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "Jakub Pachocki, Arthur Petron, Matthias Plappert, Glenn Powell, Alex Ray, et al. 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Single episode policy transfer", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 115, + 366, + 267, + 376 + ], + "spans": [ + { + "bbox": [ + 115, + 366, + 267, + 376 + ], + "score": 1.0, + "content": "in reinforcement learning, 2019. 1, 2", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 106, + 354, + 505, + 376 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 504, + 407 + ], + "lines": [ + { + "bbox": [ + 105, + 382, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 398 + ], + "score": 1.0, + "content": "Denis Yarats, Amy Zhang, Ilya Kostrikov, Brandon Amos, Joelle Pineau, and Rob Fergus. Improving", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 115, + 395, + 462, + 407 + ], + "spans": [ + { + "bbox": [ + 115, + 395, + 462, + 407 + ], + "score": 1.0, + "content": "sample efficiency in model-free reinforcement learning from images, 2019. 2, 4, 5, 17", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 382, + 505, + 407 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 414, + 504, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 428 + ], + "score": 1.0, + "content": "Richard Zhang, Phillip Isola, and Alexei A Efros. Colorful image colorization. In European", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 116, + 425, + 372, + 437 + ], + "spans": [ + { + "bbox": [ + 116, + 425, + 372, + 437 + ], + "score": 1.0, + "content": "conference on computer vision, pp. 649–666. Springer, 2016. 2", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 412, + 506, + 437 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 459 + ], + "score": 1.0, + "content": "Richard Zhang, Phillip Isola, and Alexei A Efros. Split-brain autoencoders: Unsupervised learning", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 115, + 455, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 115, + 455, + 506, + 468 + ], + "score": 1.0, + "content": "by cross-channel prediction. In Proceedings of the IEEE Conference on Computer Vision and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 115, + 466, + 300, + 478 + ], + "spans": [ + { + "bbox": [ + 115, + 466, + 300, + 478 + ], + "score": 1.0, + "content": "Pattern Recognition, pp. 1058–1067, 2017. 2", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 442, + 506, + 478 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 382, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 383, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 383, + 96 + ], + "score": 1.0, + "content": "A PERFORMANCE ON THE TRAINING ENVIRONMENT", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 107, + 506, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 108, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 108, + 505, + 119 + ], + "score": 1.0, + "content": "Historically, agents have commonly been trained and evaluated in the same environment when", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 118, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 505, + 130 + ], + "score": 1.0, + "content": "benchmarking RL algorithms exclusively in simulation. Although such an evaluation procedure does", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 130, + 505, + 143 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 505, + 143 + ], + "score": 1.0, + "content": "not consider generalization, it is still a useful metric for comparison of sample efficiency and stability", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 505, + 153 + ], + "score": 1.0, + "content": "of algorithms. For completeness, we also evaluate our method and baselines in this setting on both", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 506, + 164 + ], + "score": 1.0, + "content": "DMControl and CRLMaze. DMControl results are reported in Table 7 and results on the CRLMaze", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 162, + 506, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 506, + 175 + ], + "score": 1.0, + "content": "environment are shown in Table 8. In this setting, we also compare to an additional baseline on", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 173, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 506, + 186 + ], + "score": 1.0, + "content": "DMControl: a blind SAC agent that operates purely on its previous actions. The performance of a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 184, + 506, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 506, + 197 + ], + "score": 1.0, + "content": "blind agent indicates to which degree a given task benefits from visual information. We find that,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 195, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 506, + 208 + ], + "score": 1.0, + "content": "while PAD improves generalization to novel environments, performance is virtually unchanged when", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 207, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 505, + 218 + ], + "score": 1.0, + "content": "evaluated on the same environment as in training. We conjecture that this is because the algorithm", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 217, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 506, + 230 + ], + "score": 1.0, + "content": "already is adapted to the training environment and any continued training on the same data distribution", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 228, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 506, + 240 + ], + "score": 1.0, + "content": "thus has little influence. We further emphasize that, even when evaluated on the training environment,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 239, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 397, + 250 + ], + "score": 1.0, + "content": "PAD still outperforms baselines on most tasks. For example, we observe a", + "type": "text" + }, + { + "bbox": [ + 397, + 239, + 418, + 250 + ], + "score": 0.87, + "content": "15 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 239, + 505, + 250 + ], + "score": 1.0, + "content": "relative improvement", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 250, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 262 + ], + "score": 1.0, + "content": "over SAC on the Finger, spin task. We hypothesize that this gain in performance is because the self-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 261, + 507, + 274 + ], + "spans": [ + { + "bbox": [ + 104, + 261, + 507, + 274 + ], + "score": 1.0, + "content": "supervised objective improves learning by constraining the intermediate representation of policies.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "score": 1.0, + "content": "A blind agent is no better than random on this particular task, which would suggest that agents", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 283, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 505, + 296 + ], + "score": 1.0, + "content": "benefit substantially from visual information in Finger, spin. Therefore, learning a good intermediate", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 294, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 506, + 306 + ], + "score": 1.0, + "content": "representation of that information is highly beneficial to the RL objective, which we find PAD to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "score": 1.0, + "content": "facilitate through its self-supervised learning framework. Likewise, the SAC baseline only achieves a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 315, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 126, + 326 + ], + "score": 0.86, + "content": "51 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 315, + 506, + 328 + ], + "score": 1.0, + "content": "improvement over the blind agent on Cartpole, balance, which indicates that extracting visual", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 327, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 506, + 340 + ], + "score": 1.0, + "content": "information from observations is not as crucial on this task. Consequently, both PAD and baselines", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 338, + 272, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 272, + 350 + ], + "score": 1.0, + "content": "achieve similar performance on this task.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 11.5 + }, + { + "type": "table", + "bbox": [ + 155, + 405, + 453, + 515 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 362, + 505, + 396 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "score": 1.0, + "content": "Table 7. Episodic return on the training environment for each of the 9 tasks considered in DMControl,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 457, + 385 + ], + "score": 1.0, + "content": "mean and std. dev. for 10 seeds. Best method on each task is in bold and blue compares", + "type": "text" + }, + { + "bbox": [ + 457, + 373, + 484, + 383 + ], + "score": 0.84, + "content": "+ \\mathrm { I D M }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 383, + 479, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 479, + 397 + ], + "score": 1.0, + "content": "and without PAD. It is shown that PAD hurts minimally when the environment is unchanged.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "table_body", + "bbox": [ + 155, + 405, + 453, + 515 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 155, + 405, + 453, + 515 + ], + "spans": [ + { + "bbox": [ + 155, + 405, + 453, + 515 + ], + "score": 0.983, + "html": "
Training env.BlindSAC+DR+IDM+IDM (PAD)
Walker, walk235±17847±71756±71911±24895±28
Walker, stand388±10959±11928±36966±8956±20
Cartpole, swingup132±41850±28807±36849±30845±34
Cartpole,balance646±131978±22971±30982±20979±21
Ball in cup, catch150±96725±355469±339919±118910±129
Finger, spin3±2809±138686±295928±45927±45
Finger, turn_easy172±27462±146243±124462±152455±160
Cheetah, run264±75387±74195±46384±88380±91
Reacher, easy107±11264±11392±45390±126365±114
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CRLMazeRandomA2C+DR+IDM+IDM (PAD)+Rot+Rot (PAD)
Training env.-868±34371±198-355±93585±246-416±135729±148681±99
", + "type": "table", + "image_path": "19c0693ab395a83584be5ba43fa0acda3ed1d09f22e138b75faf8392ab844913.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 106, + 575, + 504, + 586.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 106, + 586.0, + 504, + 597.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 106, + 597.0, + 504, + 608.0 + ], + "spans": [], + "index": 34 + } + ] + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 107, + 628, + 359, + 641 + ], + "lines": [ + { + "bbox": [ + 104, + 627, + 361, + 643 + ], + "spans": [ + { + "bbox": [ + 104, + 627, + 361, + 643 + ], + "score": 1.0, + "content": "B LEARNING CURVES ON DEEPMIND CONTROL", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "All methods are trained until convergence (500,000 frames) on DMControl. While we do not", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "consider the sample efficiency of our method and baselines in this study, we report learning curves", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 146, + 689 + ], + "score": 1.0, + "content": "for SAC,", + "type": "text" + }, + { + "bbox": [ + 147, + 677, + 193, + 687 + ], + "score": 0.67, + "content": "\\mathrm { S A C + I D M }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "and SAC trained with domain randomization on three tasks in Figure 5 for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "completeness. SAC trained with and without an IDM are similar in terms of sample efficiency and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 698, + 507, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 507, + 713 + ], + "score": 1.0, + "content": "final performance, whereas domain randomization consistently displays worse sample efficiency,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "larger variation between seeds, and converges to sub-optimal performance in two out of the three", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 720, + 160, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 160, + 732 + ], + "score": 1.0, + "content": "tasks shown.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 382, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 383, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 383, + 96 + ], + "score": 1.0, + "content": "A PERFORMANCE ON THE TRAINING ENVIRONMENT", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 107, + 506, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 108, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 108, + 505, + 119 + ], + "score": 1.0, + "content": "Historically, agents have commonly been trained and evaluated in the same environment when", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 118, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 505, + 130 + ], + "score": 1.0, + "content": "benchmarking RL algorithms exclusively in simulation. Although such an evaluation procedure does", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 130, + 505, + 143 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 505, + 143 + ], + "score": 1.0, + "content": "not consider generalization, it is still a useful metric for comparison of sample efficiency and stability", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 505, + 153 + ], + "score": 1.0, + "content": "of algorithms. For completeness, we also evaluate our method and baselines in this setting on both", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 506, + 164 + ], + "score": 1.0, + "content": "DMControl and CRLMaze. DMControl results are reported in Table 7 and results on the CRLMaze", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 162, + 506, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 506, + 175 + ], + "score": 1.0, + "content": "environment are shown in Table 8. In this setting, we also compare to an additional baseline on", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 173, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 506, + 186 + ], + "score": 1.0, + "content": "DMControl: a blind SAC agent that operates purely on its previous actions. The performance of a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 184, + 506, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 506, + 197 + ], + "score": 1.0, + "content": "blind agent indicates to which degree a given task benefits from visual information. We find that,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 195, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 506, + 208 + ], + "score": 1.0, + "content": "while PAD improves generalization to novel environments, performance is virtually unchanged when", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 207, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 505, + 218 + ], + "score": 1.0, + "content": "evaluated on the same environment as in training. We conjecture that this is because the algorithm", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 217, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 506, + 230 + ], + "score": 1.0, + "content": "already is adapted to the training environment and any continued training on the same data distribution", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 228, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 506, + 240 + ], + "score": 1.0, + "content": "thus has little influence. We further emphasize that, even when evaluated on the training environment,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 239, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 397, + 250 + ], + "score": 1.0, + "content": "PAD still outperforms baselines on most tasks. For example, we observe a", + "type": "text" + }, + { + "bbox": [ + 397, + 239, + 418, + 250 + ], + "score": 0.87, + "content": "15 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 239, + 505, + 250 + ], + "score": 1.0, + "content": "relative improvement", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 250, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 262 + ], + "score": 1.0, + "content": "over SAC on the Finger, spin task. We hypothesize that this gain in performance is because the self-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 261, + 507, + 274 + ], + "spans": [ + { + "bbox": [ + 104, + 261, + 507, + 274 + ], + "score": 1.0, + "content": "supervised objective improves learning by constraining the intermediate representation of policies.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "score": 1.0, + "content": "A blind agent is no better than random on this particular task, which would suggest that agents", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 283, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 505, + 296 + ], + "score": 1.0, + "content": "benefit substantially from visual information in Finger, spin. Therefore, learning a good intermediate", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 294, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 506, + 306 + ], + "score": 1.0, + "content": "representation of that information is highly beneficial to the RL objective, which we find PAD to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "score": 1.0, + "content": "facilitate through its self-supervised learning framework. Likewise, the SAC baseline only achieves a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 315, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 126, + 326 + ], + "score": 0.86, + "content": "51 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 315, + 506, + 328 + ], + "score": 1.0, + "content": "improvement over the blind agent on Cartpole, balance, which indicates that extracting visual", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 327, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 506, + 340 + ], + "score": 1.0, + "content": "information from observations is not as crucial on this task. Consequently, both PAD and baselines", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 338, + 272, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 272, + 350 + ], + "score": 1.0, + "content": "achieve similar performance on this task.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 108, + 507, + 350 + ] + }, + { + "type": "table", + "bbox": [ + 155, + 405, + 453, + 515 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 362, + 505, + 396 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "score": 1.0, + "content": "Table 7. Episodic return on the training environment for each of the 9 tasks considered in DMControl,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 457, + 385 + ], + "score": 1.0, + "content": "mean and std. dev. for 10 seeds. Best method on each task is in bold and blue compares", + "type": "text" + }, + { + "bbox": [ + 457, + 373, + 484, + 383 + ], + "score": 0.84, + "content": "+ \\mathrm { I D M }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 383, + 479, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 479, + 397 + ], + "score": 1.0, + "content": "and without PAD. It is shown that PAD hurts minimally when the environment is unchanged.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "table_body", + "bbox": [ + 155, + 405, + 453, + 515 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 155, + 405, + 453, + 515 + ], + "spans": [ + { + "bbox": [ + 155, + 405, + 453, + 515 + ], + "score": 0.983, + "html": "
Training env.BlindSAC+DR+IDM+IDM (PAD)
Walker, walk235±17847±71756±71911±24895±28
Walker, stand388±10959±11928±36966±8956±20
Cartpole, swingup132±41850±28807±36849±30845±34
Cartpole,balance646±131978±22971±30982±20979±21
Ball in cup, catch150±96725±355469±339919±118910±129
Finger, spin3±2809±138686±295928±45927±45
Finger, turn_easy172±27462±146243±124462±152455±160
Cheetah, run264±75387±74195±46384±88380±91
Reacher, easy107±11264±11392±45390±126365±114
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CRLMazeRandomA2C+DR+IDM+IDM (PAD)+Rot+Rot (PAD)
Training env.-868±34371±198-355±93585±246-416±135729±148681±99
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While we do not", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "consider the sample efficiency of our method and baselines in this study, we report learning curves", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 146, + 689 + ], + "score": 1.0, + "content": "for SAC,", + "type": "text" + }, + { + "bbox": [ + 147, + 677, + 193, + 687 + ], + "score": 0.67, + "content": "\\mathrm { S A C + I D M }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "and SAC trained with domain randomization on three tasks in Figure 5 for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "completeness. SAC trained with and without an IDM are similar in terms of sample efficiency and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 698, + 507, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 507, + 713 + ], + "score": 1.0, + "content": "final performance, whereas domain randomization consistently displays worse sample efficiency,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "larger variation between seeds, and converges to sub-optimal performance in two out of the three", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 720, + 160, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 160, + 732 + ], + "score": 1.0, + "content": "tasks shown.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 655, + 507, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 119, + 81, + 488, + 204 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 119, + 81, + 488, + 204 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 119, + 81, + 488, + 204 + ], + "spans": [ + { + "bbox": [ + 119, + 81, + 488, + 204 + ], + "score": 0.969, + "type": "image", + "image_path": "9988caa8b9a55fc3afee7ef0f724429fd455ed6f365e57a9ccd860a250478099.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 119, + 81, + 488, + 122.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 119, + 122.0, + 488, + 163.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 119, + 163.0, + 488, + 204.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 208, + 506, + 275 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 208, + 507, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 440, + 221 + ], + "score": 1.0, + "content": "Figure 5. 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Episodic return is", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 230, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 236, + 243 + ], + "score": 1.0, + "content": "averaged across 10 seeds and the", + "type": "text" + }, + { + "bbox": [ + 236, + 231, + 256, + 241 + ], + "score": 0.88, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 230, + 506, + 243 + ], + "score": 1.0, + "content": "confidence intervals are visualized as shaded regions. 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We empirically find the difference between updating", + "type": "text" + }, + { + "bbox": [ + 476, + 349, + 487, + 358 + ], + "score": 0.86, + "content": "\\pi _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 311, + 371 + ], + "score": 1.0, + "content": "keeping it fixed negligible, and we choose to update", + "type": "text" + }, + { + "bbox": [ + 311, + 360, + 322, + 369 + ], + "score": 0.86, + "content": "\\pi _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "by default since its gradients are computed by", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 369, + 224, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 224, + 382 + ], + "score": 1.0, + "content": "back-propagation regardless.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 315, + 506, + 382 + ] + }, + { + "type": "table", + "bbox": [ + 178, + 440, + 428, + 492 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 389, + 506, + 434 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 390, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 506, + 401 + ], + "score": 1.0, + "content": "Table 9. Episodic return in test environments with randomized colors, mean and std. dev. for 10 seeds.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 401, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 264, + 413 + ], + "score": 1.0, + "content": "All methods use SAC. 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Walker, walk406±29452±38468±47
Walker, stand743±37802±41797±46
Cartpole, swingup585±73623±57630±63
", + "type": "table", + "image_path": "c266f3001c177fdb22a13cff9448f751770c90dd9138aea3b9fd7faa826e8279.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 178, + 440, + 428, + 457.3333333333333 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 178, + 457.3333333333333, + 428, + 474.66666666666663 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 178, + 474.66666666666663, + 428, + 491.99999999999994 + ], + "spans": [], + "index": 22 + } + ] + } + ], + "index": 19.25 + }, + { + "type": "title", + "bbox": [ + 106, + 510, + 367, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 508, + 367, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 367, + 524 + ], + "score": 1.0, + "content": "D COMPARISON TO ADAPTATION WITH REWARDS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 529, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "score": 1.0, + "content": "While our method does not require data collected prior to deployment and does not assume access", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 540, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 104, + 540, + 506, + 554 + ], + "score": 1.0, + "content": "to a reward signal, we additionally compare our method to a na¨ıve fine-tuning approach using", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "transitions and rewards collected from the target environment prior to deployment. To fine-tune the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 563, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 575 + ], + "score": 1.0, + "content": "pre-trained policy using rewards, we collect datasets consisting of 1, 10, and 100 episodes in each", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "target environment using the learned policy while keeping its parameters fixed, and then subsequently", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 583, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 167, + 598 + ], + "score": 1.0, + "content": "fine-tune both", + "type": "text" + }, + { + "bbox": [ + 167, + 586, + 178, + 596 + ], + "score": 0.85, + "content": "\\pi _ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 583, + 198, + 598 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 198, + 586, + 210, + 596 + ], + "score": 0.86, + "content": "\\pi _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 583, + 506, + 598 + ], + "score": 1.0, + "content": "on the collected data, following the same training procedure as during", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "score": 1.0, + "content": "the training phase. This fine-tuning approach is analogous to Julian et al. (2020) but does not use", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "data from the original environment during adaptation. Results are shown in Table 10. We find that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "na¨ıvely fine-tuning the policy using data collected prior to deployment can improve generalization", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 627, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 642 + ], + "score": 1.0, + "content": "but requires comparably more data than PAD, as well as access to a reward signal in the target", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "environment. This finding suggests that PAD may be a more suitable method for settings where data", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 650, + 441, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 441, + 663 + ], + "score": 1.0, + "content": "from the target environment is scarce and not easily accessible prior to deployment.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 529, + 506, + 663 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 679, + 375, + 690 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 376, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 376, + 693 + ], + "score": 1.0, + "content": "E ADDITIONAL ROBOTIC MANIPULATION SAMPLES", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Figure 6 provides samples from the training and test environments for the reach robotic manipulation", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "task. Agents are trained in simulation and deployed on a real robot. Samples from the push task are", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 721, + 183, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 183, + 732 + ], + "score": 1.0, + "content": "shown in Figure 4.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 699, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 80, + 506, + 135 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 506, + 92 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 506, + 92 + ], + "score": 1.0, + "content": "Table 10. Episodic return in test environments with randomized colors, mean and std. dev. for", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 506, + 103 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 506, + 103 + ], + "score": 1.0, + "content": "10 seeds. All methods use SAC trained with an inverse dynamics model (IDM) as auxiliary task.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 505, + 115 + ], + "score": 1.0, + "content": "Our method is denoted IDM (PAD), and we compare to a na¨ıve fine-tuning approach that assumes", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 112, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 112, + 506, + 127 + ], + "score": 1.0, + "content": "access to transitions and rewards collected from 1, 10, and 100 episodes, respectively, from target", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 125, + 248, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 125, + 248, + 137 + ], + "score": 1.0, + "content": "environments prior to deployment.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "table", + "bbox": [ + 144, + 152, + 462, + 203 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 342, + 142, + 428, + 151 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 341, + 141, + 429, + 153 + ], + "spans": [ + { + "bbox": [ + 341, + 141, + 429, + 153 + ], + "score": 1.0, + "content": "Fine-tuning w/ rewards", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "table_body", + "bbox": [ + 144, + 152, + 462, + 203 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 144, + 152, + 462, + 203 + ], + "spans": [ + { + "bbox": [ + 144, + 152, + 462, + 203 + ], + "score": 0.968, + "html": "
Random colorsIDMIDM (PAD)1 episode10 episodes100 episodes
Walker, walk406±29468±47395±78489±104561±62
Walker, stand743±37797±46661±65728±44784±31
Cartpole, swingup585±73630±63538±53605±51650±58
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Samples from the reach robotic manipulation task. The task is to move the robot gripper to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 350, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 505, + 362 + ], + "score": 1.0, + "content": "the location of the red disc. Agents are trained in setting (a) and evaluated in settings (b-d) on a real", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 361, + 332, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 332, + 373 + ], + "score": 1.0, + "content": "robot, taking observations from an uncalibrated camera.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 108, + 391, + 267, + 404 + ], + "lines": [ + { + "bbox": [ + 104, + 390, + 269, + 406 + ], + "spans": [ + { + "bbox": [ + 104, + 390, + 269, + 406 + ], + "score": 1.0, + "content": "F IMPLEMENTATION DETAILS", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 417, + 506, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 417, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 506, + 429 + ], + "score": 1.0, + "content": "In this section, we elaborate on implementation details for our experiments on DeepMind Control", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 429, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 506, + 441 + ], + "score": 1.0, + "content": "(DMControl) suite (Tassa et al., 2018) and CRLMaze (Lomonaco et al., 2019) for ViZDoom (Wyd-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 440, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 506, + 452 + ], + "score": 1.0, + "content": "much et al., 2018). 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All", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 324, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 506, + 336 + ], + "score": 1.0, + "content": "convolutional layers use 32 filters and all fully connected layers use a hidden size of 1024, as in", + "type": "text", + "cross_page": true + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 334, + 188, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 188, + 348 + ], + "score": 1.0, + "content": "Yarats et al. 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HyperparameterValue
Frame rendering3 ×100×100
Frame after crop3×84×84
Stacked frames3
Action repeat2 (finger)
8 (cartpole) 4(otherwise)
Discount factor y0.99
Episode length1,000
Learning algorithmSoft Actor-Critic
Self-supervised taskInverse Dynamics Model
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Replay buffer size500,000
Optimizer(πe,πä,π)Adam (β=0.9,β=0.999)
Optimizer (α)Adam(β=0.5,β=0.999)
Learning rate (πe,π,π$)3e-4 (cheetah)
Learning rate (α)le-3 (otherwise) 1e-4
Batch size128
Batch size (test-time)32
πe,π update freq.2
πe,π update freq. (test-time)1
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HyperparameterValue
Frame rendering3×100×100
Frame after crop3×84×84
Stacked frames3
Action repeat4
Discount factor y0.99
Episode length1,000
Learning algorithmAdvantage Actor-Critic
Self-supervised taskRotation Prediction
Number of training episodes1,000 (dom. rand.) 500 (otherwise)
Number of processes20
OptimizerAdam (β=0.9,β=0.999)
Learning rate1e-4
Learning rate (test-time)1e-5
Batch size20
32
Batch size (test-time) π,πloss coefficient0.5
1
πe,πloss coefficient (test-time)1
πe,π update freq. πe,π update freq.(test-time)1
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As the action spaces of both DMControl and robotic manipulation", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 397, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 408 + ], + "score": 1.0, + "content": "are continuous, the policy learned by SAC outputs the mean and variance of a Gaussian distribution", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "score": 1.0, + "content": "over actions. CRLMaze has a discrete action space and the policy learned by A2C thus learns a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 419, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 429 + ], + "score": 1.0, + "content": "soft-max distribution over actions. 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We use a frame stack of", + "type": "text" + }, + { + "bbox": [ + 307, + 490, + 332, + 500 + ], + "score": 0.9, + "content": "k = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 490, + 505, + 501 + ], + "score": 1.0, + "content": "frames for DMControl and CRLMaze, and", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 126, + 513 + ], + "score": 1.0, + "content": "only", + "type": "text" + }, + { + "bbox": [ + 127, + 501, + 151, + 511 + ], + "score": 0.89, + "content": "k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "frame for robotic manipulation. For completeness, we detail all hyperparameters used for", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 511, + 390, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 390, + 523 + ], + "score": 1.0, + "content": "the DMControl and CRLMaze environments in Table 11 and Table 12.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45.5 + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 639 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "score": 1.0, + "content": "Data augmentation. 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We evaluate our method and baselines by episodic return", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "of an agent trained in a single environment and tested in a collection of test environments, each with", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "distinct changes from the training environment. We assume no reward signal at test-time and agents", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "are expected to generalize without pre-training or resetting in the new environment. Therefore, we", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "make updates to the policy using a self-supervised objective, and we train using observations from", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "the environment in an online manner without memory, i.e. we make one update per step using the", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 106, + 721, + 207, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 207, + 732 + ], + "score": 1.0, + "content": "most-recent observation.", + "type": "text" + } + ], + "index": 64 + } + ], + "index": 61 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 111, + 294, + 300 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 81, + 295, + 102 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 80, + 296, + 92 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 296, + 92 + ], + "score": 1.0, + "content": "Table 11. Hyperparameters used for the DM-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 91, + 242, + 102 + ], + "spans": [ + { + "bbox": [ + 107, + 91, + 242, + 102 + ], + "score": 1.0, + "content": "Control (Tassa et al., 2018) tasks.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 111, + 294, + 300 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 111, + 294, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 294, + 300 + ], + "score": 0.978, + "html": "
HyperparameterValue
Frame rendering3 ×100×100
Frame after crop3×84×84
Stacked frames3
Action repeat2 (finger)
8 (cartpole) 4(otherwise)
Discount factor y0.99
Episode length1,000
Learning algorithmSoft Actor-Critic
Self-supervised taskInverse Dynamics Model
Number of training steps500,000
Replay buffer size500,000
Optimizer(πe,πä,π)Adam (β=0.9,β=0.999)
Optimizer (α)Adam(β=0.5,β=0.999)
Learning rate (πe,π,π$)3e-4 (cheetah)
Learning rate (α)le-3 (otherwise) 1e-4
Batch size128
Batch size (test-time)32
πe,π update freq.2
πe,π update freq. (test-time)1
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HyperparameterValue
Frame rendering3×100×100
Frame after crop3×84×84
Stacked frames3
Action repeat4
Discount factor y0.99
Episode length1,000
Learning algorithmAdvantage Actor-Critic
Self-supervised taskRotation Prediction
Number of training episodes1,000 (dom. rand.) 500 (otherwise)
Number of processes20
OptimizerAdam (β=0.9,β=0.999)
Learning rate1e-4
Learning rate (test-time)1e-5
Batch size20
32
Batch size (test-time) π,πloss coefficient0.5
1
πe,πloss coefficient (test-time)1
πe,π update freq. πe,π update freq.(test-time)1
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We use Soft Actor-Critic (SAC) (Haarnoja et al., 2018) for DMControl and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 386 + ], + "score": 1.0, + "content": "robotic manipulation, and Advantage Actor-Critic (A2C) for CRLMaze. Network outputs depend on", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "the task and learning algorithm. As the action spaces of both DMControl and robotic manipulation", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 397, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 408 + ], + "score": 1.0, + "content": "are continuous, the policy learned by SAC outputs the mean and variance of a Gaussian distribution", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "score": 1.0, + "content": "over actions. CRLMaze has a discrete action space and the policy learned by A2C thus learns a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 419, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 429 + ], + "score": 1.0, + "content": "soft-max distribution over actions. For details on the critics learned by SAC and A2C, the reader is", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 428, + 387, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 387, + 442 + ], + "score": 1.0, + "content": "referred to Haarnoja et al. (2018) and Mnih et al. (2016), respectively.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 363, + 506, + 442 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 457, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 507, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 507, + 469 + ], + "score": 1.0, + "content": "Hyperparameters. When applicable, we adopt our hyperparameters from Yarats et al. (2019) (DM-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "Control, robotic manipulation) and Lomonaco et al. (2019) (CRLMaze). For the robotic manipulation", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "experiments, our implementation closely follows that of DMControl, only differing by number of", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 490, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 307, + 501 + ], + "score": 1.0, + "content": "frames in an observation. We use a frame stack of", + "type": "text" + }, + { + "bbox": [ + 307, + 490, + 332, + 500 + ], + "score": 0.9, + "content": "k = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 490, + 505, + 501 + ], + "score": 1.0, + "content": "frames for DMControl and CRLMaze, and", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 126, + 513 + ], + "score": 1.0, + "content": "only", + "type": "text" + }, + { + "bbox": [ + 127, + 501, + 151, + 511 + ], + "score": 0.89, + "content": "k = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "frame for robotic manipulation. For completeness, we detail all hyperparameters used for", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 511, + 390, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 390, + 523 + ], + "score": 1.0, + "content": "the DMControl and CRLMaze environments in Table 11 and Table 12.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 456, + 507, + 523 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 639 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "score": 1.0, + "content": "Data augmentation. Random cropping is a commonly used data augmentation used in computer", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "vision systems (Krizhevsky et al., 2012; Szegedy et al., 2015) but has only recently gained interest", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "score": 1.0, + "content": "as a stochastic regularization technique in the RL literature (Srinivas et al., 2020; Kostrikov et al.,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 570, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 586 + ], + "score": 1.0, + "content": "2020; Laskin et al., 2020). We adopt the random crop proposed in Srinivas et al. 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This has the added benefits of regularization while still preserving spatio-temporal", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "patterns between frames. When learning an inverse dynamics model, we apply the same crop to all", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 466, + 629 + ], + "score": 1.0, + "content": "frames of a given observation but apply two different crops to the consecutive observations", + "type": "text" + }, + { + "bbox": [ + 466, + 616, + 505, + 628 + ], + "score": 0.91, + "content": "\\left( \\mathbf { s } _ { t } , \\mathbf { s } _ { t + 1 } \\right)", + "type": "inline_equation" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 626, + 209, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 194, + 640 + ], + "score": 1.0, + "content": "used to predict action", + "type": "text" + }, + { + "bbox": [ + 194, + 629, + 204, + 638 + ], + "score": 0.85, + "content": "\\mathbf { a } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 626, + 209, + 640 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 53, + "bbox_fs": [ + 105, + 540, + 506, + 640 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "Policy Adaptation during Deployment. We evaluate our method and baselines by episodic return", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "of an agent trained in a single environment and tested in a collection of test environments, each with", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "distinct changes from the training environment. We assume no reward signal at test-time and agents", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "are expected to generalize without pre-training or resetting in the new environment. 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As such, we apply both of these techniques when performing", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "Policy Adaptation during Deployment and use a batch size of 32. 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When using the policy to take", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 365, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 365, + 149 + ], + "score": 1.0, + "content": "actions, however, inputs to the policy are simply center-cropped.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 83, + 505, + 149 + ] + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git a/parse/train/pk4q0SD_r1X/pk4q0SD_r1X.md b/parse/train/pk4q0SD_r1X/pk4q0SD_r1X.md new file mode 100644 index 0000000000000000000000000000000000000000..72c009d6ac11d7dfaa5e4a953bb23a0008f524b5 --- /dev/null +++ b/parse/train/pk4q0SD_r1X/pk4q0SD_r1X.md @@ -0,0 +1,311 @@ +# COCO-LM: Correcting and Contrasting Text Sequences for Language Model Pretraining + +Yu Meng1∗, Chenyan Xiong2, Payal Bajaj2, Saurabh Tiwary2, Paul Bennett2, Jiawei Han1, Xia Song2 + +1 University of Illinois at Urbana-Champaign 2 Microsoft 1 {yumeng5,hanj}@illinois.edu 2 {chenyan.xiong,payal.bajaj,satiwary, paul.n.bennett,xiaso}@microsoft.com + +# Abstract + +We present a self-supervised learning framework, COCO-LM, that pretrains Language Models by COrrecting and COntrasting corrupted text sequences. Following ELECTRA-style pretraining, COCO-LM employs an auxiliary language model to corrupt text sequences, upon which it constructs two new tasks for pretraining the main model. The first token-level task, Corrective Language Modeling, is to detect and correct tokens replaced by the auxiliary model, in order to better capture token-level semantics. The second sequence-level task, Sequence Contrastive Learning, is to align text sequences originated from the same source input while ensuring uniformity in the representation space. Experiments on GLUE and SQuAD demonstrate that COCO-LM not only outperforms recent state-of-the-art pretrained models in accuracy, but also improves pretraining efficiency. It achieves the MNLI accuracy of ELECTRA with $5 0 \%$ of its pretraining GPU hours. With the same pretraining steps of standard base/large-sized models, COCO-LM outperforms the previous best models by $1 +$ GLUE average points. + +# 1 Introduction + +Pretrained language models (PLMs) have reshaped the way AI systems process natural language [11, 36, 39, 40]. Before task-specific training, it is now a common practice to first pretrain the deep neural networks, often Transformers [53], via a self-supervised token-level language modeling task [29, 31, 40]. Whether it is autoregressive [39], permutational [62], or masked language modeling (MLM) [11], the Transformer networks are pretrained to recover some omitted tokens using the rest of input texts. Then the language semantics captured during pretraining are conveyed to downstream tasks via the pretrained Transformer parameters [5, 8, 44]. + +Recent research [14, 16, 25, 43] observed several challenges in this self-supervised learning framework. One challenge is its efficiency. After pretrained for a while with the standard token-level language modeling, the networks have already captured the basic language patterns, making a large fraction of pretraining signals no longer informative. Linear improvement in the model effectiveness often requires exponentially more pretraining compute and parameters [25], which is unsustainable. Another challenge is the anisotropy of text representations from pretrained models. The sequence representations from many pretrained models are quite irregular [30, 43] and require dedicated fine-tuning approaches to be useful in sequence-level applications [32, 60]. + +Clark et al. [7] proposed a new pretraining strategy, ELECTRA, that uses an auxiliary language model (“generator”) to replace tokens in input texts and pretrains the main Transformer (“discriminator”) to detect replaced tokens. This improves the pretraining efficiency and effectiveness, but pretraining via binary classification hinders the model’s usage on applications requiring language modeling capability (e.g., prompt-based learning [15, 28, 46]). It could further distort the representation space as the Transformers are pretrained to output the same “non-replacement” label for all actual tokens. + +In this paper, we present a new self-supervised learning approach, COCO-LM, that pretrains Language Models by COrrecting and COntrasting corrupted text sequences. Following ELECTRA-style pretraining, COCO-LM employs an auxiliary model to corrupt the input texts, upon which it introduces two new pretraining tasks for the main Transformer, one at token level and one at sequence level. The token-level task, corrective language modeling (CLM), pretrains the main Transformer to detect and correct the tokens in the corrupted sequences. It uses a multi-task setup to combine the benefits of replaced token detection and language modeling. The sequence-level task, sequence contrastive learning (SCL), pretrains the model to align text sequences originated from the same source sequence and enforce uniformity of the representation space. + +In our experiments on GLUE [54] and $\mathrm { S Q u A D }$ [41] benchmarks, COCO-LM not only outperforms state-of-the-art pretraining approaches in effectiveness, but also significantly improves the pretraining efficiency. Under the same setting, COCO-LM matches the MNLI accuracy of RoBERTa and ELECTRA with $6 0 \%$ and $5 0 \%$ of their GPU hours in pretraining, respectively. When pretrained with the same number of steps, COCO-LM outperforms the previous best models by $1 +$ GLUE average points under the standard base/large-sized model evaluations. With 367 million parameters, COCO$\mathrm { L M _ { L a r g e + + } }$ reaches the MNLI accuracy of Megatron3.9B [49], one of the largest BERT-style model with 3.9 billion parameters. Our analyses provide further insights on the advantage of CLM in learning token representations and its effectiveness in prompted-based fine-tuning, as well as the benefit of SCL in ensuring alignment and uniformity in the representation space for better generalization1. + +# 2 Related Work + +Various token-level tasks have been used to pretrain language models. The most classic auto-regressive language modeling is to predict a token given all the previous tokens, or all subsequent ones [36, 39]. BERT uses masked language modeling (MLM) that recovers randomly masked tokens using the rest input. XLNet proposes permutation language modeling that conducts MLM in an autoregressive manner [62]. UniLM uses pseudo MLM which unifies autoregressive and MLM tasks [1, 13]. + +Sequence-level tasks are also explored, which often pretrain the model to predict certain cooccurrences of sequence pairs. For example, next sentence prediction [11], sentence ordering [27] and previous sentence prediction [56] concatenate two sentences (either correlated or random), and train the Transformer to classify the pair. + +Empirically, MLM is still among the most effective tasks to pretrain encoders [29, 31, 40]. RoBERTa [31] found the sentence-level task in BERT not benefitial and discarded it. BART [29] and T5 [40] both observed that MLM is often the most effective task. The empirical advantages of other pretraining tasks are more task-specific, for example, entity related masks for knowledge intensive applications [20, 24], and sequence-level tasks for long form text modeling [42]. + +Instead of randomly altering texts, ELECTRA [7] uses a smaller auxiliary Transformer pretrained by MLM to replace some tokens in the text sequences using its language modeling probability, and pretrains the main Transformer to detect the replaced tokens. ELECTRA achieves state-of-the-art accuracy in many language tasks [7]. Later, Clark et el. [6] developed ELECTRIC, which pretrains encoders by contrasting original tokens against negatives sampled from a cloze model. ELECTRIC re-enables the language modeling capability but underperforms ELECTRA in downstream tasks. + +Our work is also related to contrastive learning which has shown great success in visual representation learning [4, 22, 34]. Its effectiveness of in language is more observed in the fine-tuning stage, for example, in sentence representation [16], dense retrieval [60], and GLUE fine-tuning [19]. + +# 3 Method + +We present the preliminaries of PLMs, their challenges, and the new COCO-LM framework. + +# 3.1 Preliminary on Language Model Pretraining + +In this work we focus on pretraining BERT-style bidirectional Transformer encoders [11] that are widely used in language representation tasks. We first recap the masked language modeling (MLM) task introduced by BERT [11] and then discuss the pretraining framework of ELECTRA [7]. + +BERT Pretraining uses the masked language modeling task (MLM) [11], which is to take an input sequence $X ^ { \mathrm { o r i g } } = [ x _ { 1 } ^ { \mathrm { o r i g } } , \dotsc , x _ { i } ^ { \mathrm { o r i g } } , \dotsc , x _ { n } ^ { \mathrm { o r i g } } ]$ , with $1 5 \%$ random tokens replaced by [MASK] symbols (e.g., the $i$ -th token), and train the model to predict the original tokens at the masked positions: + +$$ +\left[ x _ { 1 } ^ { \mathrm { o r i g } } , \dots , \ [ \mathrm { M \AA S K } ] _ { i } , \dots , x _ { n } ^ { \mathrm { o r i g } } \right] \xrightarrow { \mathrm { T r a n s f o r m e r } } H \xrightarrow { \mathrm { M L M H e a d } } p _ { \mathrm { M L M } } ( x | h _ { i } ) , +$$ + +where the Transformer generates contextualized representations ${ \pmb H } = \{ h _ { i } \} _ { i = 1 } ^ { n }$ . The MLM Head predicts the masked token from the vocabulary $V$ using the hidden representation $\boldsymbol { h } _ { i }$ and token embeddings $_ { \textbf { \em x } }$ . The pretraining minimizes the MLM loss on the set of masked positions $\mathcal { M }$ . Specifically, + +$$ +p _ { \mathrm { M L M } } ( x | h _ { i } ) = \frac { \exp ( x ^ { \top } h _ { i } ) } { \sum _ { x _ { t } \in V } \exp ( x _ { t } ^ { \top } h _ { i } ) } ; \quad \mathcal { L } _ { \mathrm { M L M } } = \mathbb { E } \left( - \sum _ { i \in \mathcal { M } } \log p _ { \mathrm { M L M } } \left( x _ { i } ^ { \mathrm { o r i g } } \middle | h _ { i } \right) \right) . +$$ + +ELECTRA Pretraining uses two Transformers, a “generator” pretrained by MLM, and a “discriminator” pretrained using the generator’s outputs. We refer them as auxiliary and main Transformers, as the former is discarded after pretraining and the latter may be trained by “generative” tasks too. + +The auxiliary model outputs a corrupted sequence $X ^ { \mathrm { M L M } }$ by sampling from its predicted probability: + +$$ +x _ { i } ^ { \mathrm { M L M } } \sim p _ { \mathrm { M L M } } \left( x | h _ { i } \right) , \mathrm { i f } i \in \mathcal { M } ; \quad x _ { i } ^ { \mathrm { M L M } } = x _ { i } ^ { \mathrm { o r i g } } , \mathrm { e l s e } . +$$ + +The masked positions are replaced by sampled tokens considered plausible in context by the auxiliary Transformer, which are more deceiving than random replacements. ELECTRA uses a skinnier auxiliary network (e.g., hidden dimension is $1 / 3$ of the main model) to control the signal difficulty. + +The main Transformer takes $X ^ { \mathrm { M L M } }$ and classifies the replaced tokens: + +$$ +\begin{array} { r } { X ^ { \mathrm { M L M } } \xrightarrow { \mathrm { M a i n ~ T r a n s f o r m e r } } \pmb { H } \xrightarrow { \mathrm { R T D ~ H e a d } } p _ { \mathrm { R T D } } \left( \mathbb { 1 } \big ( x _ { i } ^ { \mathrm { M L M } } = x _ { i } ^ { \mathrm { o r i g } } \big ) \big | h _ { i } \right) , } \end{array} +$$ + +where $\mathbb { 1 } ( \cdot )$ is the indicator function. The Replaced Token Detection (RTD) head uses a sigmoid linear layer to output the binary probability, and the main Transformer is trained with binary cross entropy loss. The RTD task is trained on all tokens instead of masked ones and improves efficiency. + +The two Transformers are pretrained jointly. The auxiliary model gradually generates more realistic replacement tokens and the main model learns to better detect them. This forms a natural learning curriculum and significantly improves ELECTRA’s accuracy in downstream tasks [7]. + +# 3.2 Challenges of ELECTRA-Style Pretraining + +Missing Language Modeling Benefits. The classification task in ELECTRA is simpler and more stable [61], but raises two challenges. The first is the lack of language modeling capability which is a necessity in some tasks [6]. For example, prompt-based learning requires a language model to generate labels [15, 33, 45, 46]. The second is that the binary classification task may not be sufficient to capture certain word-level semantics that are critical for token-level tasks. + +Squeezing Representation Space. Another challenge is that the representations from Transformer-based language models often reside in a narrow cone, where two random sentences have high similarity scores (lack of uniformity), + +![](images/3a8c91a226e2aa036c614aa7d53c2b811d005dba76544c5617acff0d9b62c12a.jpg) +Figure 1: Cosine similarity distributions of random/similar sequence pairs using [CLS] embeddings from pretrained models. Histograms/curves are distribution bins/kernel density estimates. + +and closely related sentences may have more different representations (lack of alignment) [14, 16, 30]. + +![](images/88292be09312bfcedea184a72ece038e7b7b44db60e61df6cc39ca285f523c38.jpg) +Figure 2: The overview of COCO-LM. The auxiliary Transformer is pretrained by MLM. Its corrupted text sequence is used as the main Transformer’s pretraining input in Corrective Language Modeling and paired with the cropped original sequence for Sequence Contrastive Learning. + +Figure 1 illustrates such behaviors with random sentence pairs (from pretraining corpus) and semantically similar pairs (those annotated with maximum similarity from STS-B [3]). With RoBERTa, the cosine similarities of most random sentence pairs are near 0.8, bigger than many semantically similar pairs. The representation space from ELECTRA is even more squeezed. Nearly all sentence pairs, both random and similar ones, have around 0.9 cosine similarity. This may not be surprising as ELECTRA is pretrained to predict the same output (“non-replacement”) for all tokens in these sequences. The irregular representation space raises the risk of degeneration [37, 55] and often necessitates sophisticated post-adjustment or fine-tuning to improve the sequence representations [16, 30, 32, 60]. + +# 3.3 COCO-LM Pretraining + +COCO-LM also employs an auxiliary Transformer to construct the corrupted text sequence, as in Eqn. (1), but it introduces two new pretraining tasks upon the corrupted sequences to address the challenges previously described. In the rest of this section, we present these two tasks and then the detailed configurations of COCO-LM. Its framework is illustrated in Figure 2. + +Corrective Language Modeling (CLM) trains the main Transformer to recover the original tokens, given the corrupted text sequence XMLM: + +$$ +\begin{array} { r } { X ^ { \mathrm { M L M } } \xrightarrow { \mathrm { M a i n ~ T r a n s f o r m e r } } H \xrightarrow { \mathrm { C L M H e a d } } p _ { \mathrm { C L M } } ( x | h _ { i } ) . } \end{array} +$$ + +The CLM Head uses the hidden representations $\pmb { H }$ to output a language modeling probability, instead of a binary classification score. The forward pass of the CLM Head is the same as All-Token MLM, a variation of ELECTRA [7] that consists of a language modeling layer and a binary classification layer for the copy mechanism: + +$$ +\begin{array} { r l } & { p _ { \mathrm { L M } } ( x _ { i } | h _ { i } ) = \mathbb { 1 } \left( x _ { i } = x _ { i } ^ { \mathrm { M L M } } \right) p _ { \mathrm { c o p y } } ( 1 | h _ { i } ) + p _ { \mathrm { c o p y } } ( 0 | h _ { i } ) \frac { \exp ( x _ { i } ^ { \top } h _ { i } ) } { \sum _ { x _ { t } \in V } \exp ( x _ { t } ^ { \top } h _ { i } ) } , } \\ & { p _ { \mathrm { c o p y } } ( y _ { i } | h _ { i } ) = \exp ( y _ { i } \cdot w _ { \mathrm { c o p y } } ^ { \top } h _ { i } ) / \left( \exp ( w _ { \mathrm { c o p y } } ^ { \top } h _ { i } ) + 1 \right) , } \end{array} +$$ + +where ${ \pmb w } _ { \mathrm { c o p y } }$ is a learnable weight and $p _ { \mathrm { c o p y } } ( y _ { i } | h _ { i } )$ is the copy mechanism ( $y _ { i } = 1$ when the input token is original and can be directly copied to the output; $y _ { i } = 0$ when the input token needs to be corrected to another token from the vocabulary). + +In ELECTRA, All-Token MLM performs worse than RTD [7]. Language modeling on the corrupted text sequence $X ^ { \mathrm { M L M } }$ is hard as the replaced tokens from the auxiliary model are more deceiving than [MASK]. To improve the language model learning, different from All-Token MLM, CLM employs a + +multi-task setup that combines the RTD task to explicitly train the copy mechanism $p _ { \mathrm { c o p y } } ( \cdot )$ + +$$ +\begin{array} { l } { \mathcal { L } _ { \mathrm { c o p y } } = - \mathbb { E } \left( \displaystyle \sum _ { i = 1 } ^ { n } \mathbb { 1 } \left( x _ { i } ^ { \mathrm { M L M } } = x _ { i } ^ { \mathrm { o r i g } } \right) \log p _ { \mathrm { c o p y } } ( 1 | h _ { i } ) + \mathbb { 1 } \left( x _ { i } ^ { \mathrm { M L M } } \neq x _ { i } ^ { \mathrm { o r i g } } \right) \log p _ { \mathrm { c o p y } } ( 0 | h _ { i } ) \right) , \mathrm { ~ } \forall i \mathrm { ~ c o p y ~ } ( \mathbb { E } ) , } \\ { \mathcal { L } _ { \mathrm { L M } } = - \mathbb { E } \left( \displaystyle \sum _ { i \in \mathcal { M } } \log p _ { \mathrm { L M } } \left( x _ { i } ^ { \mathrm { o r i g } } | h _ { i } \right) \right) } \\ { \displaystyle \qquad = - \mathbb { E } \left( \displaystyle \sum _ { i \in \mathcal { M } } \log \left( \mathbb { 1 } \left( x _ { i } ^ { \mathrm { M L M } } = x _ { i } ^ { \mathrm { o r i g } } \right) p _ { \mathrm { c o p y } } ^ { \mathrm { s g } } ( 1 | h _ { i } ) + p _ { \mathrm { c o p y } } ^ { \mathrm { s g } } ( 0 | h _ { i } ) \frac { \exp ( x _ { i } ^ { \top } h _ { i } ) } { \sum _ { x _ { t } \in V } \exp ( x _ { t } ^ { \top } h _ { i } ) } \right) \right) , \mathrm { ~ } } \\ { \mathcal { L } _ { \mathrm { C L M } } = \lambda _ { \mathrm { c o p y } } \mathcal { L } _ { \mathrm { c o p y } } + \mathcal { L } _ { \mathrm { L M } } . } \end{array} +$$ + +The hyperparameter $\lambda _ { \mathrm { c o p y } }$ balances the weights of the two tasks. The binary cross entropy loss in Eqn. (2) explicitly trains the copy probability. We also use stop gradient (sg) to decouple the gradient backpropagation to $p _ { \mathrm { c o p y } } ( \cdot )$ from the LM task. This way, the main Transformer first learns the easier classification task and then uses it to help learn the harder LM task. The binary classification task is trained on all tokens while the language modeling task is trained only on masked positions. + +CLM combines the advantages of MLM and ELECTRA: The main Transformer is trained on all tokens with the help of the binary classification task while also being able to predict words, thus enjoying the efficiency benefits of ELECTRA and preserving the language modeling benefits. + +Sequence Contrastive Learning (SCL) forms a contrastive learning objective upon the sequence embeddings to learn more robust representations. Broadly, contrastive learning is to align a positive pair of instances, often different views of the same information [4, 34], in contrast to unrelated negative instances [22, 60]. The different views are often obtained by applying data augmentations on the same input, for example, rotation, cropping, and blurring on visual representations [4, 34], so that the neural networks can learn representations robust to these data alterations. + +In COCO-LM, the corrupted sequence $X ^ { \mathrm { M L M } }$ already provides a form of data augmentation. We pair it with another augmentation, $X ^ { \mathrm { c r o p } }$ , a randomly cropped contiguous span of $X ^ { \mathrm { o r i g } }$ (the length of $X ^ { \mathrm { c r o p } }$ is $9 0 \%$ of $X ^ { \mathrm { o r i g } }$ so that the major sequence meaning is preserved), to construct the positive pair and to contrast with random negatives. + +Specifically, a training batch $B$ in SCL includes a random set of corrupted and cropped sequences: 1contrastive pair $B = \{ ( X _ { 1 } ^ { \mathrm { M L M } } , X _ { 1 } ^ { \mathrm { c r o p } } ) , \dots , ( X _ { N } ^ { \mathrm { M L M } } , X _ { N } ^ { \mathrm { c r o p } } ) \}$ $( X , X ^ { + } )$ N N consists of either $( X _ { k } ^ { \mathrm { M L M } } , X _ { k } ^ { \mathrm { c r o p } } )$ , with $X _ { k } ^ { \mathrm { M L M } }$ and or $X _ { k } ^ { \mathrm { c r o p } }$ $( \ddot { X } _ { k } ^ { \mathrm { c r o p } } , X _ { k } ^ { \mathrm { M L M } } )$ originated from $X _ { k } ^ { \mathrm { o r i g } }$ . A positive trast). The negative instances are all the remaining sequences in the batch $\ddot { B } ^ { - } = B \setminus \{ ( X , X ^ { + } ) \}$ contrastive loss is formulated as: + +$$ +\begin{array} { r l r } { { \mathcal { L } _ { \mathrm { S C L } } = - \mathbb { E } ( \log \frac { \exp ( \cos ( s , s ^ { + } ) / \tau ) } { \exp ( \cos ( s , s ^ { + } ) / \tau ) + \sum _ { X ^ { - } \in B ^ { - } } \exp ( \cos ( s , s ^ { - } ) / \tau ) } ) , } } \\ & { } & { = - \mathbb { E } ( \cos ( s , s ^ { + } ) / \tau - \log ( \exp ( \cos ( s , s ^ { + } ) / \tau ) + \sum _ { X ^ { - } \in B ^ { - } } \exp ( \cos ( s , s ^ { - } ) / \tau ) ) ) , } \end{array} +$$ + +where $s , s ^ { + } , s ^ { - }$ are the representations of $X , X ^ { + } , X ^ { - }$ , respectively, from the main Transformer (i.e., $\boldsymbol { h } _ { \mathrm { [ C L S ] } } ,$ ). The similarity metric is cosine similarity (cos) and the temperature $\tau$ is set to $1$ . + +As shown in Wang et al. [55], the first term in Eqn. (3) $( \cos ( s , s ^ { + } ) )$ improves alignment of the space. It encourages representations to be robust to the corruptions and the alterations on the original text. The second term in Eqn. (3) promotes uniformity. It pushes unrelated sequences apart in the representation space and ensures low cosine similarity between random data points. Several studies have observed improved generalization ability from better alignment and uniformity [16, 37, 55]. + +Aligning $X ^ { \mathrm { M L M } }$ with $X ^ { \mathrm { c r o p } }$ requires the main Transformer to produce sequence representations robust to both token-level (i.e., MLM replacements) and sequence-level (i.e., cropping) alterations. The model is thus encouraged to reason more using partially altered sequences to recover the original information. + +Overall Training. COCO-LM uses the following loss function: + +$$ +\mathcal { L } _ { \mathrm { C O C O - L M } } = \mathcal { L } _ { \mathrm { M L M } } ^ { \mathrm { A u x . } } + \mathcal { L } _ { \mathrm { C L M } } ^ { \mathrm { M a i n } } + \mathcal { L } _ { \mathrm { S C L } } ^ { \mathrm { M a i n } } . +$$ + +The auxiliary Transformer is pretrained by masked language modeling (MLM) and generates corrupted sequences. The main Transformer is pretrained to correct the corruption (CLM) and to contrast the corrupted sequences with the cropped sequences (SCL). The two Transformers are pretrained jointly with the loss in Eqn. (4). The main Transformer is used in downstream applications. + +Network Configurations. Similar to ELECTRA, the auxiliary Transformer is smaller than the main model, but we use different configurations in the auxiliary model: (1) We reduce the number of layers to $1 / 3$ or $1 / 4$ (under base or large model setup, respectively) but keep its hidden dimension the same with the main model, instead of shrinking its hidden dimensions; (2) We disable dropout in it when sampling replacement tokens. We find such configurations empirically more effective and use them as the backbone of COCO-LM. The main Transformer follows the standard architecture of BERT/ELECTRA and can be easily adopted by downstream application pipelines with almost no changes. + +# 4 Experimental Setup + +Pretraining Settings. We employ three standard settings, base, base $^ { + + }$ , and large $^ { + + }$ . Base is the $\mathbf { B E R T _ { B a s e } }$ training configuration [11]: Pretraining on Wikipedia and BookCorpus [63] (16 GB of texts) for 256 million samples on 512 token sequences (125K batches with 2048 batch size). We use the same corpus and 32, 768 uncased BPE vocabulary [47] as with TUPE [26]. + +$B a s e + +$ trains the base size model with larger corpora and/or more training steps. Following recent research [1, 31, 62], we add in OpenWebText [18], CC-News [31], and STORIES [52], to a total of 160 GB texts, and train for 4 billion (with 2048 batch size) samples [31]. We follow the prepossessing of UniLMV2 [1] and use 64, 000 cased BPE vocabulary. + +$L a r g e { + + }$ uses the same training corpora as $b a s e + +$ and pretrains for 4 billion samples (2048 batch size). Its Transformer configuration is the same with BERTLarge [11]. + +Model Architecture. Our base/base $^ { + + }$ model uses the $\mathbf { B E R T _ { B a s e } }$ architecture [11]: 12 layer Transformer, 768 hidden size, plus T5 relative position encoding [40]. Our large $^ { + + }$ model is the same with $\mathrm { B E R T _ { L a r g e } }$ , 24 layer and 1024 hidden size, plus T5 relative position encoding [40]. Our auxiliary network uses the same hidden size but a shallow 4-layer Transformer in base/base $^ { + + }$ and a 6-layer one in $l a r g e + +$ . When generating $X ^ { \mathrm { M L M } }$ we disable dropout in the auxiliary model. + +Downstream Tasks. We use the tasks included in GLUE [54] and $\mathrm { S Q u A D } 2 . 0$ reading compression [41]. Please refer to Appendix A for more details about GLUE tasks. Standard hyperparameter search in fine-tuning is performed, and the search space can be found in Appendix B. The fine-tuning protocols use the open-source implementation of TUPE [26]. The reported results are the median of five random seeds on GLUE and SQuAD. + +Baselines. We compare with various pretrained models in each setting. To reduce the variance in data processing/environments, we also pretrain and fine-tune RoBERTa and ELECTRA under exactly the same setting with COCO-LM, marked with “(Ours)”. All numbers unless marked by “(Ours)” are from reported results in recent research (more details in Appendix C). + +Implementation Details. Our implementation builds upon the open-source implementation from MC-BERT [61] and fairseq [35]. More implementation details are mentioned in Appendix D. + +# 5 Evaluation Results + +Three groups of experiments are conducted to evaluate COCO-LM and its two new pretraining tasks. + +# 5.1 Overall Results and Ablations + +Overall Results are listed in Table 1. Under all three settings, COCO-LM outperforms all recent state-of-the-art pretraining models on GLUE average and SQuAD. It improves the state-of-the-art GLUE score by about one point under all three settings. COCO-LM also enjoys better parameter efficiency. Using less than $1 0 \%$ of Megatron’s parameters, $\mathrm { C O C O - L M _ { L a r g e + + } }$ matches the MNLI accuracy of Megatron3.9B, one of the largest pretrained BERT-style encoders. + +
ModelParamsGLUE Single TaskSQuAD 2.0
MNLI-(m/mm)QQPQNLISST-2CoLARTEMRPCSTS-BAVGEMF1
Base Setting: BERT Base Size,Wikipedia + Book Corpus (16GB)
BERT[11]110M84.5/-91.391.793.258.968.687.389.583.173.776.3
RoBERTa [31]125M84.7/-1192.71111179.7
XLNet [62]110M85.8/85.41192.71111178.581.3
ELECTRA [7]110M86.0/85.390.091.993.464.370.884.989.183.780.583.3
MC-BERT [61]110M85.7/85.289.791.392.362.175.086.088.083.7
DeBERTa [23]134M86.3/86.2二一一一一一一79.382.5
TUPE [26]110M86.2/86.291.392.293.363.673.689.989.284.911
RoBERTa (Ours)110M85.8/85.591.392.093.760.168.287.388.583.377.780.5
ELECTRA (Ours)110M86.9/86.791.992.693.666.275.188.289.785.579.782.6
COCO-LM110M88.5/88.392.093.193.263.984.891.490.387.282.485.2
Base++ Seting: BERT Base Size,Bigger Training Data,and/or More Training Steps
XLNet [62]110M86.8/-91.491.794.760.274.088.289.584.680.2
RoBERTa [31]125M87.6/-91.992.894.863.678.790.291.286.480.583.7
UniLMV2[1]110M88.5/-91.793.595.165.281.391.891.087.183.386.1
DeBERTa [23]134M88.8/88.5一一一1一一一83.186.2
CLEAR [59]110M86.7/-90.092.994.564.378.389.289.885.7一1
COCO-LM134M90.2/90.092.294.294.667.387.491.291.888.685.488.1
Large++ Setting: BERTLarge Size,Bigger Training Data,and More Training Steps
XLNet [62]360M90.8/90.892.394.997.069.085.990.892.589.287.990.6
RoBERTa [31]356M90.2/90.292.294.796.468.086.690.992.488.986.589.4
ELECTRA[7]335M90.9/-92.495.096.969.188.090.892.689.488.090.6
DeBERTa [23]384M91.1/91.192.395.396.870.5111188.090.7
COCO-LM367M91.4/91.692.895.796.973.991.092.292.790.888.291.0
Megatron1.3B [49]1.3B90.9/91.092.611一187.190.2
Megatron3.9B [49]3.9B91.4/91.492.711188.591.2
+ +Table 1: Results on GLUE and SQuAD 2.0 development set. All results are single-task, single-model fine-tuning. Results not available in public reports are marked as “–”. DeBERTa reported RTE, MRPC and STS-B results by fine-tuning from MNLI checkpoints which are not single-task results. We use Spearman correlation for STS, Matthews correlation for CoLA, and accuracy for the rest on GLUE. AVG is the average of the eight tasks on GLUE. All baseline results unless marked by (Ours) are reported by previous research. + +
ModelParamsMNLI-(m/mm)QQPQNLISST-2CoLARTEMRPCSTS-BAVG
Base/Base++ Setting: BERT Base Size
BERTBase110M84.6/83.489.290.593.552.166.484.885.880.8
ELECTRABase++110M88.5/88.089.593.196.064.675.288.190.285.6
COCO-LMBase++134M89.8/89.389.894.295.668.682.388.590.387.4
Large/Large++ Seting: BERT Large Size
BERTLarge335M86.7/85.989.392.794.960.570.185.486.583.2
ELECTRALarge++335M90.7/90.290.495.596.768.186.189.291.788.5
COCO-LMLarge++367M91.6/91.190.595.896.770.589.288.491.889.3
+ +Table 2: GLUE test set results obtained from the GLUE leaderboard. We perform hyperparameter search for each task with ten random seeds and use the best development set model for test predictions. All results are from vanilla single-task fine-tuning (no ensemble, task-specific tricks, etc.). + +Table 2 shows GLUE test set results which further confirm the advantages of COCO-LM over previous methods. + +Efficiency. In downstream tasks, the efficiency of COCO-LM is the same with BERT. In pretraining, the auxiliary model and SCL introduce extra cost. However, as shown in Figure 3, COCO-LM is more efficient in GPU hours. It outperforms RoBERTa & ELECTRA by $1 +$ points on MNLI with the same GPU hours and reaches their accuracy with around $6 0 \%$ & $5 0 \%$ GPU hours, respectively. + +Ablation Studies. Table 3 shows the ablations of COCO-LM under the base setting on GLUE DEV. + +Pretraining Task. With only RTD, our backbone model with the shallow auxiliary Transformer is quite effective. CLM and SCL both provide additional improvements on MNLI and GLUE average. Their advantages are better observed on different tasks, for example, CLM on MNLI-mm and SCL on RTE and MRPC. Combining the two in COCO-LM provides better overall effectiveness. In later experiments, we further analyze the benefits of these two tasks. + +
GroupMethodMNLI-(m/mm)QQPQNLISST-2CoLARTEMRPCSTS-BAVG
COCO-LMBase88.5/88.392.093.193.263.984.891.490.387.2
Pretraining TaskRTDOnly88.4/88.292.193.592.767.380.589.090.986.8
CLMOnly88.6/88.492.093.293.767.480.190.090.486.9
SCL +RTD88.6/88.292.193.593.864.382.790.290.686.9
Network Setingw/o. Rel-Pos w. ELECTRA's Auxiliary88.2/87.792.293.493.768.882.791.290.687.6
Training88.0/87.791.992.793.564.381.289.589.786.3
w.Random Replacements84.9/84.791.491.191.441.670.087.387.180.6
Signalw. Converged Auxiliary88.3/88.192.092.894.364.278.390.490.286.3
CLM SetupAll-Token LM Only87.2/87.092.693.788.589.784.7
CLM w/o. Copy88.0/87.991.8 91.893.194.460.6 66.674.0 76.989.590.186.3
CLM w/o. Stop-grad88.5/88.292.092.994.366.580.990.090.686.9
+ +Table 3: Ablations on GLUE Dev. that eliminate (w/o.), keep (Only) or switch (w.) one component. + +![](images/7c7b853b454b46cee7bf8429edacf37572ac7613714aa9383792bc931834deea.jpg) +Figure 3: $\mathrm { C O C O - L M _ { B a s e } }$ on MNLI Dev. ( $y$ -axes) at different pretraining hours on four DGX-2 nodes (64 V100 GPUs). The final training hours and accuracy of RoBERTa (Ours) and ELECTRA (Ours) measured in the same settings are marked. + +![](images/b0838b4ee1160733d8845b67109fdb9dc7efa9acb20b3820901e16797ecc717f.jpg) +Figure 4: The performance of COCO- $\mathbf { \cdot L M _ { B a s e } }$ when pretrained with different crop fractions. The $x$ -axis is the fraction of $X ^ { \mathrm { o r i g } }$ being kept (no cropping is $1 0 0 \%$ ). + +Architecture. Removing relative position encoding (Rel-Pos) leads to better numbers on some tasks but significantly hurts MNLI. Using a shallow auxiliary network and keeping the same hidden dimension (768) is more effective than ELECTRA’s 12-layer but 256-hidden dimension generator. + +Pretraining Signal Construction. Using randomly replaced tokens to corrupt text sequence hurts significantly. Using a converged auxiliary network to pretrain the main model also hurts. It is better to pretrain the two Transformers together, as the auxiliary model gradually increases the difficulty of the corrupted sequences and provides a natural learning curriculum for the main Transformer. + +CLM Setup. Disabling the multi-task learning and using All-Token MLM [7] reduces model accuracy. The copy mechanism is effective. The benefits of the stop gradient operation are more on stability (preventing training divergence). + +# 5.2 Analyses of Contrastive Learning with SCL + +This group of experiments analyzes the behavior of SCL. All experiments use the base setting. + +Ablation on Data Augmentation. Figure 4 shows the effects of the cropping operation when forming positive SCL pairs with the corrupted sequence. Using the original sequence results in worse GLUE accuracy. It is less informative as the model no longer needs to learn representations robust to sequence-level alteration. Cropping too much (e.g., only keeping $7 0 \%$ of the original sequence), may hurt as it can alter the semantics too much. Empirically a simple alteration works the best, similar to the observations in recent research [4, 16, 22]. + +Alignment and Uniformity. Figure 5 plots the distribution of cosine similarities between random sequence pairs and similar ones using representations pretrained by COCO-LM. The representation space from COCO-LM is drastically different from those in Figure 1. With COCO-LM, similar pairs are more aligned and random pairs are distributed more uniformly. Many similar pairs have near 1 cosine similarity and are clearly separated from random pairs which center around 0. The t-SNE [9] plot in Figure 6 further demonstrates the benefits of SCL. The similar sentence pairs (marked by same shapes) are aligned closer when pretrained with SCL. Their average cosine similarity is 0.925 when pretrained with SCL, while is 0.863 without SCL. This better alignment and uniformity is achieved by COCO-LM with SCL via pretraining, without using task-specific data nor supervised labels. + +![](images/82aaaec6538ad34bc36ab524aaa756ff25c2f6723407c61d051220eb759a68b2.jpg) +Figure 5: Cosine similarity of sequence pairs randomly sampled from pretraining corpus and most similar pairs from STS-B using [CLS] from COCO- $\mathrm { L M } _ { \mathrm { B a s e } }$ . + +![](images/c1d97d0fc27a4c3c0d227b63fc15ba16a052a5c2bfaf1aa791b9ae2c82470dd6.jpg) +Figure 6: The t-SNE of sequence representations learned with or without SCL. The points are sampled from the most semantically similar sentences pairs from STS-B (with 5-score labels). The [CLS] embeddings are not fine-tuned. Some randomly selected similar pairs are marked by same shapes. + +![](images/992113321700f003cba33a4d9ae2e2e1ae1bb00ef61b749c875f454276b56dcf.jpg) +Figure 7: Analyses of SCL. Figs. (a) and (b) show the average cosine similarity between the [CLS] embeddings of positive and negative contrastive pairs during pretraining. Figs. (c) and (d) show the few-shot accuracy on MNLI with different fractions of MNLI training set used ( $\scriptstyle { \dot { x } }$ -axes). The error bars mark the max/min and the solid lines are the average of five fine-tuning runs. + +Regularizing the Representation Learning for Better Few-Shot Ability. One would expect any pretrained Transformers to easily align a pair of corrupted sequence and cropped sequence as the two share about $8 0 \%$ tokens. However, as shown in Figure 7a, that is not the case: Without SCL, the cosine similarity of the positive pairs is even lower than random negatives. SCL is necessary to regularize the representation space and to reduce the risk of degeneration (Figure 7b). + +Similar to empirical observations and theoretical analyses in recent research [14, 16, 55], a more regularized representation space results in better generalization ability in scenarios with limited labels. Figure $\mathrm { 7 c }$ and 7d show the results when COCO-LM are trained (via standard fine-tuning) with only a fraction of MNLI labels. The improvements brought by SCL are more significant when fewer fine-tuning labels are available. With $1 \%$ MNLI labels, pretraining with SCL improves MNLI- $. \mathrm { m } / \mathrm { m m }$ accuracy by $0 . 8 / 0 . 5$ compared to that without SCL. Using only $1 0 \% / 2 0 \%$ labels, COCO-LM with SCL reaches similar MNLI accuracy with RoBERTa (Ours)/ELECTRA (Ours) fine-tuned with all labels, respectively. + +# 5.3 Analyses of Language Modeling with CLM + +The last group of experiments studies the effectiveness and benefits of CLM. + +Ablations on Training Configurations. Figure 8 illustrates pretraining process with CLM and All-Token MLM. The plots demonstrate the difficulty of language modeling upon corrupted text sequences. It is quite an unbalanced task. For the majority of the tokens (Original) the task is simply to copy its input at the same position. For the replaced tokens $( 7 - 8 \%$ total), however, the model needs to detect the abnormality brought by the auxiliary model and recover the original token. Implicitly training the copy mechanism as part of the hard LM task is not effective: The copy accuracy of All-Token MLM is much lower, and thus the LM head may confuse original tokens with replaced ones. As shown in Table 3 and ELECTRA [7], pretraining with All-Token MLM performs worse than using the RTD task, though the latter is equivalent to only training the copy mechanism. The multi-task learning of CLM is necessary for the main Transformer to stably learn the language modeling task upon the corrupted text sequence. + +![](images/7f4f1d35a5619b20f4bb5dab0e62dc152a989762d0e164bcb37c0492f45877fe.jpg) +Figure 8: The copying accuracy and the language modeling accuracy $y$ -axes) of CLM and All-Token MLM at different pretraining steps ( $x$ -axes, in 10K scale). The accuracy is averaged on tokens that are replaced by the auxiliary Transformer (Replaced) or those from the original input text (Original). + +Prompt-Based Fine-Tuning with CLM. Table 4 includes the prompt-based fine-tuning experiments on MNLI for RoBERTa and COCO-LM under $b a s e + +$ and large $^ { + + }$ sizes, following the same few-shot manual prompt fine-tuning with demonstration setup in LM-BFF [15]. We use $\{ 3 e - 6 , 4 e - 6 , 5 e - 6 \}$ for the learning rate search of COCO-LM base++/large++ model, with everything else kept same as described in LM-BFF. With exactly the same pipeline, COCOLM outperforms RoBERTa under both $b a s e + +$ and $l a r g e + +$ sizes by significant margins on MNLI$\mathrm { m } / \mathrm { m m }$ . Such observations are interesting as COCOLM’s main Transformer does not even see any + +
ModelMNLI-mMNLI-mm
RoBERTaBase++60.1 (1.5)61.8 (1.2)
COCO-LMBase++66.5 (2.1)68.0 (2.3)
RoBERTaLarge++70.7 (1.3)72.0 (1.2)
COCO-LMLarge++72.0 (1.5)73.3 (1.1)
+ +Table 4: Few-shot prompt-based fine-tuning using RoBERTa and COCO-LM trained on 16 samples per class. Mean (and standard deviation) accuracy results over 5 different splits on MNLI- $. \mathrm { m } / \mathrm { m m }$ are shown. + +[MASK] tokens during pretraining but still performs well on predicting masked tokens for promptbased learning. Note that ELECTRA and COCO-LM variants without the CLM task are not applicable: Their main Transformers are not pretrained by language modeling tasks (thus no language modeling capability is learned to generate prompt label words). This points out the importance, if not necessity, of COCO-LM in the family of ELECTRA-style pretraining models. With the benefits and rapid developments of prompt-based approaches, the lack of language modeling capability is going to limit the potential of ELECTRA’s self-supervised learning framework in many real-world scenarios. COCO-LM not only addresses this limitation but also provides better prompt-based learning results. + +# 6 Conclusions and Future Work + +In this paper, we present COCO-LM, which pretrains language models using Corrective Language Modeling and Sequence Contrastive Learning upon corrupted text sequences. With standard pretraining data and Transformer architectures, COCO-LM improves the accuracy on the GLUE and SQuAD benchmarks, while also being more efficient in utilizing pretraining computing resources and network parameters. + +One limitation of this work is that the contrastive pairs are constructed by simple cropping and MLM replacements. Recent studies have shown the effectiveness of advanced data augmentation techniques in fine-tuning language models [16, 38, 51]. A future research direction is to explore better ways to construct contrastive pairs in language model pretraining. + +Despite the empirical advantage of this auxiliary-main dual model framework, the auxiliary Transformer training is not influenced by the main Transformer nor learns to generate the optimal pretraining signals for the main model. To better understand and tailor the training of the auxiliary model to the main model is another important future research direction. + +# Acknowledgments + +We sincerely thank Guolin Ke for discussions and advice on model implementation. We also thank anonymous reviewers for valuable and insightful feedback, especially the suggestion of adding prompt-based fine-tuning experiments. + +# References + +[1] Hangbo Bao, Li Dong, Furu Wei, Wenhui Wang, Nan Yang, Xiaodong Liu, Yu Wang, Songhao Piao, Jianfeng Gao, Ming Zhou, and Hsiao-Wuen Hon. UniLMv2: Pseudo-masked language models for unified language model pre-training. In ICML, 2020. 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In ICCV, 2015. \ No newline at end of file diff --git a/parse/train/pk4q0SD_r1X/pk4q0SD_r1X_content_list.json b/parse/train/pk4q0SD_r1X/pk4q0SD_r1X_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..44d24dcf9ba3fee417ebb1a7138513c10aa30ca9 --- /dev/null +++ b/parse/train/pk4q0SD_r1X/pk4q0SD_r1X_content_list.json @@ -0,0 +1,1380 @@ +[ + { + "type": "text", + "text": "COCO-LM: Correcting and Contrasting Text Sequences for Language Model Pretraining ", + "text_level": 1, + "bbox": [ + 223, + 122, + 776, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Yu Meng1∗, Chenyan Xiong2, Payal Bajaj2, Saurabh Tiwary2, Paul Bennett2, Jiawei Han1, Xia Song2 ", + "bbox": [ + 285, + 224, + 710, + 255 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 University of Illinois at Urbana-Champaign 2 Microsoft 1 {yumeng5,hanj}@illinois.edu 2 {chenyan.xiong,payal.bajaj,satiwary, paul.n.bennett,xiaso}@microsoft.com ", + "bbox": [ + 308, + 256, + 691, + 311 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 347, + 535, + 363 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We present a self-supervised learning framework, COCO-LM, that pretrains Language Models by COrrecting and COntrasting corrupted text sequences. Following ELECTRA-style pretraining, COCO-LM employs an auxiliary language model to corrupt text sequences, upon which it constructs two new tasks for pretraining the main model. The first token-level task, Corrective Language Modeling, is to detect and correct tokens replaced by the auxiliary model, in order to better capture token-level semantics. The second sequence-level task, Sequence Contrastive Learning, is to align text sequences originated from the same source input while ensuring uniformity in the representation space. Experiments on GLUE and SQuAD demonstrate that COCO-LM not only outperforms recent state-of-the-art pretrained models in accuracy, but also improves pretraining efficiency. It achieves the MNLI accuracy of ELECTRA with $5 0 \\%$ of its pretraining GPU hours. With the same pretraining steps of standard base/large-sized models, COCO-LM outperforms the previous best models by $1 +$ GLUE average points. ", + "bbox": [ + 233, + 378, + 766, + 573 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 597, + 310, + 614 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Pretrained language models (PLMs) have reshaped the way AI systems process natural language [11, 36, 39, 40]. Before task-specific training, it is now a common practice to first pretrain the deep neural networks, often Transformers [53], via a self-supervised token-level language modeling task [29, 31, 40]. Whether it is autoregressive [39], permutational [62], or masked language modeling (MLM) [11], the Transformer networks are pretrained to recover some omitted tokens using the rest of input texts. Then the language semantics captured during pretraining are conveyed to downstream tasks via the pretrained Transformer parameters [5, 8, 44]. ", + "bbox": [ + 174, + 628, + 825, + 727 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recent research [14, 16, 25, 43] observed several challenges in this self-supervised learning framework. One challenge is its efficiency. After pretrained for a while with the standard token-level language modeling, the networks have already captured the basic language patterns, making a large fraction of pretraining signals no longer informative. Linear improvement in the model effectiveness often requires exponentially more pretraining compute and parameters [25], which is unsustainable. Another challenge is the anisotropy of text representations from pretrained models. The sequence representations from many pretrained models are quite irregular [30, 43] and require dedicated fine-tuning approaches to be useful in sequence-level applications [32, 60]. ", + "bbox": [ + 174, + 732, + 825, + 843 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Clark et al. [7] proposed a new pretraining strategy, ELECTRA, that uses an auxiliary language model (“generator”) to replace tokens in input texts and pretrains the main Transformer (“discriminator”) to detect replaced tokens. This improves the pretraining efficiency and effectiveness, but pretraining via binary classification hinders the model’s usage on applications requiring language modeling capability (e.g., prompt-based learning [15, 28, 46]). It could further distort the representation space as the Transformers are pretrained to output the same “non-replacement” label for all actual tokens. ", + "bbox": [ + 178, + 849, + 823, + 877 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 147 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we present a new self-supervised learning approach, COCO-LM, that pretrains Language Models by COrrecting and COntrasting corrupted text sequences. Following ELECTRA-style pretraining, COCO-LM employs an auxiliary model to corrupt the input texts, upon which it introduces two new pretraining tasks for the main Transformer, one at token level and one at sequence level. The token-level task, corrective language modeling (CLM), pretrains the main Transformer to detect and correct the tokens in the corrupted sequences. It uses a multi-task setup to combine the benefits of replaced token detection and language modeling. The sequence-level task, sequence contrastive learning (SCL), pretrains the model to align text sequences originated from the same source sequence and enforce uniformity of the representation space. ", + "bbox": [ + 174, + 152, + 825, + 279 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In our experiments on GLUE [54] and $\\mathrm { S Q u A D }$ [41] benchmarks, COCO-LM not only outperforms state-of-the-art pretraining approaches in effectiveness, but also significantly improves the pretraining efficiency. Under the same setting, COCO-LM matches the MNLI accuracy of RoBERTa and ELECTRA with $6 0 \\%$ and $5 0 \\%$ of their GPU hours in pretraining, respectively. When pretrained with the same number of steps, COCO-LM outperforms the previous best models by $1 +$ GLUE average points under the standard base/large-sized model evaluations. With 367 million parameters, COCO$\\mathrm { L M _ { L a r g e + + } }$ reaches the MNLI accuracy of Megatron3.9B [49], one of the largest BERT-style model with 3.9 billion parameters. Our analyses provide further insights on the advantage of CLM in learning token representations and its effectiveness in prompted-based fine-tuning, as well as the benefit of SCL in ensuring alignment and uniformity in the representation space for better generalization1. ", + "bbox": [ + 174, + 284, + 825, + 422 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 441, + 321, + 459 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Various token-level tasks have been used to pretrain language models. The most classic auto-regressive language modeling is to predict a token given all the previous tokens, or all subsequent ones [36, 39]. BERT uses masked language modeling (MLM) that recovers randomly masked tokens using the rest input. XLNet proposes permutation language modeling that conducts MLM in an autoregressive manner [62]. UniLM uses pseudo MLM which unifies autoregressive and MLM tasks [1, 13]. ", + "bbox": [ + 174, + 474, + 825, + 544 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Sequence-level tasks are also explored, which often pretrain the model to predict certain cooccurrences of sequence pairs. For example, next sentence prediction [11], sentence ordering [27] and previous sentence prediction [56] concatenate two sentences (either correlated or random), and train the Transformer to classify the pair. ", + "bbox": [ + 174, + 550, + 825, + 606 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Empirically, MLM is still among the most effective tasks to pretrain encoders [29, 31, 40]. RoBERTa [31] found the sentence-level task in BERT not benefitial and discarded it. BART [29] and T5 [40] both observed that MLM is often the most effective task. The empirical advantages of other pretraining tasks are more task-specific, for example, entity related masks for knowledge intensive applications [20, 24], and sequence-level tasks for long form text modeling [42]. ", + "bbox": [ + 174, + 612, + 825, + 681 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Instead of randomly altering texts, ELECTRA [7] uses a smaller auxiliary Transformer pretrained by MLM to replace some tokens in the text sequences using its language modeling probability, and pretrains the main Transformer to detect the replaced tokens. ELECTRA achieves state-of-the-art accuracy in many language tasks [7]. Later, Clark et el. [6] developed ELECTRIC, which pretrains encoders by contrasting original tokens against negatives sampled from a cloze model. ELECTRIC re-enables the language modeling capability but underperforms ELECTRA in downstream tasks. ", + "bbox": [ + 174, + 688, + 825, + 771 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our work is also related to contrastive learning which has shown great success in visual representation learning [4, 22, 34]. Its effectiveness of in language is more observed in the fine-tuning stage, for example, in sentence representation [16], dense retrieval [60], and GLUE fine-tuning [19]. ", + "bbox": [ + 176, + 777, + 823, + 820 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 Method ", + "text_level": 1, + "bbox": [ + 174, + 839, + 269, + 856 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We present the preliminaries of PLMs, their challenges, and the new COCO-LM framework. ", + "bbox": [ + 173, + 871, + 777, + 886 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3.1 Preliminary on Language Model Pretraining ", + "text_level": 1, + "bbox": [ + 176, + 90, + 524, + 106 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this work we focus on pretraining BERT-style bidirectional Transformer encoders [11] that are widely used in language representation tasks. We first recap the masked language modeling (MLM) task introduced by BERT [11] and then discuss the pretraining framework of ELECTRA [7]. ", + "bbox": [ + 174, + 116, + 825, + 159 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "BERT Pretraining uses the masked language modeling task (MLM) [11], which is to take an input sequence $X ^ { \\mathrm { o r i g } } = [ x _ { 1 } ^ { \\mathrm { o r i g } } , \\dotsc , x _ { i } ^ { \\mathrm { o r i g } } , \\dotsc , x _ { n } ^ { \\mathrm { o r i g } } ]$ , with $1 5 \\%$ random tokens replaced by [MASK] symbols (e.g., the $i$ -th token), and train the model to predict the original tokens at the masked positions: ", + "bbox": [ + 173, + 164, + 825, + 209 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/076614958bda0182580d886177a2a3a70e46a46adc6509ef0a4acc38dc0cbf99.jpg", + "text": "$$\n\\left[ x _ { 1 } ^ { \\mathrm { o r i g } } , \\dots , \\ [ \\mathrm { M \\AA S K } ] _ { i } , \\dots , x _ { n } ^ { \\mathrm { o r i g } } \\right] \\xrightarrow { \\mathrm { T r a n s f o r m e r } } H \\xrightarrow { \\mathrm { M L M H e a d } } p _ { \\mathrm { M L M } } ( x | h _ { i } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 267, + 215, + 728, + 242 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where the Transformer generates contextualized representations ${ \\pmb H } = \\{ h _ { i } \\} _ { i = 1 } ^ { n }$ . The MLM Head predicts the masked token from the vocabulary $V$ using the hidden representation $\\boldsymbol { h } _ { i }$ and token embeddings $_ { \\textbf { \\em x } }$ . The pretraining minimizes the MLM loss on the set of masked positions $\\mathcal { M }$ . Specifically, ", + "bbox": [ + 174, + 250, + 826, + 291 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/44ef3bd6eb3f680f4bf25edbebc27dd93b45fdc9f04b6736a5ee5b7e2b588d26.jpg", + "text": "$$\np _ { \\mathrm { M L M } } ( x | h _ { i } ) = \\frac { \\exp ( x ^ { \\top } h _ { i } ) } { \\sum _ { x _ { t } \\in V } \\exp ( x _ { t } ^ { \\top } h _ { i } ) } ; \\quad \\mathcal { L } _ { \\mathrm { M L M } } = \\mathbb { E } \\left( - \\sum _ { i \\in \\mathcal { M } } \\log p _ { \\mathrm { M L M } } \\left( x _ { i } ^ { \\mathrm { o r i g } } \\middle | h _ { i } \\right) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 223, + 296, + 772, + 340 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "ELECTRA Pretraining uses two Transformers, a “generator” pretrained by MLM, and a “discriminator” pretrained using the generator’s outputs. We refer them as auxiliary and main Transformers, as the former is discarded after pretraining and the latter may be trained by “generative” tasks too. ", + "bbox": [ + 173, + 353, + 825, + 396 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The auxiliary model outputs a corrupted sequence $X ^ { \\mathrm { M L M } }$ by sampling from its predicted probability: ", + "bbox": [ + 176, + 400, + 823, + 416 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/b824c400834632d12c6a490dad58c221e879bf27313a16ad8af9d62808aa078e.jpg", + "text": "$$\nx _ { i } ^ { \\mathrm { M L M } } \\sim p _ { \\mathrm { M L M } } \\left( x | h _ { i } \\right) , \\mathrm { i f } i \\in \\mathcal { M } ; \\quad x _ { i } ^ { \\mathrm { M L M } } = x _ { i } ^ { \\mathrm { o r i g } } , \\mathrm { e l s e } .\n$$", + "text_format": "latex", + "bbox": [ + 312, + 422, + 686, + 443 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The masked positions are replaced by sampled tokens considered plausible in context by the auxiliary Transformer, which are more deceiving than random replacements. ELECTRA uses a skinnier auxiliary network (e.g., hidden dimension is $1 / 3$ of the main model) to control the signal difficulty. ", + "bbox": [ + 173, + 449, + 826, + 491 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The main Transformer takes $X ^ { \\mathrm { M L M } }$ and classifies the replaced tokens: ", + "bbox": [ + 176, + 496, + 632, + 511 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/2eb7d2c92d65747022a0ee7e813aa301ee14c1cca99b57bcf4cfff3a6d9189e3.jpg", + "text": "$$\n\\begin{array} { r } { X ^ { \\mathrm { M L M } } \\xrightarrow { \\mathrm { M a i n ~ T r a n s f o r m e r } } \\pmb { H } \\xrightarrow { \\mathrm { R T D ~ H e a d } } p _ { \\mathrm { R T D } } \\left( \\mathbb { 1 } \\big ( x _ { i } ^ { \\mathrm { M L M } } = x _ { i } ^ { \\mathrm { o r i g } } \\big ) \\big | h _ { i } \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 284, + 517, + 712, + 545 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathbb { 1 } ( \\cdot )$ is the indicator function. The Replaced Token Detection (RTD) head uses a sigmoid linear layer to output the binary probability, and the main Transformer is trained with binary cross entropy loss. The RTD task is trained on all tokens instead of masked ones and improves efficiency. ", + "bbox": [ + 174, + 551, + 823, + 593 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The two Transformers are pretrained jointly. The auxiliary model gradually generates more realistic replacement tokens and the main model learns to better detect them. This forms a natural learning curriculum and significantly improves ELECTRA’s accuracy in downstream tasks [7]. ", + "bbox": [ + 173, + 598, + 826, + 642 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 Challenges of ELECTRA-Style Pretraining ", + "text_level": 1, + "bbox": [ + 173, + 657, + 514, + 672 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Missing Language Modeling Benefits. The classification task in ELECTRA is simpler and more stable [61], but raises two challenges. The first is the lack of language modeling capability which is a necessity in some tasks [6]. For example, prompt-based learning requires a language model to generate labels [15, 33, 45, 46]. The second is that the binary classification task may not be sufficient to capture certain word-level semantics that are critical for token-level tasks. ", + "bbox": [ + 174, + 683, + 485, + 821 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Squeezing Representation Space. Another challenge is that the representations from Transformer-based language models often reside in a narrow cone, where two random sentences have high similarity scores (lack of uniformity), ", + "bbox": [ + 173, + 828, + 483, + 897 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/3a8c91a226e2aa036c614aa7d53c2b811d005dba76544c5617acff0d9b62c12a.jpg", + "image_caption": [ + "Figure 1: Cosine similarity distributions of random/similar sequence pairs using [CLS] embeddings from pretrained models. Histograms/curves are distribution bins/kernel density estimates. " + ], + "image_footnote": [], + "bbox": [ + 496, + 699, + 828, + 805 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "and closely related sentences may have more different representations (lack of alignment) [14, 16, 30]. ", + "bbox": [ + 176, + 897, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/88292be09312bfcedea184a72ece038e7b7b44db60e61df6cc39ca285f523c38.jpg", + "image_caption": [ + "Figure 2: The overview of COCO-LM. The auxiliary Transformer is pretrained by MLM. Its corrupted text sequence is used as the main Transformer’s pretraining input in Corrective Language Modeling and paired with the cropped original sequence for Sequence Contrastive Learning. " + ], + "image_footnote": [], + "bbox": [ + 174, + 92, + 825, + 231 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Figure 1 illustrates such behaviors with random sentence pairs (from pretraining corpus) and semantically similar pairs (those annotated with maximum similarity from STS-B [3]). With RoBERTa, the cosine similarities of most random sentence pairs are near 0.8, bigger than many semantically similar pairs. The representation space from ELECTRA is even more squeezed. Nearly all sentence pairs, both random and similar ones, have around 0.9 cosine similarity. This may not be surprising as ELECTRA is pretrained to predict the same output (“non-replacement”) for all tokens in these sequences. The irregular representation space raises the risk of degeneration [37, 55] and often necessitates sophisticated post-adjustment or fine-tuning to improve the sequence representations [16, 30, 32, 60]. ", + "bbox": [ + 173, + 325, + 826, + 438 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 COCO-LM Pretraining ", + "text_level": 1, + "bbox": [ + 174, + 469, + 379, + 486 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "COCO-LM also employs an auxiliary Transformer to construct the corrupted text sequence, as in Eqn. (1), but it introduces two new pretraining tasks upon the corrupted sequences to address the challenges previously described. In the rest of this section, we present these two tasks and then the detailed configurations of COCO-LM. Its framework is illustrated in Figure 2. ", + "bbox": [ + 173, + 502, + 825, + 559 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Corrective Language Modeling (CLM) trains the main Transformer to recover the original tokens, given the corrupted text sequence XMLM: ", + "bbox": [ + 174, + 565, + 825, + 594 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/17aeff6497bb1d60361a2ffa28441b12de20093398264297d4fe56e5aa290f81.jpg", + "text": "$$\n\\begin{array} { r } { X ^ { \\mathrm { M L M } } \\xrightarrow { \\mathrm { M a i n ~ T r a n s f o r m e r } } H \\xrightarrow { \\mathrm { C L M H e a d } } p _ { \\mathrm { C L M } } ( x | h _ { i } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 338, + 617, + 658, + 638 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The CLM Head uses the hidden representations $\\pmb { H }$ to output a language modeling probability, instead of a binary classification score. The forward pass of the CLM Head is the same as All-Token MLM, a variation of ELECTRA [7] that consists of a language modeling layer and a binary classification layer for the copy mechanism: ", + "bbox": [ + 173, + 660, + 826, + 717 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/6e99146a7d8af754b940a8246882a6816802df00b87f58c77e7850208f6a9fb3.jpg", + "text": "$$\n\\begin{array} { r l } & { p _ { \\mathrm { L M } } ( x _ { i } | h _ { i } ) = \\mathbb { 1 } \\left( x _ { i } = x _ { i } ^ { \\mathrm { M L M } } \\right) p _ { \\mathrm { c o p y } } ( 1 | h _ { i } ) + p _ { \\mathrm { c o p y } } ( 0 | h _ { i } ) \\frac { \\exp ( x _ { i } ^ { \\top } h _ { i } ) } { \\sum _ { x _ { t } \\in V } \\exp ( x _ { t } ^ { \\top } h _ { i } ) } , } \\\\ & { p _ { \\mathrm { c o p y } } ( y _ { i } | h _ { i } ) = \\exp ( y _ { i } \\cdot w _ { \\mathrm { c o p y } } ^ { \\top } h _ { i } ) / \\left( \\exp ( w _ { \\mathrm { c o p y } } ^ { \\top } h _ { i } ) + 1 \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 739, + 753, + 800 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where ${ \\pmb w } _ { \\mathrm { c o p y } }$ is a learnable weight and $p _ { \\mathrm { c o p y } } ( y _ { i } | h _ { i } )$ is the copy mechanism ( $y _ { i } = 1$ when the input token is original and can be directly copied to the output; $y _ { i } = 0$ when the input token needs to be corrected to another token from the vocabulary). ", + "bbox": [ + 174, + 820, + 823, + 863 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In ELECTRA, All-Token MLM performs worse than RTD [7]. Language modeling on the corrupted text sequence $X ^ { \\mathrm { M L M } }$ is hard as the replaced tokens from the auxiliary model are more deceiving than [MASK]. To improve the language model learning, different from All-Token MLM, CLM employs a ", + "bbox": [ + 174, + 868, + 825, + 911 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "multi-task setup that combines the RTD task to explicitly train the copy mechanism $p _ { \\mathrm { c o p y } } ( \\cdot )$ ", + "bbox": [ + 169, + 90, + 776, + 107 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/48e1788e8b54b0cbed2da5f27924dd575db2bcb592bc657951c88fe854e1300c.jpg", + "text": "$$\n\\begin{array} { l } { \\mathcal { L } _ { \\mathrm { c o p y } } = - \\mathbb { E } \\left( \\displaystyle \\sum _ { i = 1 } ^ { n } \\mathbb { 1 } \\left( x _ { i } ^ { \\mathrm { M L M } } = x _ { i } ^ { \\mathrm { o r i g } } \\right) \\log p _ { \\mathrm { c o p y } } ( 1 | h _ { i } ) + \\mathbb { 1 } \\left( x _ { i } ^ { \\mathrm { M L M } } \\neq x _ { i } ^ { \\mathrm { o r i g } } \\right) \\log p _ { \\mathrm { c o p y } } ( 0 | h _ { i } ) \\right) , \\mathrm { ~ } \\forall i \\mathrm { ~ c o p y ~ } ( \\mathbb { E } ) , } \\\\ { \\mathcal { L } _ { \\mathrm { L M } } = - \\mathbb { E } \\left( \\displaystyle \\sum _ { i \\in \\mathcal { M } } \\log p _ { \\mathrm { L M } } \\left( x _ { i } ^ { \\mathrm { o r i g } } | h _ { i } \\right) \\right) } \\\\ { \\displaystyle \\qquad = - \\mathbb { E } \\left( \\displaystyle \\sum _ { i \\in \\mathcal { M } } \\log \\left( \\mathbb { 1 } \\left( x _ { i } ^ { \\mathrm { M L M } } = x _ { i } ^ { \\mathrm { o r i g } } \\right) p _ { \\mathrm { c o p y } } ^ { \\mathrm { s g } } ( 1 | h _ { i } ) + p _ { \\mathrm { c o p y } } ^ { \\mathrm { s g } } ( 0 | h _ { i } ) \\frac { \\exp ( x _ { i } ^ { \\top } h _ { i } ) } { \\sum _ { x _ { t } \\in V } \\exp ( x _ { t } ^ { \\top } h _ { i } ) } \\right) \\right) , \\mathrm { ~ } } \\\\ { \\mathcal { L } _ { \\mathrm { C L M } } = \\lambda _ { \\mathrm { c o p y } } \\mathcal { L } _ { \\mathrm { c o p y } } + \\mathcal { L } _ { \\mathrm { L M } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 184, + 111, + 807, + 258 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The hyperparameter $\\lambda _ { \\mathrm { c o p y } }$ balances the weights of the two tasks. The binary cross entropy loss in Eqn. (2) explicitly trains the copy probability. We also use stop gradient (sg) to decouple the gradient backpropagation to $p _ { \\mathrm { c o p y } } ( \\cdot )$ from the LM task. This way, the main Transformer first learns the easier classification task and then uses it to help learn the harder LM task. The binary classification task is trained on all tokens while the language modeling task is trained only on masked positions. ", + "bbox": [ + 173, + 262, + 825, + 333 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "CLM combines the advantages of MLM and ELECTRA: The main Transformer is trained on all tokens with the help of the binary classification task while also being able to predict words, thus enjoying the efficiency benefits of ELECTRA and preserving the language modeling benefits. ", + "bbox": [ + 173, + 338, + 825, + 381 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Sequence Contrastive Learning (SCL) forms a contrastive learning objective upon the sequence embeddings to learn more robust representations. Broadly, contrastive learning is to align a positive pair of instances, often different views of the same information [4, 34], in contrast to unrelated negative instances [22, 60]. The different views are often obtained by applying data augmentations on the same input, for example, rotation, cropping, and blurring on visual representations [4, 34], so that the neural networks can learn representations robust to these data alterations. ", + "bbox": [ + 173, + 386, + 825, + 470 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In COCO-LM, the corrupted sequence $X ^ { \\mathrm { M L M } }$ already provides a form of data augmentation. We pair it with another augmentation, $X ^ { \\mathrm { c r o p } }$ , a randomly cropped contiguous span of $X ^ { \\mathrm { o r i g } }$ (the length of $X ^ { \\mathrm { c r o p } }$ is $9 0 \\%$ of $X ^ { \\mathrm { o r i g } }$ so that the major sequence meaning is preserved), to construct the positive pair and to contrast with random negatives. ", + "bbox": [ + 173, + 474, + 825, + 532 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Specifically, a training batch $B$ in SCL includes a random set of corrupted and cropped sequences: 1contrastive pair $B = \\{ ( X _ { 1 } ^ { \\mathrm { M L M } } , X _ { 1 } ^ { \\mathrm { c r o p } } ) , \\dots , ( X _ { N } ^ { \\mathrm { M L M } } , X _ { N } ^ { \\mathrm { c r o p } } ) \\}$ $( X , X ^ { + } )$ N N consists of either $( X _ { k } ^ { \\mathrm { M L M } } , X _ { k } ^ { \\mathrm { c r o p } } )$ , with $X _ { k } ^ { \\mathrm { M L M } }$ and or $X _ { k } ^ { \\mathrm { c r o p } }$ $( \\ddot { X } _ { k } ^ { \\mathrm { c r o p } } , X _ { k } ^ { \\mathrm { M L M } } )$ originated from $X _ { k } ^ { \\mathrm { o r i g } }$ . A positive trast). The negative instances are all the remaining sequences in the batch $\\ddot { B } ^ { - } = B \\setminus \\{ ( X , X ^ { + } ) \\}$ contrastive loss is formulated as: ", + "bbox": [ + 173, + 537, + 825, + 613 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/0468e73383b732e24a23d25d539ca44cc4d7d1296b11c731e78c4534179ebef4.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\mathcal { L } _ { \\mathrm { S C L } } = - \\mathbb { E } ( \\log \\frac { \\exp ( \\cos ( s , s ^ { + } ) / \\tau ) } { \\exp ( \\cos ( s , s ^ { + } ) / \\tau ) + \\sum _ { X ^ { - } \\in B ^ { - } } \\exp ( \\cos ( s , s ^ { - } ) / \\tau ) } ) , } } \\\\ & { } & { = - \\mathbb { E } ( \\cos ( s , s ^ { + } ) / \\tau - \\log ( \\exp ( \\cos ( s , s ^ { + } ) / \\tau ) + \\sum _ { X ^ { - } \\in B ^ { - } } \\exp ( \\cos ( s , s ^ { - } ) / \\tau ) ) ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 187, + 613, + 789, + 695 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $s , s ^ { + } , s ^ { - }$ are the representations of $X , X ^ { + } , X ^ { - }$ , respectively, from the main Transformer (i.e., $\\boldsymbol { h } _ { \\mathrm { [ C L S ] } } ,$ ). The similarity metric is cosine similarity (cos) and the temperature $\\tau$ is set to $1$ . ", + "bbox": [ + 173, + 699, + 826, + 728 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "As shown in Wang et al. [55], the first term in Eqn. (3) $( \\cos ( s , s ^ { + } ) )$ improves alignment of the space. It encourages representations to be robust to the corruptions and the alterations on the original text. The second term in Eqn. (3) promotes uniformity. It pushes unrelated sequences apart in the representation space and ensures low cosine similarity between random data points. Several studies have observed improved generalization ability from better alignment and uniformity [16, 37, 55]. ", + "bbox": [ + 173, + 733, + 825, + 804 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Aligning $X ^ { \\mathrm { M L M } }$ with $X ^ { \\mathrm { c r o p } }$ requires the main Transformer to produce sequence representations robust to both token-level (i.e., MLM replacements) and sequence-level (i.e., cropping) alterations. The model is thus encouraged to reason more using partially altered sequences to recover the original information. ", + "bbox": [ + 174, + 809, + 825, + 866 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Overall Training. COCO-LM uses the following loss function: ", + "bbox": [ + 173, + 871, + 593, + 887 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/151d5b1e75c7750cf345e470a3d88a0143c6fd346e8a419d39c2358543a28291.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { C O C O - L M } } = \\mathcal { L } _ { \\mathrm { M L M } } ^ { \\mathrm { A u x . } } + \\mathcal { L } _ { \\mathrm { C L M } } ^ { \\mathrm { M a i n } } + \\mathcal { L } _ { \\mathrm { S C L } } ^ { \\mathrm { M a i n } } .\n$$", + "text_format": "latex", + "bbox": [ + 375, + 890, + 622, + 910 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The auxiliary Transformer is pretrained by masked language modeling (MLM) and generates corrupted sequences. The main Transformer is pretrained to correct the corruption (CLM) and to contrast the corrupted sequences with the cropped sequences (SCL). The two Transformers are pretrained jointly with the loss in Eqn. (4). The main Transformer is used in downstream applications. ", + "bbox": [ + 174, + 90, + 825, + 147 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Network Configurations. Similar to ELECTRA, the auxiliary Transformer is smaller than the main model, but we use different configurations in the auxiliary model: (1) We reduce the number of layers to $1 / 3$ or $1 / 4$ (under base or large model setup, respectively) but keep its hidden dimension the same with the main model, instead of shrinking its hidden dimensions; (2) We disable dropout in it when sampling replacement tokens. We find such configurations empirically more effective and use them as the backbone of COCO-LM. The main Transformer follows the standard architecture of BERT/ELECTRA and can be easily adopted by downstream application pipelines with almost no changes. ", + "bbox": [ + 174, + 152, + 825, + 265 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 Experimental Setup ", + "text_level": 1, + "bbox": [ + 176, + 287, + 372, + 305 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Pretraining Settings. We employ three standard settings, base, base $^ { + + }$ , and large $^ { + + }$ . Base is the $\\mathbf { B E R T _ { B a s e } }$ training configuration [11]: Pretraining on Wikipedia and BookCorpus [63] (16 GB of texts) for 256 million samples on 512 token sequences (125K batches with 2048 batch size). We use the same corpus and 32, 768 uncased BPE vocabulary [47] as with TUPE [26]. ", + "bbox": [ + 174, + 321, + 825, + 377 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "$B a s e + +$ trains the base size model with larger corpora and/or more training steps. Following recent research [1, 31, 62], we add in OpenWebText [18], CC-News [31], and STORIES [52], to a total of 160 GB texts, and train for 4 billion (with 2048 batch size) samples [31]. We follow the prepossessing of UniLMV2 [1] and use 64, 000 cased BPE vocabulary. ", + "bbox": [ + 174, + 383, + 825, + 439 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "$L a r g e { + + }$ uses the same training corpora as $b a s e + +$ and pretrains for 4 billion samples (2048 batch size). Its Transformer configuration is the same with BERTLarge [11]. ", + "bbox": [ + 174, + 445, + 820, + 474 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Model Architecture. Our base/base $^ { + + }$ model uses the $\\mathbf { B E R T _ { B a s e } }$ architecture [11]: 12 layer Transformer, 768 hidden size, plus T5 relative position encoding [40]. Our large $^ { + + }$ model is the same with $\\mathrm { B E R T _ { L a r g e } }$ , 24 layer and 1024 hidden size, plus T5 relative position encoding [40]. Our auxiliary network uses the same hidden size but a shallow 4-layer Transformer in base/base $^ { + + }$ and a 6-layer one in $l a r g e + +$ . When generating $X ^ { \\mathrm { M L M } }$ we disable dropout in the auxiliary model. ", + "bbox": [ + 174, + 479, + 825, + 550 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Downstream Tasks. We use the tasks included in GLUE [54] and $\\mathrm { S Q u A D } 2 . 0$ reading compression [41]. Please refer to Appendix A for more details about GLUE tasks. Standard hyperparameter search in fine-tuning is performed, and the search space can be found in Appendix B. The fine-tuning protocols use the open-source implementation of TUPE [26]. The reported results are the median of five random seeds on GLUE and SQuAD. ", + "bbox": [ + 174, + 555, + 825, + 626 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Baselines. We compare with various pretrained models in each setting. To reduce the variance in data processing/environments, we also pretrain and fine-tune RoBERTa and ELECTRA under exactly the same setting with COCO-LM, marked with “(Ours)”. All numbers unless marked by “(Ours)” are from reported results in recent research (more details in Appendix C). ", + "bbox": [ + 174, + 632, + 825, + 688 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Implementation Details. Our implementation builds upon the open-source implementation from MC-BERT [61] and fairseq [35]. More implementation details are mentioned in Appendix D. ", + "bbox": [ + 173, + 694, + 823, + 722 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 Evaluation Results ", + "text_level": 1, + "bbox": [ + 174, + 746, + 362, + 762 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Three groups of experiments are conducted to evaluate COCO-LM and its two new pretraining tasks. ", + "bbox": [ + 173, + 779, + 825, + 795 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 Overall Results and Ablations ", + "text_level": 1, + "bbox": [ + 176, + 814, + 421, + 829 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Overall Results are listed in Table 1. Under all three settings, COCO-LM outperforms all recent state-of-the-art pretraining models on GLUE average and SQuAD. It improves the state-of-the-art GLUE score by about one point under all three settings. COCO-LM also enjoys better parameter efficiency. Using less than $1 0 \\%$ of Megatron’s parameters, $\\mathrm { C O C O - L M _ { L a r g e + + } }$ matches the MNLI accuracy of Megatron3.9B, one of the largest pretrained BERT-style encoders. ", + "bbox": [ + 174, + 842, + 825, + 911 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/60cefa43ec47ef9531239e335ac10e76eda4144c99fd0af584e146167233b9d7.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
ModelParamsGLUE Single TaskSQuAD 2.0
MNLI-(m/mm)QQPQNLISST-2CoLARTEMRPCSTS-BAVGEMF1
Base Setting: BERT Base Size,Wikipedia + Book Corpus (16GB)
BERT[11]110M84.5/-91.391.793.258.968.687.389.583.173.776.3
RoBERTa [31]125M84.7/-1192.71111179.7
XLNet [62]110M85.8/85.41192.71111178.581.3
ELECTRA [7]110M86.0/85.390.091.993.464.370.884.989.183.780.583.3
MC-BERT [61]110M85.7/85.289.791.392.362.175.086.088.083.7
DeBERTa [23]134M86.3/86.2二一一一一一一79.382.5
TUPE [26]110M86.2/86.291.392.293.363.673.689.989.284.911
RoBERTa (Ours)110M85.8/85.591.392.093.760.168.287.388.583.377.780.5
ELECTRA (Ours)110M86.9/86.791.992.693.666.275.188.289.785.579.782.6
COCO-LM110M88.5/88.392.093.193.263.984.891.490.387.282.485.2
Base++ Seting: BERT Base Size,Bigger Training Data,and/or More Training Steps
XLNet [62]110M86.8/-91.491.794.760.274.088.289.584.680.2
RoBERTa [31]125M87.6/-91.992.894.863.678.790.291.286.480.583.7
UniLMV2[1]110M88.5/-91.793.595.165.281.391.891.087.183.386.1
DeBERTa [23]134M88.8/88.5一一一1一一一83.186.2
CLEAR [59]110M86.7/-90.092.994.564.378.389.289.885.7一1
COCO-LM134M90.2/90.092.294.294.667.387.491.291.888.685.488.1
Large++ Setting: BERTLarge Size,Bigger Training Data,and More Training Steps
XLNet [62]360M90.8/90.892.394.997.069.085.990.892.589.287.990.6
RoBERTa [31]356M90.2/90.292.294.796.468.086.690.992.488.986.589.4
ELECTRA[7]335M90.9/-92.495.096.969.188.090.892.689.488.090.6
DeBERTa [23]384M91.1/91.192.395.396.870.5111188.090.7
COCO-LM367M91.4/91.692.895.796.973.991.092.292.790.888.291.0
Megatron1.3B [49]1.3B90.9/91.092.611一187.190.2
Megatron3.9B [49]3.9B91.4/91.492.711188.591.2
", + "bbox": [ + 174, + 89, + 823, + 409 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 1: Results on GLUE and SQuAD 2.0 development set. All results are single-task, single-model fine-tuning. Results not available in public reports are marked as “–”. DeBERTa reported RTE, MRPC and STS-B results by fine-tuning from MNLI checkpoints which are not single-task results. We use Spearman correlation for STS, Matthews correlation for CoLA, and accuracy for the rest on GLUE. AVG is the average of the eight tasks on GLUE. All baseline results unless marked by (Ours) are reported by previous research. ", + "bbox": [ + 173, + 412, + 825, + 497 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/fd3062d90e85ad4b2a1ecd68665ab53b140aae24bc83d4255bf47749796908bf.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
ModelParamsMNLI-(m/mm)QQPQNLISST-2CoLARTEMRPCSTS-BAVG
Base/Base++ Setting: BERT Base Size
BERTBase110M84.6/83.489.290.593.552.166.484.885.880.8
ELECTRABase++110M88.5/88.089.593.196.064.675.288.190.285.6
COCO-LMBase++134M89.8/89.389.894.295.668.682.388.590.387.4
Large/Large++ Seting: BERT Large Size
BERTLarge335M86.7/85.989.392.794.960.570.185.486.583.2
ELECTRALarge++335M90.7/90.290.495.596.768.186.189.291.788.5
COCO-LMLarge++367M91.6/91.190.595.896.770.589.288.491.889.3
", + "bbox": [ + 174, + 513, + 823, + 642 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 2: GLUE test set results obtained from the GLUE leaderboard. We perform hyperparameter search for each task with ten random seeds and use the best development set model for test predictions. All results are from vanilla single-task fine-tuning (no ensemble, task-specific tricks, etc.). ", + "bbox": [ + 174, + 652, + 825, + 695 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 2 shows GLUE test set results which further confirm the advantages of COCO-LM over previous methods. ", + "bbox": [ + 174, + 724, + 823, + 752 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Efficiency. In downstream tasks, the efficiency of COCO-LM is the same with BERT. In pretraining, the auxiliary model and SCL introduce extra cost. However, as shown in Figure 3, COCO-LM is more efficient in GPU hours. It outperforms RoBERTa & ELECTRA by $1 +$ points on MNLI with the same GPU hours and reaches their accuracy with around $6 0 \\%$ & $5 0 \\%$ GPU hours, respectively. ", + "bbox": [ + 174, + 758, + 825, + 814 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Ablation Studies. Table 3 shows the ablations of COCO-LM under the base setting on GLUE DEV. ", + "bbox": [ + 173, + 820, + 823, + 835 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Pretraining Task. With only RTD, our backbone model with the shallow auxiliary Transformer is quite effective. CLM and SCL both provide additional improvements on MNLI and GLUE average. Their advantages are better observed on different tasks, for example, CLM on MNLI-mm and SCL on RTE and MRPC. Combining the two in COCO-LM provides better overall effectiveness. In later experiments, we further analyze the benefits of these two tasks. ", + "bbox": [ + 174, + 842, + 825, + 911 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/d2b58bb6c1c54f09e25b712c6e2942447aa58b38e114da9cd3eed138883fa24a.jpg", + "table_caption": [], + "table_footnote": [ + "Table 3: Ablations on GLUE Dev. that eliminate (w/o.), keep (Only) or switch (w.) one component. " + ], + "table_body": "
GroupMethodMNLI-(m/mm)QQPQNLISST-2CoLARTEMRPCSTS-BAVG
COCO-LMBase88.5/88.392.093.193.263.984.891.490.387.2
Pretraining TaskRTDOnly88.4/88.292.193.592.767.380.589.090.986.8
CLMOnly88.6/88.492.093.293.767.480.190.090.486.9
SCL +RTD88.6/88.292.193.593.864.382.790.290.686.9
Network Setingw/o. Rel-Pos w. ELECTRA's Auxiliary88.2/87.792.293.493.768.882.791.290.687.6
Training88.0/87.791.992.793.564.381.289.589.786.3
w.Random Replacements84.9/84.791.491.191.441.670.087.387.180.6
Signalw. Converged Auxiliary88.3/88.192.092.894.364.278.390.490.286.3
CLM SetupAll-Token LM Only87.2/87.092.693.788.589.784.7
CLM w/o. Copy88.0/87.991.8 91.893.194.460.6 66.674.0 76.989.590.186.3
CLM w/o. Stop-grad88.5/88.292.092.994.366.580.990.090.686.9
", + "bbox": [ + 174, + 88, + 821, + 239 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/7c7b853b454b46cee7bf8429edacf37572ac7613714aa9383792bc931834deea.jpg", + "image_caption": [ + "Figure 3: $\\mathrm { C O C O - L M _ { B a s e } }$ on MNLI Dev. ( $y$ -axes) at different pretraining hours on four DGX-2 nodes (64 V100 GPUs). The final training hours and accuracy of RoBERTa (Ours) and ELECTRA (Ours) measured in the same settings are marked. " + ], + "image_footnote": [], + "bbox": [ + 179, + 282, + 580, + 404 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/b0838b4ee1160733d8845b67109fdb9dc7efa9acb20b3820901e16797ecc717f.jpg", + "image_caption": [ + "Figure 4: The performance of COCO- $\\mathbf { \\cdot L M _ { B a s e } }$ when pretrained with different crop fractions. The $x$ -axis is the fraction of $X ^ { \\mathrm { o r i g } }$ being kept (no cropping is $1 0 0 \\%$ ). " + ], + "image_footnote": [], + "bbox": [ + 612, + 281, + 820, + 388 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Architecture. Removing relative position encoding (Rel-Pos) leads to better numbers on some tasks but significantly hurts MNLI. Using a shallow auxiliary network and keeping the same hidden dimension (768) is more effective than ELECTRA’s 12-layer but 256-hidden dimension generator. ", + "bbox": [ + 176, + 502, + 820, + 545 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Pretraining Signal Construction. Using randomly replaced tokens to corrupt text sequence hurts significantly. Using a converged auxiliary network to pretrain the main model also hurts. It is better to pretrain the two Transformers together, as the auxiliary model gradually increases the difficulty of the corrupted sequences and provides a natural learning curriculum for the main Transformer. ", + "bbox": [ + 174, + 551, + 825, + 607 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "CLM Setup. Disabling the multi-task learning and using All-Token MLM [7] reduces model accuracy. The copy mechanism is effective. The benefits of the stop gradient operation are more on stability (preventing training divergence). ", + "bbox": [ + 174, + 613, + 825, + 655 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2 Analyses of Contrastive Learning with SCL ", + "text_level": 1, + "bbox": [ + 173, + 676, + 514, + 691 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "This group of experiments analyzes the behavior of SCL. All experiments use the base setting. ", + "bbox": [ + 169, + 704, + 794, + 719 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Ablation on Data Augmentation. Figure 4 shows the effects of the cropping operation when forming positive SCL pairs with the corrupted sequence. Using the original sequence results in worse GLUE accuracy. It is less informative as the model no longer needs to learn representations robust to sequence-level alteration. Cropping too much (e.g., only keeping $7 0 \\%$ of the original sequence), may hurt as it can alter the semantics too much. Empirically a simple alteration works the best, similar to the observations in recent research [4, 16, 22]. ", + "bbox": [ + 174, + 724, + 825, + 809 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Alignment and Uniformity. Figure 5 plots the distribution of cosine similarities between random sequence pairs and similar ones using representations pretrained by COCO-LM. The representation space from COCO-LM is drastically different from those in Figure 1. With COCO-LM, similar pairs are more aligned and random pairs are distributed more uniformly. Many similar pairs have near 1 cosine similarity and are clearly separated from random pairs which center around 0. The t-SNE [9] plot in Figure 6 further demonstrates the benefits of SCL. The similar sentence pairs (marked by same shapes) are aligned closer when pretrained with SCL. Their average cosine similarity is 0.925 when pretrained with SCL, while is 0.863 without SCL. This better alignment and uniformity is achieved by COCO-LM with SCL via pretraining, without using task-specific data nor supervised labels. ", + "bbox": [ + 174, + 814, + 825, + 911 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/82aaaec6538ad34bc36ab524aaa756ff25c2f6723407c61d051220eb759a68b2.jpg", + "image_caption": [ + "Figure 5: Cosine similarity of sequence pairs randomly sampled from pretraining corpus and most similar pairs from STS-B using [CLS] from COCO- $\\mathrm { L M } _ { \\mathrm { B a s e } }$ . " + ], + "image_footnote": [], + "bbox": [ + 179, + 75, + 374, + 176 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/c1d97d0fc27a4c3c0d227b63fc15ba16a052a5c2bfaf1aa791b9ae2c82470dd6.jpg", + "image_caption": [ + "Figure 6: The t-SNE of sequence representations learned with or without SCL. The points are sampled from the most semantically similar sentences pairs from STS-B (with 5-score labels). The [CLS] embeddings are not fine-tuned. Some randomly selected similar pairs are marked by same shapes. " + ], + "image_footnote": [], + "bbox": [ + 419, + 75, + 816, + 190 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/992113321700f003cba33a4d9ae2e2e1ae1bb00ef61b749c875f454276b56dcf.jpg", + "image_caption": [ + "Figure 7: Analyses of SCL. Figs. (a) and (b) show the average cosine similarity between the [CLS] embeddings of positive and negative contrastive pairs during pretraining. Figs. (c) and (d) show the few-shot accuracy on MNLI with different fractions of MNLI training set used ( $\\scriptstyle { \\dot { x } }$ -axes). The error bars mark the max/min and the solid lines are the average of five fine-tuning runs. " + ], + "image_footnote": [], + "bbox": [ + 173, + 285, + 826, + 388 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 472, + 821, + 500 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Regularizing the Representation Learning for Better Few-Shot Ability. One would expect any pretrained Transformers to easily align a pair of corrupted sequence and cropped sequence as the two share about $8 0 \\%$ tokens. However, as shown in Figure 7a, that is not the case: Without SCL, the cosine similarity of the positive pairs is even lower than random negatives. SCL is necessary to regularize the representation space and to reduce the risk of degeneration (Figure 7b). ", + "bbox": [ + 174, + 506, + 825, + 575 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Similar to empirical observations and theoretical analyses in recent research [14, 16, 55], a more regularized representation space results in better generalization ability in scenarios with limited labels. Figure $\\mathrm { 7 c }$ and 7d show the results when COCO-LM are trained (via standard fine-tuning) with only a fraction of MNLI labels. The improvements brought by SCL are more significant when fewer fine-tuning labels are available. With $1 \\%$ MNLI labels, pretraining with SCL improves MNLI- $. \\mathrm { m } / \\mathrm { m m }$ accuracy by $0 . 8 / 0 . 5$ compared to that without SCL. Using only $1 0 \\% / 2 0 \\%$ labels, COCO-LM with SCL reaches similar MNLI accuracy with RoBERTa (Ours)/ELECTRA (Ours) fine-tuned with all labels, respectively. ", + "bbox": [ + 174, + 582, + 825, + 693 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.3 Analyses of Language Modeling with CLM ", + "text_level": 1, + "bbox": [ + 174, + 709, + 513, + 724 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The last group of experiments studies the effectiveness and benefits of CLM. ", + "bbox": [ + 176, + 734, + 674, + 750 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Ablations on Training Configurations. Figure 8 illustrates pretraining process with CLM and All-Token MLM. The plots demonstrate the difficulty of language modeling upon corrupted text sequences. It is quite an unbalanced task. For the majority of the tokens (Original) the task is simply to copy its input at the same position. For the replaced tokens $( 7 - 8 \\%$ total), however, the model needs to detect the abnormality brought by the auxiliary model and recover the original token. Implicitly training the copy mechanism as part of the hard LM task is not effective: The copy accuracy of All-Token MLM is much lower, and thus the LM head may confuse original tokens with replaced ones. As shown in Table 3 and ELECTRA [7], pretraining with All-Token MLM performs worse than using the RTD task, though the latter is equivalent to only training the copy mechanism. The multi-task learning of CLM is necessary for the main Transformer to stably learn the language modeling task upon the corrupted text sequence. ", + "bbox": [ + 174, + 756, + 825, + 909 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/7f4f1d35a5619b20f4bb5dab0e62dc152a989762d0e164bcb37c0492f45877fe.jpg", + "image_caption": [ + "Figure 8: The copying accuracy and the language modeling accuracy $y$ -axes) of CLM and All-Token MLM at different pretraining steps ( $x$ -axes, in 10K scale). The accuracy is averaged on tokens that are replaced by the auxiliary Transformer (Replaced) or those from the original input text (Original). " + ], + "image_footnote": [], + "bbox": [ + 174, + 88, + 823, + 189 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Prompt-Based Fine-Tuning with CLM. Table 4 includes the prompt-based fine-tuning experiments on MNLI for RoBERTa and COCO-LM under $b a s e + +$ and large $^ { + + }$ sizes, following the same few-shot manual prompt fine-tuning with demonstration setup in LM-BFF [15]. We use $\\{ 3 e - 6 , 4 e - 6 , 5 e - 6 \\}$ for the learning rate search of COCO-LM base++/large++ model, with everything else kept same as described in LM-BFF. With exactly the same pipeline, COCOLM outperforms RoBERTa under both $b a s e + +$ and $l a r g e + +$ sizes by significant margins on MNLI$\\mathrm { m } / \\mathrm { m m }$ . Such observations are interesting as COCOLM’s main Transformer does not even see any ", + "bbox": [ + 174, + 267, + 517, + 446 + ], + "page_idx": 9 + }, + { + "type": "table", + "img_path": "images/b69ebfa31e564bad91527412c0c106a5b7acd7bf62724bb1f125afb7db4b1808.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
ModelMNLI-mMNLI-mm
RoBERTaBase++60.1 (1.5)61.8 (1.2)
COCO-LMBase++66.5 (2.1)68.0 (2.3)
RoBERTaLarge++70.7 (1.3)72.0 (1.2)
COCO-LMLarge++72.0 (1.5)73.3 (1.1)
", + "bbox": [ + 532, + 270, + 821, + 347 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Table 4: Few-shot prompt-based fine-tuning using RoBERTa and COCO-LM trained on 16 samples per class. Mean (and standard deviation) accuracy results over 5 different splits on MNLI- $. \\mathrm { m } / \\mathrm { m m }$ are shown. ", + "bbox": [ + 529, + 357, + 823, + 426 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "[MASK] tokens during pretraining but still performs well on predicting masked tokens for promptbased learning. Note that ELECTRA and COCO-LM variants without the CLM task are not applicable: Their main Transformers are not pretrained by language modeling tasks (thus no language modeling capability is learned to generate prompt label words). This points out the importance, if not necessity, of COCO-LM in the family of ELECTRA-style pretraining models. With the benefits and rapid developments of prompt-based approaches, the lack of language modeling capability is going to limit the potential of ELECTRA’s self-supervised learning framework in many real-world scenarios. COCO-LM not only addresses this limitation but also provides better prompt-based learning results. ", + "bbox": [ + 174, + 446, + 826, + 558 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 Conclusions and Future Work ", + "text_level": 1, + "bbox": [ + 174, + 582, + 457, + 599 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this paper, we present COCO-LM, which pretrains language models using Corrective Language Modeling and Sequence Contrastive Learning upon corrupted text sequences. With standard pretraining data and Transformer architectures, COCO-LM improves the accuracy on the GLUE and SQuAD benchmarks, while also being more efficient in utilizing pretraining computing resources and network parameters. ", + "bbox": [ + 174, + 616, + 825, + 686 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "One limitation of this work is that the contrastive pairs are constructed by simple cropping and MLM replacements. Recent studies have shown the effectiveness of advanced data augmentation techniques in fine-tuning language models [16, 38, 51]. A future research direction is to explore better ways to construct contrastive pairs in language model pretraining. ", + "bbox": [ + 174, + 693, + 825, + 748 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Despite the empirical advantage of this auxiliary-main dual model framework, the auxiliary Transformer training is not influenced by the main Transformer nor learns to generate the optimal pretraining signals for the main model. To better understand and tailor the training of the auxiliary model to the main model is another important future research direction. ", + "bbox": [ + 174, + 755, + 825, + 810 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgments ", + "text_level": 1, + "bbox": [ + 176, + 835, + 328, + 852 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We sincerely thank Guolin Ke for discussions and advice on model implementation. We also thank anonymous reviewers for valuable and insightful feedback, especially the suggestion of adding prompt-based fine-tuning experiments. ", + "bbox": [ + 176, + 869, + 823, + 911 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 90, + 266, + 106 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "[1] Hangbo Bao, Li Dong, Furu Wei, Wenhui Wang, Nan Yang, Xiaodong Liu, Yu Wang, Songhao Piao, Jianfeng Gao, Ming Zhou, and Hsiao-Wuen Hon. UniLMv2: Pseudo-masked language models for unified language model pre-training. In ICML, 2020. 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The first token-level task, Corrective Language Modeling, is to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 355, + 469, + 367 + ], + "spans": [ + { + "bbox": [ + 141, + 355, + 469, + 367 + ], + "score": 1.0, + "content": "detect and correct tokens replaced by the auxiliary model, in order to better capture", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 366, + 470, + 379 + ], + "spans": [ + { + "bbox": [ + 141, + 366, + 470, + 379 + ], + "score": 1.0, + "content": "token-level semantics. 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For example, next sentence prediction [11], sentence ordering [27]", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 458, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 506, + 471 + ], + "score": 1.0, + "content": "and previous sentence prediction [56] concatenate two sentences (either correlated or random), and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 469, + 271, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 271, + 481 + ], + "score": 1.0, + "content": "train the Transformer to classify the pair.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 436, + 507, + 481 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 485, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 486, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 506, + 497 + ], + "score": 1.0, + "content": "Empirically, MLM is still among the most effective tasks to pretrain encoders [29, 31, 40].", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "score": 1.0, + "content": "RoBERTa [31] found the sentence-level task in BERT not benefitial and discarded it. 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The empirical advantages of other", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 517, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 532 + ], + "score": 1.0, + "content": "pretraining tasks are more task-specific, for example, entity related masks for knowledge intensive", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 528, + 429, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 429, + 542 + ], + "score": 1.0, + "content": "applications [20, 24], and sequence-level tasks for long form text modeling [42].", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 486, + 506, + 542 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 505, + 611 + ], + "lines": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "Instead of randomly altering texts, ELECTRA [7] uses a smaller auxiliary Transformer pretrained", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 556, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 570 + ], + "score": 1.0, + "content": "by MLM to replace some tokens in the text sequences using its language modeling probability, and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 567, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 104, + 567, + 506, + 580 + ], + "score": 1.0, + "content": "pretrains the main Transformer to detect the replaced tokens. 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Its effectiveness of in language is more observed in the fine-tuning stage, for", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 638, + 468, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 468, + 650 + ], + "score": 1.0, + "content": "example, in sentence representation [16], dense retrieval [60], and GLUE fine-tuning [19].", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 616, + 506, + 650 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 665, + 165, + 678 + ], + "lines": [ + { + "bbox": [ + 104, + 663, + 168, + 681 + ], + "spans": [ + { + "bbox": [ + 104, + 663, + 168, + 681 + ], + "score": 1.0, + "content": "3 Method", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 106, + 690, + 476, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 689, + 478, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 478, + 704 + ], + "score": 1.0, + "content": "We present the preliminaries of PLMs, their challenges, and the new COCO-LM framework.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 689, + 478, + 704 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 72, + 321, + 84 + ], + "lines": [ + { + "bbox": [ + 105, + 71, + 322, + 87 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 322, + 87 + ], + "score": 1.0, + "content": "3.1 Preliminary on Language Model Pretraining", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 92, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "score": 1.0, + "content": "In this work we focus on pretraining BERT-style bidirectional Transformer encoders [11] that are", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "score": 1.0, + "content": "widely used in language representation tasks. 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Similar to ELECTRA, the auxiliary Transformer is smaller than the main", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 132, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 506, + 146 + ], + "score": 1.0, + "content": "model, but we use different configurations in the auxiliary model: (1) We reduce the number of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 144, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 143, + 157 + ], + "score": 1.0, + "content": "layers to", + "type": "text" + }, + { + "bbox": [ + 144, + 144, + 160, + 155 + ], + "score": 0.58, + "content": "1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 144, + 172, + 157 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 172, + 144, + 189, + 156 + ], + "score": 0.53, + "content": "1 / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 144, + 505, + 157 + ], + "score": 1.0, + "content": "(under base or large model setup, respectively) but keep its hidden dimension", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 155, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 505, + 167 + ], + "score": 1.0, + "content": "the same with the main model, instead of shrinking its hidden dimensions; (2) We disable dropout in", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "it when sampling replacement tokens. We find such configurations empirically more effective and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "score": 1.0, + "content": "use them as the backbone of COCO-LM. The main Transformer follows the standard architecture of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 186, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 201 + ], + "score": 1.0, + "content": "BERT/ELECTRA and can be easily adopted by downstream application pipelines with almost no", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 197, + 145, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 145, + 213 + ], + "score": 1.0, + "content": "changes.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 108, + 228, + 228, + 242 + ], + "lines": [ + { + "bbox": [ + 104, + 226, + 230, + 247 + ], + "spans": [ + { + "bbox": [ + 104, + 226, + 230, + 247 + ], + "score": 1.0, + "content": "4 Experimental Setup", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 255, + 505, + 299 + ], + "lines": [ + { + "bbox": [ + 105, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 383, + 268 + ], + "score": 1.0, + "content": "Pretraining Settings. We employ three standard settings, base, base", + "type": "text" + }, + { + "bbox": [ + 384, + 257, + 398, + 266 + ], + "score": 0.48, + "content": "^ { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 255, + 440, + 268 + ], + "score": 1.0, + "content": ", and large", + "type": "text" + }, + { + "bbox": [ + 440, + 257, + 454, + 266 + ], + "score": 0.32, + "content": "^ { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 255, + 505, + 268 + ], + "score": 1.0, + "content": ". Base is the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 266, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 147, + 277 + ], + "score": 0.64, + "content": "\\mathbf { B E R T _ { B a s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 266, + 506, + 279 + ], + "score": 1.0, + "content": "training configuration [11]: Pretraining on Wikipedia and BookCorpus [63] (16 GB of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 277, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 506, + 290 + ], + "score": 1.0, + "content": "texts) for 256 million samples on 512 token sequences (125K batches with 2048 batch size). We use", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 288, + 423, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 423, + 301 + ], + "score": 1.0, + "content": "the same corpus and 32, 768 uncased BPE vocabulary [47] as with TUPE [26].", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 304, + 505, + 348 + ], + "lines": [ + { + "bbox": [ + 106, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 141, + 315 + ], + "score": 0.73, + "content": "B a s e + +", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "trains the base size model with larger corpora and/or more training steps. Following recent", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "research [1, 31, 62], we add in OpenWebText [18], CC-News [31], and STORIES [52], to a total of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 340 + ], + "score": 1.0, + "content": "160 GB texts, and train for 4 billion (with 2048 batch size) samples [31]. We follow the prepossessing", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 337, + 334, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 334, + 350 + ], + "score": 1.0, + "content": "of UniLMV2 [1] and use 64, 000 cased BPE vocabulary.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 353, + 502, + 376 + ], + "lines": [ + { + "bbox": [ + 106, + 353, + 504, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 145, + 365 + ], + "score": 0.33, + "content": "L a r g e { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 353, + 280, + 366 + ], + "score": 1.0, + "content": "uses the same training corpora as", + "type": "text" + }, + { + "bbox": [ + 281, + 354, + 315, + 364 + ], + "score": 0.57, + "content": "b a s e + +", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 353, + 504, + 366 + ], + "score": 1.0, + "content": "and pretrains for 4 billion samples (2048 batch", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 364, + 383, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 383, + 377 + ], + "score": 1.0, + "content": "size). Its Transformer configuration is the same with BERTLarge [11].", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 380, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 507, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 253, + 394 + ], + "score": 1.0, + "content": "Model Architecture. Our base/base", + "type": "text" + }, + { + "bbox": [ + 253, + 382, + 267, + 391 + ], + "score": 0.44, + "content": "^ { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 380, + 330, + 394 + ], + "score": 1.0, + "content": "model uses the", + "type": "text" + }, + { + "bbox": [ + 330, + 381, + 370, + 392 + ], + "score": 0.86, + "content": "\\mathbf { B E R T _ { B a s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 380, + 507, + 394 + ], + "score": 1.0, + "content": "architecture [11]: 12 layer Trans-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 391, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 413, + 405 + ], + "score": 1.0, + "content": "former, 768 hidden size, plus T5 relative position encoding [40]. Our large", + "type": "text" + }, + { + "bbox": [ + 413, + 393, + 428, + 402 + ], + "score": 0.63, + "content": "^ { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 391, + 505, + 405 + ], + "score": 1.0, + "content": "model is the same", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 126, + 415 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 403, + 168, + 415 + ], + "score": 0.36, + "content": "\\mathrm { B E R T _ { L a r g e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 402, + 505, + 415 + ], + "score": 1.0, + "content": ", 24 layer and 1024 hidden size, plus T5 relative position encoding [40]. Our auxiliary", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 412, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 435, + 426 + ], + "score": 1.0, + "content": "network uses the same hidden size but a shallow 4-layer Transformer in base/base", + "type": "text" + }, + { + "bbox": [ + 435, + 415, + 450, + 424 + ], + "score": 0.58, + "content": "^ { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 412, + 505, + 426 + ], + "score": 1.0, + "content": "and a 6-layer", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 421, + 445, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 133, + 438 + ], + "score": 1.0, + "content": "one in", + "type": "text" + }, + { + "bbox": [ + 134, + 425, + 169, + 436 + ], + "score": 0.3, + "content": "l a r g e + +", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 421, + 244, + 438 + ], + "score": 1.0, + "content": ". When generating", + "type": "text" + }, + { + "bbox": [ + 244, + 423, + 271, + 435 + ], + "score": 0.88, + "content": "X ^ { \\mathrm { M L M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 421, + 445, + 438 + ], + "score": 1.0, + "content": "we disable dropout in the auxiliary model.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 381, + 453 + ], + "score": 1.0, + "content": "Downstream Tasks. We use the tasks included in GLUE [54] and", + "type": "text" + }, + { + "bbox": [ + 381, + 441, + 432, + 452 + ], + "score": 0.4, + "content": "\\mathrm { S Q u A D } 2 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "reading compres-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "sion [41]. Please refer to Appendix A for more details about GLUE tasks. Standard hyperparameter", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "score": 1.0, + "content": "search in fine-tuning is performed, and the search space can be found in Appendix B. The fine-tuning", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "protocols use the open-source implementation of TUPE [26]. The reported results are the median of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 484, + 275, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 275, + 496 + ], + "score": 1.0, + "content": "five random seeds on GLUE and SQuAD.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "score": 1.0, + "content": "Baselines. We compare with various pretrained models in each setting. To reduce the variance in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "data processing/environments, we also pretrain and fine-tune RoBERTa and ELECTRA under exactly", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "the same setting with COCO-LM, marked with “(Ours)”. All numbers unless marked by “(Ours)” are", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 534, + 386, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 386, + 546 + ], + "score": 1.0, + "content": "from reported results in recent research (more details in Appendix C).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 106, + 550, + 504, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "Implementation Details. Our implementation builds upon the open-source implementation from", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 560, + 481, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 481, + 573 + ], + "score": 1.0, + "content": "MC-BERT [61] and fairseq [35]. More implementation details are mentioned in Appendix D.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "title", + "bbox": [ + 107, + 591, + 222, + 604 + ], + "lines": [ + { + "bbox": [ + 104, + 589, + 223, + 606 + ], + "spans": [ + { + "bbox": [ + 104, + 589, + 223, + 606 + ], + "score": 1.0, + "content": "5 Evaluation Results", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 106, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "Three groups of experiments are conducted to evaluate COCO-LM and its two new pretraining tasks.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "title", + "bbox": [ + 108, + 645, + 258, + 657 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 259, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 259, + 659 + ], + "score": 1.0, + "content": "5.1 Overall Results and Ablations", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "Overall Results are listed in Table 1. Under all three settings, COCO-LM outperforms all recent", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "state-of-the-art pretraining models on GLUE average and SQuAD. It improves the state-of-the-art", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "GLUE score by about one point under all three settings. COCO-LM also enjoys better parameter", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 218, + 714 + ], + "score": 1.0, + "content": "efficiency. 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The main Transformer is pretrained to correct the corruption (CLM) and to contrast", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 506, + 108 + ], + "score": 1.0, + "content": "the corrupted sequences with the cropped sequences (SCL). The two Transformers are pretrained", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 473, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 473, + 118 + ], + "score": 1.0, + "content": "jointly with the loss in Eqn. (4). The main Transformer is used in downstream applications.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 73, + 507, + 118 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 121, + 505, + 210 + ], + "lines": [ + { + "bbox": [ + 105, + 122, + 506, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 506, + 134 + ], + "score": 1.0, + "content": "Network Configurations. 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We use", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 288, + 423, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 423, + 301 + ], + "score": 1.0, + "content": "the same corpus and 32, 768 uncased BPE vocabulary [47] as with TUPE [26].", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 255, + 506, + 301 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 304, + 505, + 348 + ], + "lines": [ + { + "bbox": [ + 106, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 141, + 315 + ], + "score": 0.73, + "content": "B a s e + +", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "trains the base size model with larger corpora and/or more training steps. Following recent", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "research [1, 31, 62], we add in OpenWebText [18], CC-News [31], and STORIES [52], to a total of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 340 + ], + "score": 1.0, + "content": "160 GB texts, and train for 4 billion (with 2048 batch size) samples [31]. 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Its Transformer configuration is the same with BERTLarge [11].", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 106, + 353, + 504, + 377 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 380, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 507, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 253, + 394 + ], + "score": 1.0, + "content": "Model Architecture. Our base/base", + "type": "text" + }, + { + "bbox": [ + 253, + 382, + 267, + 391 + ], + "score": 0.44, + "content": "^ { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 380, + 330, + 394 + ], + "score": 1.0, + "content": "model uses the", + "type": "text" + }, + { + "bbox": [ + 330, + 381, + 370, + 392 + ], + "score": 0.86, + "content": "\\mathbf { B E R T _ { B a s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 380, + 507, + 394 + ], + "score": 1.0, + "content": "architecture [11]: 12 layer Trans-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 391, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 413, + 405 + ], + "score": 1.0, + "content": "former, 768 hidden size, plus T5 relative position encoding [40]. Our large", + "type": "text" + }, + { + "bbox": [ + 413, + 393, + 428, + 402 + ], + "score": 0.63, + "content": "^ { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 391, + 505, + 405 + ], + "score": 1.0, + "content": "model is the same", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 126, + 415 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 403, + 168, + 415 + ], + "score": 0.36, + "content": "\\mathrm { B E R T _ { L a r g e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 402, + 505, + 415 + ], + "score": 1.0, + "content": ", 24 layer and 1024 hidden size, plus T5 relative position encoding [40]. 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When generating", + "type": "text" + }, + { + "bbox": [ + 244, + 423, + 271, + 435 + ], + "score": 0.88, + "content": "X ^ { \\mathrm { M L M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 421, + 445, + 438 + ], + "score": 1.0, + "content": "we disable dropout in the auxiliary model.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 380, + 507, + 438 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 381, + 453 + ], + "score": 1.0, + "content": "Downstream Tasks. We use the tasks included in GLUE [54] and", + "type": "text" + }, + { + "bbox": [ + 381, + 441, + 432, + 452 + ], + "score": 0.4, + "content": "\\mathrm { S Q u A D } 2 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "reading compres-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "sion [41]. Please refer to Appendix A for more details about GLUE tasks. Standard hyperparameter", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "score": 1.0, + "content": "search in fine-tuning is performed, and the search space can be found in Appendix B. The fine-tuning", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "protocols use the open-source implementation of TUPE [26]. The reported results are the median of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 484, + 275, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 275, + 496 + ], + "score": 1.0, + "content": "five random seeds on GLUE and SQuAD.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 440, + 506, + 496 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "score": 1.0, + "content": "Baselines. We compare with various pretrained models in each setting. To reduce the variance in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "data processing/environments, we also pretrain and fine-tune RoBERTa and ELECTRA under exactly", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "the same setting with COCO-LM, marked with “(Ours)”. All numbers unless marked by “(Ours)” are", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 534, + 386, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 386, + 546 + ], + "score": 1.0, + "content": "from reported results in recent research (more details in Appendix C).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 501, + 506, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 550, + 504, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "Implementation Details. Our implementation builds upon the open-source implementation from", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 560, + 481, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 481, + 573 + ], + "score": 1.0, + "content": "MC-BERT [61] and fairseq [35]. More implementation details are mentioned in Appendix D.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 550, + 505, + 573 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 591, + 222, + 604 + ], + "lines": [ + { + "bbox": [ + 104, + 589, + 223, + 606 + ], + "spans": [ + { + "bbox": [ + 104, + 589, + 223, + 606 + ], + "score": 1.0, + "content": "5 Evaluation Results", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 106, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "Three groups of experiments are conducted to evaluate COCO-LM and its two new pretraining tasks.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40, + "bbox_fs": [ + 106, + 617, + 506, + 631 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 645, + 258, + 657 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 259, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 259, + 659 + ], + "score": 1.0, + "content": "5.1 Overall Results and Ablations", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "Overall Results are listed in Table 1. Under all three settings, COCO-LM outperforms all recent", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "state-of-the-art pretraining models on GLUE average and SQuAD. It improves the state-of-the-art", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "GLUE score by about one point under all three settings. COCO-LM also enjoys better parameter", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 218, + 714 + ], + "score": 1.0, + "content": "efficiency. 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ModelParamsGLUE Single TaskSQuAD 2.0
MNLI-(m/mm)QQPQNLISST-2CoLARTEMRPCSTS-BAVGEMF1
Base Setting: BERT Base Size,Wikipedia + Book Corpus (16GB)
BERT[11]110M84.5/-91.391.793.258.968.687.389.583.173.776.3
RoBERTa [31]125M84.7/-1192.71111179.7
XLNet [62]110M85.8/85.41192.71111178.581.3
ELECTRA [7]110M86.0/85.390.091.993.464.370.884.989.183.780.583.3
MC-BERT [61]110M85.7/85.289.791.392.362.175.086.088.083.7
DeBERTa [23]134M86.3/86.2二一一一一一一79.382.5
TUPE [26]110M86.2/86.291.392.293.363.673.689.989.284.911
RoBERTa (Ours)110M85.8/85.591.392.093.760.168.287.388.583.377.780.5
ELECTRA (Ours)110M86.9/86.791.992.693.666.275.188.289.785.579.782.6
COCO-LM110M88.5/88.392.093.193.263.984.891.490.387.282.485.2
Base++ Seting: BERT Base Size,Bigger Training Data,and/or More Training Steps
XLNet [62]110M86.8/-91.491.794.760.274.088.289.584.680.2
RoBERTa [31]125M87.6/-91.992.894.863.678.790.291.286.480.583.7
UniLMV2[1]110M88.5/-91.793.595.165.281.391.891.087.183.386.1
DeBERTa [23]134M88.8/88.5一一一1一一一83.186.2
CLEAR [59]110M86.7/-90.092.994.564.378.389.289.885.7一1
COCO-LM134M90.2/90.092.294.294.667.387.491.291.888.685.488.1
Large++ Setting: BERTLarge Size,Bigger Training Data,and More Training Steps
XLNet [62]360M90.8/90.892.394.997.069.085.990.892.589.287.990.6
RoBERTa [31]356M90.2/90.292.294.796.468.086.690.992.488.986.589.4
ELECTRA[7]335M90.9/-92.495.096.969.188.090.892.689.488.090.6
DeBERTa [23]384M91.1/91.192.395.396.870.5111188.090.7
COCO-LM367M91.4/91.692.895.796.973.991.092.292.790.888.291.0
Megatron1.3B [49]1.3B90.9/91.092.611一187.190.2
Megatron3.9B [49]3.9B91.4/91.492.711188.591.2
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ModelParamsMNLI-(m/mm)QQPQNLISST-2CoLARTEMRPCSTS-BAVG
Base/Base++ Setting: BERT Base Size
BERTBase110M84.6/83.489.290.593.552.166.484.885.880.8
ELECTRABase++110M88.5/88.089.593.196.064.675.288.190.285.6
COCO-LMBase++134M89.8/89.389.894.295.668.682.388.590.387.4
Large/Large++ Seting: BERT Large Size
BERTLarge335M86.7/85.989.392.794.960.570.185.486.583.2
ELECTRALarge++335M90.7/90.290.495.596.768.186.189.291.788.5
COCO-LMLarge++367M91.6/91.190.595.896.770.589.288.491.889.3
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It outperforms RoBERTa & ELECTRA by", + "type": "text" + }, + { + "bbox": [ + 401, + 624, + 415, + 634 + ], + "score": 0.81, + "content": "1 +", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 623, + 505, + 635 + ], + "score": 1.0, + "content": "points on MNLI with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 634, + 503, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 349, + 647 + ], + "score": 1.0, + "content": "the same GPU hours and reaches their accuracy with around", + "type": "text" + }, + { + "bbox": [ + 349, + 634, + 369, + 645 + ], + "score": 0.88, + "content": "6 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 634, + 380, + 647 + ], + "score": 1.0, + "content": "&", + "type": "text" + }, + { + "bbox": [ + 380, + 634, + 400, + 645 + ], + "score": 0.88, + "content": "5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 634, + 503, + 647 + ], + "score": 1.0, + "content": "GPU hours, respectively.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 650, + 504, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 506, + 663 + ], + "score": 1.0, + "content": "Ablation Studies. 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ModelParamsGLUE Single TaskSQuAD 2.0
MNLI-(m/mm)QQPQNLISST-2CoLARTEMRPCSTS-BAVGEMF1
Base Setting: BERT Base Size,Wikipedia + Book Corpus (16GB)
BERT[11]110M84.5/-91.391.793.258.968.687.389.583.173.776.3
RoBERTa [31]125M84.7/-1192.71111179.7
XLNet [62]110M85.8/85.41192.71111178.581.3
ELECTRA [7]110M86.0/85.390.091.993.464.370.884.989.183.780.583.3
MC-BERT [61]110M85.7/85.289.791.392.362.175.086.088.083.7
DeBERTa [23]134M86.3/86.2二一一一一一一79.382.5
TUPE [26]110M86.2/86.291.392.293.363.673.689.989.284.911
RoBERTa (Ours)110M85.8/85.591.392.093.760.168.287.388.583.377.780.5
ELECTRA (Ours)110M86.9/86.791.992.693.666.275.188.289.785.579.782.6
COCO-LM110M88.5/88.392.093.193.263.984.891.490.387.282.485.2
Base++ Seting: BERT Base Size,Bigger Training Data,and/or More Training Steps
XLNet [62]110M86.8/-91.491.794.760.274.088.289.584.680.2
RoBERTa [31]125M87.6/-91.992.894.863.678.790.291.286.480.583.7
UniLMV2[1]110M88.5/-91.793.595.165.281.391.891.087.183.386.1
DeBERTa [23]134M88.8/88.5一一一1一一一83.186.2
CLEAR [59]110M86.7/-90.092.994.564.378.389.289.885.7一1
COCO-LM134M90.2/90.092.294.294.667.387.491.291.888.685.488.1
Large++ Setting: BERTLarge Size,Bigger Training Data,and More Training Steps
XLNet [62]360M90.8/90.892.394.997.069.085.990.892.589.287.990.6
RoBERTa [31]356M90.2/90.292.294.796.468.086.690.992.488.986.589.4
ELECTRA[7]335M90.9/-92.495.096.969.188.090.892.689.488.090.6
DeBERTa [23]384M91.1/91.192.395.396.870.5111188.090.7
COCO-LM367M91.4/91.692.895.796.973.991.092.292.790.888.291.0
Megatron1.3B [49]1.3B90.9/91.092.611一187.190.2
Megatron3.9B [49]3.9B91.4/91.492.711188.591.2
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ModelParamsMNLI-(m/mm)QQPQNLISST-2CoLARTEMRPCSTS-BAVG
Base/Base++ Setting: BERT Base Size
BERTBase110M84.6/83.489.290.593.552.166.484.885.880.8
ELECTRABase++110M88.5/88.089.593.196.064.675.288.190.285.6
COCO-LMBase++134M89.8/89.389.894.295.668.682.388.590.387.4
Large/Large++ Seting: BERT Large Size
BERTLarge335M86.7/85.989.392.794.960.570.185.486.583.2
ELECTRALarge++335M90.7/90.290.495.596.768.186.189.291.788.5
COCO-LMLarge++367M91.6/91.190.595.896.770.589.288.491.889.3
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It outperforms RoBERTa & ELECTRA by", + "type": "text" + }, + { + "bbox": [ + 401, + 624, + 415, + 634 + ], + "score": 0.81, + "content": "1 +", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 623, + 505, + 635 + ], + "score": 1.0, + "content": "points on MNLI with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 634, + 503, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 349, + 647 + ], + "score": 1.0, + "content": "the same GPU hours and reaches their accuracy with around", + "type": "text" + }, + { + "bbox": [ + 349, + 634, + 369, + 645 + ], + "score": 0.88, + "content": "6 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 634, + 380, + 647 + ], + "score": 1.0, + "content": "&", + "type": "text" + }, + { + "bbox": [ + 380, + 634, + 400, + 645 + ], + "score": 0.88, + "content": "5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 634, + 503, + 647 + ], + "score": 1.0, + "content": "GPU hours, respectively.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 600, + 507, + 647 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 650, + 504, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 506, + 663 + ], + "score": 1.0, + "content": "Ablation Studies. Table 3 shows the ablations of COCO-LM under the base setting on GLUE DEV.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 106, + 649, + 506, + 663 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 678 + ], + "score": 1.0, + "content": "Pretraining Task. With only RTD, our backbone model with the shallow auxiliary Transformer is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 677, + 507, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 507, + 692 + ], + "score": 1.0, + "content": "quite effective. CLM and SCL both provide additional improvements on MNLI and GLUE average.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "Their advantages are better observed on different tasks, for example, CLM on MNLI-mm and SCL", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "on RTE and MRPC. Combining the two in COCO-LM provides better overall effectiveness. In later", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 711, + 361, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 361, + 723 + ], + "score": 1.0, + "content": "experiments, we further analyze the benefits of these two tasks.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 667, + 507, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 107, + 70, + 503, + 190 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 107, + 70, + 503, + 190 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 70, + 503, + 190 + ], + "spans": [ + { + "bbox": [ + 107, + 70, + 503, + 190 + ], + "score": 0.98, + "html": "
GroupMethodMNLI-(m/mm)QQPQNLISST-2CoLARTEMRPCSTS-BAVG
COCO-LMBase88.5/88.392.093.193.263.984.891.490.387.2
Pretraining TaskRTDOnly88.4/88.292.193.592.767.380.589.090.986.8
CLMOnly88.6/88.492.093.293.767.480.190.090.486.9
SCL +RTD88.6/88.292.193.593.864.382.790.290.686.9
Network Setingw/o. Rel-Pos w. ELECTRA's Auxiliary88.2/87.792.293.493.768.882.791.290.687.6
Training88.0/87.791.992.793.564.381.289.589.786.3
w.Random Replacements84.9/84.791.491.191.441.670.087.387.180.6
Signalw. Converged Auxiliary88.3/88.192.092.894.364.278.390.490.286.3
CLM SetupAll-Token LM Only87.2/87.092.693.788.589.784.7
CLM w/o. Copy88.0/87.991.8 91.893.194.460.6 66.674.0 76.989.590.186.3
CLM w/o. Stop-grad88.5/88.292.092.994.366.580.990.090.686.9
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Using randomly replaced tokens to corrupt text sequence hurts", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "significantly. Using a converged auxiliary network to pretrain the main model also hurts. It is better", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "score": 1.0, + "content": "to pretrain the two Transformers together, as the auxiliary model gradually increases the difficulty of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 470, + 481, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 481, + 482 + ], + "score": 1.0, + "content": "the corrupted sequences and provides a natural learning curriculum for the main Transformer.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 486, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 499 + ], + "score": 1.0, + "content": "CLM Setup. 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GroupMethodMNLI-(m/mm)QQPQNLISST-2CoLARTEMRPCSTS-BAVG
COCO-LMBase88.5/88.392.093.193.263.984.891.490.387.2
Pretraining TaskRTDOnly88.4/88.292.193.592.767.380.589.090.986.8
CLMOnly88.6/88.492.093.293.767.480.190.090.486.9
SCL +RTD88.6/88.292.193.593.864.382.790.290.686.9
Network Setingw/o. Rel-Pos w. ELECTRA's Auxiliary88.2/87.792.293.493.768.882.791.290.687.6
Training88.0/87.791.992.793.564.381.289.589.786.3
w.Random Replacements84.9/84.791.491.191.441.670.087.387.180.6
Signalw. Converged Auxiliary88.3/88.192.092.894.364.278.390.490.286.3
CLM SetupAll-Token LM Only87.2/87.092.693.788.589.784.7
CLM w/o. Copy88.0/87.991.8 91.893.194.460.6 66.674.0 76.989.590.186.3
CLM w/o. Stop-grad88.5/88.292.092.994.366.580.990.090.686.9
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To better understand and tailor the training of the auxiliary model to", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 631, + 356, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 356, + 643 + ], + "score": 1.0, + "content": "the main model is another important future research direction.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47.5 + }, + { + "type": "title", + "bbox": [ + 108, + 662, + 201, + 675 + ], + "lines": [ + { + "bbox": [ + 106, + 660, + 203, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 203, + 677 + ], + "score": 1.0, + "content": "Acknowledgments", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 50 + }, + { + "type": "text", + "bbox": [ + 108, + 689, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "We sincerely thank Guolin Ke for discussions and advice on model implementation. 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We also thank", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 699, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 714 + ], + "score": 1.0, + "content": "anonymous reviewers for valuable and insightful feedback, especially the suggestion of adding", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 711, + 264, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 264, + 724 + ], + "score": 1.0, + "content": "prompt-based fine-tuning experiments.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 52, + "bbox_fs": [ + 105, + 689, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 72, + 163, + 84 + ], + "lines": [ + { + "bbox": [ + 106, + 70, + 165, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 70, + 165, + 86 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 105, + 86, + 507, + 724 + ], + "lines": [ + { + "bbox": [ + 110, + 90, + 506, + 101 + ], + "spans": [ + { + "bbox": [ + 110, + 90, + 506, + 101 + ], + "score": 1.0, + "content": "[1] Hangbo Bao, Li Dong, Furu Wei, Wenhui Wang, Nan Yang, Xiaodong Liu, Yu Wang, Songhao Piao,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 125, + 99, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 125, + 99, + 506, + 113 + ], + "score": 1.0, + "content": "Jianfeng Gao, Ming Zhou, and Hsiao-Wuen Hon. 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ModelParamsGLUE Single TaskSQuAD 2.0
MNLI-(m/mm)QQPQNLISST-2CoLARTEMRPCSTS-BAVGEMF1
Base Setting: BERT Base Size,Wikipedia + Book Corpus (16GB)
BERT[11]110M84.5/-91.391.793.258.968.687.389.583.173.776.3
RoBERTa [31]125M84.7/-1192.71111179.7
XLNet [62]110M85.8/85.41192.71111178.581.3
ELECTRA [7]110M86.0/85.390.091.993.464.370.884.989.183.780.583.3
MC-BERT [61]110M85.7/85.289.791.392.362.175.086.088.083.7
DeBERTa [23]134M86.3/86.2二一一一一一一79.382.5
TUPE [26]110M86.2/86.291.392.293.363.673.689.989.284.911
RoBERTa (Ours)110M85.8/85.591.392.093.760.168.287.388.583.377.780.5
ELECTRA (Ours)110M86.9/86.791.992.693.666.275.188.289.785.579.782.6
COCO-LM110M88.5/88.392.093.193.263.984.891.490.387.282.485.2
Base++ Seting: BERT Base Size,Bigger Training Data,and/or More Training Steps
XLNet [62]110M86.8/-91.491.794.760.274.088.289.584.680.2
RoBERTa [31]125M87.6/-91.992.894.863.678.790.291.286.480.583.7
UniLMV2[1]110M88.5/-91.793.595.165.281.391.891.087.183.386.1
DeBERTa [23]134M88.8/88.5一一一1一一一83.186.2
CLEAR [59]110M86.7/-90.092.994.564.378.389.289.885.7一1
COCO-LM134M90.2/90.092.294.294.667.387.491.291.888.685.488.1
Large++ Setting: BERTLarge Size,Bigger Training Data,and More Training Steps
XLNet [62]360M90.8/90.892.394.997.069.085.990.892.589.287.990.6
RoBERTa [31]356M90.2/90.292.294.796.468.086.690.992.488.986.589.4
ELECTRA[7]335M90.9/-92.495.096.969.188.090.892.689.488.090.6
DeBERTa [23]384M91.1/91.192.395.396.870.5111188.090.7
COCO-LM367M91.4/91.692.895.796.973.991.092.292.790.888.291.0
Megatron1.3B [49]1.3B90.9/91.092.611一187.190.2
Megatron3.9B [49]3.9B91.4/91.492.711188.591.2
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ModelParamsMNLI-(m/mm)QQPQNLISST-2CoLARTEMRPCSTS-BAVG
Base/Base++ Setting: BERT Base Size
BERTBase110M84.6/83.489.290.593.552.166.484.885.880.8
ELECTRABase++110M88.5/88.089.593.196.064.675.288.190.285.6
COCO-LMBase++134M89.8/89.389.894.295.668.682.388.590.387.4
Large/Large++ Seting: BERT Large Size
BERTLarge335M86.7/85.989.392.794.960.570.185.486.583.2
ELECTRALarge++335M90.7/90.290.495.596.768.186.189.291.788.5
COCO-LMLarge++367M91.6/91.190.595.896.770.589.288.491.889.3
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GroupMethodMNLI-(m/mm)QQPQNLISST-2CoLARTEMRPCSTS-BAVG
COCO-LMBase88.5/88.392.093.193.263.984.891.490.387.2
Pretraining TaskRTDOnly88.4/88.292.193.592.767.380.589.090.986.8
CLMOnly88.6/88.492.093.293.767.480.190.090.486.9
SCL +RTD88.6/88.292.193.593.864.382.790.290.686.9
Network Setingw/o. Rel-Pos w. ELECTRA's Auxiliary88.2/87.792.293.493.768.882.791.290.687.6
Training88.0/87.791.992.793.564.381.289.589.786.3
w.Random Replacements84.9/84.791.491.191.441.670.087.387.180.6
Signalw. Converged Auxiliary88.3/88.192.092.894.364.278.390.490.286.3
CLM SetupAll-Token LM Only87.2/87.092.693.788.589.784.7
CLM w/o. Copy88.0/87.991.8 91.893.194.460.6 66.674.0 76.989.590.186.3
CLM w/o. Stop-grad88.5/88.292.092.994.366.580.990.090.686.9
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As an example, Turing machines are synthesised from input-output examples by propagating uncertainty through a smooth relaxation of a universal Turing machine. The posterior distribution over weights is approximated using Markov chain Monte Carlo and bounds on the generalisation error of these models is estimated using the real log canonical threshold, a geometric invariant from singular learning theory. + +# 1 INTRODUCTION + +The idea of program synthesis dates back to the birth of modern computation itself (Turing, 1948) and is recognised as one of the most important open problems in computer science (Gulwani et al., 2017). However, there appear to be serious obstacles to synthesising programs by gradient descent at scale (Neelakantan et al., 2016; Kaiser & Sutskever, 2016; Bunel et al., 2016; Gaunt et al., 2016; Evans & Grefenstette, 2018; Chen et al., 2018) and these problems suggest that it would be appropriate to make a fundamental study of the geometry of loss surfaces in program synthesis, since this geometry determines the learning process. To that end, in this paper we explain a new point of view on program synthesis using the singular learning theory of Watanabe (2009) and the smooth relaxation of Turing machines from Clift & Murfet (2018). + +In broad strokes this new geometric point of view on program synthesis says: + +• Programs to be synthesised are singularities of analytic functions. If $U \subseteq \mathbb { R } ^ { d }$ is open and $K : U \longrightarrow \mathbb { R }$ is analytic, then $x \in U$ is a critical point of $K$ if $\nabla K ( x ) = 0$ and a singularity of the function $K$ if it is a critical point where $K ( x ) = 0$ . The Kolmogorov complexity of a program is related to a geometric invariant of the associated singularity called the Real Log Canonical Threshold (RLCT). This invariant controls both the generalisation error and the learning process, and is therefore an appropriate measure of “complexity” in continuous program synthesis. See Section 3. The geometry has concrete practical implications. For example, a MCMC-based approach to program synthesis will find, with high probability, a solution that is of low complexity (if it finds a solution at all). We sketch a novel point of view on the problem of “bad local minima” (Gaunt et al., 2016) based on these ideas. See Section 4. + +We demonstrate all of these principles in experiments with toy examples of synthesis problems. + +Program synthesis as inference. We use Turing machines, but mutatis mutandis everything applies to other programming languages. Let $T$ be a Turing machine with tape alphabet $\Sigma$ and set of states $Q$ and assume that on any input $x \in \Sigma ^ { * }$ the machine eventually halts with output $T ( x ) \in \Sigma ^ { * }$ . Then to the machine $T$ we may associate the set $\{ ( x , T ( x ) ) \} _ { x \in \Sigma ^ { * } } \subseteq \Sigma ^ { * } \times \Sigma ^ { * }$ . Program synthesis is the study of the inverse problem: given a subset of $\Sigma ^ { * } \times \Sigma ^ { * }$ we would like to determine (if possible) a Turing machine which computes the given outputs on the given inputs. + +If we presume given a probability distribution $q ( x )$ on $\Sigma ^ { * }$ then we can formulate this as a problem of statistical inference: given a probability distribution $q ( x , y )$ on $\Sigma ^ { * } \times \Sigma ^ { * }$ determine the most likely machine producing the observed distribution $q ( x , y ) = q ( y | x ) q ( x )$ . If we fix a universal Turing machine $\mathcal { U }$ then Turing machines can be parametrised by codes $\dot { w } \in W ^ { c o d e }$ with $\mathcal { U } ( x , w ) = T ( x )$ for all $x \in \Sigma ^ { * }$ . We let ${ \bar { p } } ( y | x , w )$ denote the probability of $\mathcal { U } ( x , w ) = y$ (which is either zero or one) + +so that solutions to the synthesis problem are in bijection with the zeros of the Kullback-Leibler divergence between the true distribution and the model + +$$ +K ( w ) = \int \int q ( y | x ) q ( x ) \log \frac { q ( y | x ) } { p ( y | x , w ) } d x d y . +$$ + +So far this is just a trivial rephrasing of the combinatorial optimisation problem of finding a Turing machine $T$ with $T ( x ) = y$ for all $( x , y )$ with $q ( x , y ) > 0$ . + +Smooth relaxation. One approach is to seek a smooth relaxation of the synthesis problem consisting of an analytic manifold $W \bar { \supseteq } W ^ { c o d e }$ and an extension of $K$ to an analytic function $K : W \longrightarrow \mathbb { R }$ so that we can search for the zeros of $K$ using gradient descent. Perhaps the most natural way to construct such a smooth relaxation is to take $W$ to be a space of probability distributions over $W ^ { \dot { c } o d e }$ and prescribe a model $p ( \boldsymbol { y } | \boldsymbol { x } , \boldsymbol { w } )$ for propagating uncertainty about codes to uncertainty about outputs (Gaunt et al., 2016; Evans & Grefenstette, 2018). The particular model we choose is based on the semantics of linear logic (Clift & Murfet, 2018). Supposing that such a smooth relaxation has been chosen together with a prior $\varphi ( w )$ over $W$ , smooth program synthesis becomes the study of the statistical learning theory of the triple $( p , q , \varphi )$ . + +There are perhaps two primary reasons to consider the smooth relaxation. Firstly, one might hope that stochastic gradient descent or techniques like Markov chain Monte Carlo will be effective means of solving the original combinatorial optimisation problem. This is not a new idea (Gulwani et al., 2017, §6) but so far its effectiveness for large programs has not been proven. Independently, one might hope to find powerful new mathematical ideas that apply to the relaxed problem and shed light on the nature of program synthesis. This is the purpose of the present paper. + +Singular learning theory. We denote by $W _ { 0 } = \{ w \in W | K ( w ) = 0 \}$ so that + +$$ +W _ { 0 } \cap W ^ { c o d e } \subseteq W _ { 0 } \subseteq W +$$ + +where $W _ { 0 } \cap W ^ { c o d e }$ is the discrete set of solutions to the original synthesis problem. We refer to these as the classical solutions. As the vanishing locus of an analytic function, $W _ { 0 }$ is an analytic space over $\mathbb { R }$ (Hironaka, 1964, $\ S 0 . 1 )$ , (Griffith & Harris, 1978) and it is interesting to study the geometry of this space near the classical solutions. Since $K$ is a Kullback-Leibler divergence it is non-negative and so it not only vanishes on $W _ { 0 }$ but $\nabla K$ also vanishes, hence every point of $W _ { 0 }$ is a singular point. + +Beyond this the geometry of $W _ { 0 }$ depends on the particular model $p ( \boldsymbol { y } | \boldsymbol { x } , \boldsymbol { w } )$ that has been chosen, but some aspects are universal: the nature of program synthesis means that typically $W _ { 0 }$ is an extended object (i.e. it contains points other than the classical solutions) and the Hessian matrix of second order partial derivatives of $K$ at a classical solution is not invertible - that is, the classical solutions are degenerate critical points of $K$ . This means that singularity theory is the appropriate branch of mathematics for studying the geometry of $W _ { 0 }$ near a classical solution. It also means that the Fisher information matrix + +$$ +I ( w ) _ { i j } = \int \int \frac { \partial } { \partial w _ { i } } \big [ \log p ( y | x , w ) \big ] \frac { \partial } { \partial w _ { j } } \big [ \log p ( y | x , w ) \big ] q ( y | x ) q ( x ) d x d y , +$$ + +is degenerate at a classical solution, so that the appropriate branch of statistical learning theory is singular learning theory (Watanabe, 2007; 2009). For an introduction to singular learning theory in the context of deep learning see (Murfet et al., 2020). + +Broadly speaking the contribution of this paper is to realise program synthesis within the framework of singular learning theory, at both a theoretical and an experimental level. In more detail the contents of the paper are: + +• We define a staged pseudo-UTM (Appendix E) which is well-suited to experiments with the ideas discussed above. Propagating uncertainty about the code through this UTM using the ideas of (Clift & Murfet, 2018) defines a triple $( p , q , \varphi )$ associated to a synthesis problem. This formally embeds program synthesis within singular learning theory. We realise this embedding in code by providing an implementation in PyTorch of this propagation of uncertainty through a UTM. Using the No-U-Turn variant of MCMC (Hoffman & Gelman, 2014) we can approximate the Bayesian posterior of any program synthesis problem (of course in practice we are limited by computational constraints in doing so). + +• We explain how the real log canonical threshold (a geometric invariant) is related to Kolmogorov complexity (Section 3). • We give a simple example (Appendix C) in which $W _ { 0 }$ contains the set of classical solutions as a proper subset and every point of $W _ { 0 }$ is a degenerate critical point of $K$ . For two simple synthesis problems detectA and parityCheck we demonstrate all of the above, using MCMC to approximate the Bayesian posterior and theorems from Watanabe (2013) to estimate the RLCT (Section 5). We discuss how $W _ { 0 }$ is an extended object and how the RLCT relates to the local dimension of $W _ { 0 }$ near a classical solution. + +# RELATED WORK + +The idea of synthesising Turing machines can be traced back to the work of Solomonoff on inductive inference (Solomonoff, 1964). A more explicit form of the problem was given in Biermann (1972) who proposed an algorithmic method. Machine learning based approaches appear in Schmidhuber (1997) and Hutter (2004), which pay particular attention to model complexity, and Gaunt et al. (2016) and Freer et al. (2014), the latter using the notion of “universal probabilistic Turing machine” (De Leeuw et al., 1956). A different probabilistic extension of a universal Turing machine was introduced in Clift & Murfet (2018) via linear logic. Studies of the singular geometry of learning models go back to Amari et al. (2003) and notably, the extensive work of Watanabe (2007; 2009). + +# 2 TURING MACHINE SYNTHESIS AS SINGULAR LEARNING + +All known approaches to program synthesis can be formulated in terms of a singular learning problem. Singular learning theory is the extension of statistical learning theory to account for the fact that the set of learned parameters $W _ { 0 }$ has the structure of an analytic space as opposed to an analytic manifold (Watanabe, 2007; 2009). It is organised around triples $( p , q , \varphi )$ consisting of a class of models $\{ p ( y | x , w ) : w \in W \}$ , a true distribution $q ( y | x )$ and a prior $\varphi$ on $W$ . + +In our approach we fix a Universal Turing Machine (UTM), denoted $\mathcal { U }$ , with a description tape (which specifies the code of the Turing machine to be executed), a work tape (simulating the tape of that Turing machine during its operation) and a state tape (simulating the state of that Turing machine). The general statistical learning problem that can be formulated using $\mathcal { U }$ is the following: given some initial string $x$ on the work tape, predict the state of the simulated machine and the contents of the work tape after some specified number of steps (Clift & Murfet, 2018, $\ S 7 . 1 )$ . For simplicity, in this paper we consider models that only predict the final state; the necessary modifications in the general case are routine. We also assume that $W$ parametrises Turing machines whose tape alphabet $\Sigma$ and set of states $Q$ have been encoded by individual symbols in the tape alphabet of $\mathcal { U }$ . Hence $\mathcal { U }$ is actually what we call a pseudo-UTM (see Appendix E). Again, treating the general case is routine and for the present purposes only introduces uninteresting complexity. + +Let $\Sigma$ denote the tape alphabet of the simulated machine, $Q$ the set of states and let $L , S , R$ stand for left, stay and right, the possible motions of the Turing machine head. We assume that $| Q | > 1$ since otherwise the synthesis problem is trivial. The set of ordinary codes $W ^ { c o d e }$ for a Turing machine sits inside a compact space of probability distributions $W$ over codes + +$$ +W ^ { c o d e } : = \prod _ { \sigma , q } \Sigma \times Q \times \{ L , S , R \} \subseteq \prod _ { \sigma , q } \Delta \Sigma \times \Delta Q \times \Delta \{ L , S , R \} = : W +$$ + +where $\Delta X$ denotes the set of probability distributions over a set $X$ , see (8), and the product is over pairs $( \sigma , q ) \in \Sigma \times Q$ .1 For example the point $\{ ( \sigma ^ { \prime } , q ^ { \prime } , d ) \} _ { \sigma , q } \in W ^ { c o d e }$ encodes the machine which when it reads $\sigma$ under the head in state $q$ writes $\sigma ^ { \prime }$ , transitions into state $q ^ { \prime }$ and moves in direction $d$ . Given $w \in W ^ { c o d e }$ let $\operatorname { s t e p } ^ { t } ( x , w ) \in Q$ denote the contents of the state tape of $\mathcal { U }$ after $t$ timesteps (of the simulated machine) when the work tape is initialised with $x$ and the description tape with $w$ . + +There is a principled extension of this operation of $\mathcal { U }$ to a smooth function + +$$ +\Delta \operatorname { s t e p } ^ { t } : { \Sigma } ^ { * } \times W \longrightarrow \Delta Q +$$ + +which propagates uncertainty about the symbols on the description tape to uncertainty about the final state and we refer to this extension as the smooth relaxation of $\mathcal { U }$ . The details are given in Appendix F but at an informal level the idea behind the relaxation is easy to understand: to sample from $\Delta \operatorname { s t e p } ^ { t } ( x , w )$ we run $\mathcal { U }$ to simulate $t$ timesteps in such a way that whenever the UTM needs to “look at” an entry on the description tape we sample from the corresponding distribution specified by $w$ .2 The significance of the particular smooth relaxation that we use is that its derivatives have a logical interpretation (Clift & Murfet, 2018, $\ S 7 . 1 \ r _ { . }$ ). + +The class of models that we consider is + +$$ +p ( y | x , w ) = \Delta \mathrm { s t e p } ^ { t } ( x , w ) +$$ + +where $t$ is fixed for simplicity in this paper. More generally we could also view $x$ as consisting of a sequence and a timeout, as is done in (Clift & Murfet, 2018, $\ S 7 . 1 )$ . The construction of this model is summarised in Figure 1. + +![](images/4a6eaec4d0956e25ce7728bf29fb6ddeee2cacf5f806fd63f0d36ae211bc6fec.jpg) +Figure 1: The state of $\mathcal { U }$ is represented by the state of the work tape, state tape and description (code) tape. The work tape is initialised with a sequence $x \in \Sigma ^ { * }$ , the code tape with $w \in W$ and the state tape with some standard initial state, the smooth relaxation $\Delta$ step of the pseudo-UTM is run for $t$ steps and the final probability distribution over states is $y$ . + +Definition 2.1 (Synthesis problem). A synthesis problem for $\mathcal { U }$ consists of a probability distribution $q ( x , y )$ over $\Sigma ^ { * } \times Q$ . We say that the synthesis problem is deterministic if there is $f : \Sigma ^ { * } \longrightarrow Q$ such that $q ( y = f ( x ) | x ) = 1$ for all $x \in \Sigma ^ { * }$ . + +Definition 2.2. The triple $( p , q , \varphi )$ associated to a synthesis problem is the model $p$ of (5) together with the true distribution $q$ and uniform prior $\varphi$ on the parameter space $W$ . The Kullback-Leibler function $K ( w )$ of the synthesis problem is defined by (1) and a solution to the synthesis problem is a point of $W _ { 0 }$ . A classical solution is a point of $W _ { 0 } \cap W ^ { c o d e }$ . + +As $\Delta \mathrm { s t e p } ^ { t }$ is a polynomial function, $K$ is analytic and so $W _ { 0 }$ is a semi-analytic space (it is cut out of the semi-analytic space $W$ by the vanishing of $K$ ). If the synthesis problem is deterministic and $q ( x )$ is uniform on some finite subset of $\Sigma ^ { * }$ then $W _ { 0 }$ is semi-algebraic (it is cut out of $W$ by polynomial equations) and all solutions lie at the boundary of the parameter space $W$ (Appendix D). However in general $W _ { 0 }$ is only semi-analytic and intersects the interior of $W$ (Example C.2). We assume that ${ \bar { q } } ( y | x )$ is realisable that is, there exists $w _ { 0 } \in W$ with $q ( y | x ) = p ( y | x , w _ { 0 } )$ . + +A triple $( p , q , \varphi )$ is regular if the model is identifiable, ie. for all inputs $x \in \mathbb { R } ^ { n }$ , the map sending $w$ to the conditional probability distribution $p ( \boldsymbol { y } | \boldsymbol { x } , \boldsymbol { w } )$ is one-to-one, and the Fisher information matrix is non-degenerate. Otherwise, the learning machine is strictly singular (Watanabe, 2009, $\ S 1 . 2 . 1 $ . Triples arising from synthesis problems are typically singular: in Example 2.5 below we show an explicit example where multiple parameters $w$ determine the same model, and in Example C.2 we give an example where the Hessian of $K$ is degenerate everywhere on $W _ { 0 }$ (Watanabe, 2009, §1.1.3). + +Remark 2.3. Non-deterministic synthesis problems arise naturally in various contexts, for example in the fitting of algorithms to the behaviour of deep reinforcement learning agents. Suppose an agent is acting in an environment with starting states encoded by $x \in \Sigma ^ { * }$ and possible episode end states by $y \in Q$ . Even if the optimal policy is known to determine a computable function $\Sigma ^ { * } \longrightarrow Q$ the statistics of the observed behaviour after finite training time will only provide a function $\Sigma ^ { * } \longrightarrow \Delta Q$ and if we wish to fit algorithms to behaviour it makes sense to deal with this uncertainty directly. + +Definition 2.4. Let $( p , q , \varphi )$ be the triple associated to a synthesis problem. The Real Log Canonical Threshold (RLCT) $\lambda$ of the synthesis problem is defined so that $- \lambda$ is the largest pole of the meromorphic extension (Atiyah, 1970) of the zeta function $\begin{array} { r } { \zeta ( z ) = \int K ( w ) ^ { z } \varphi ( w ) \hat { d w } } \end{array}$ . + +The more singular the analytic space $W _ { 0 }$ of solutions is, the smaller the RLCT. One way to think of the RLCT is as a count of the effective number of parameters near $W _ { 0 }$ (Murfet et al., 2020, $\ S 4$ ). In Section 3 we relate the RLCT to Kolmogorov complexity and in Section 5 we estimate the RLCT of the synthesis problem detectA given below, using the method explained in Appendix A. + +Example 2.5 (detectA). The deterministic synthesis problem detectA has $\Sigma = \{ \boxed \} , A , B \}$ , $Q = \{ { \mathrm { r e j e c t } } , { \mathrm { a c c e p t } } \}$ and $q ( y | x )$ is determined by the function taking in a string $x$ of $A$ ’s and $B$ ’s and returning the state accept if the string contains an $A$ and state reject otherwise. The conditional true distribution $q ( y | x )$ is realisable because this function is computed by a Turing machine. + +Two solutions are shown in Figure 2. On the left is a parameter $w _ { l } \in \ b { W } _ { 0 } \setminus \ b { W } ^ { c o d e }$ and on the right is $w _ { r } \in W _ { 0 } \cap W ^ { c o d e }$ . Varying the distributions in $w _ { l }$ that have nonzero entropy we obtain a submanifold $V \subseteq W _ { 0 }$ containing $w _ { l }$ of dimension 14. This leads by (Watanabe, 2009, Remark 7.3) to a bound on the RLCT of $\lambda \le \frac { 1 } { 2 } ( 3 0 - 1 4 ) = 8$ which is consistent with the experimental results in Table 1. This highlights that solutions need not lie at vertices of the probability simplex, and $W _ { 0 }$ may contain a high-dimensional submanifold around a given classical solution. + +![](images/94e2cb3be00f1414e5f172ce0b09cdea4b00fe6fbc48f36b360a03838c63cdbb.jpg) +Figure 2: Visualisation of two solutions for the synthesis problem detectA . + +# 2.1 THE SYNTHESIS PROCESS + +Synthesis is a problem because we do not assume that the true distribution is known: for example, if $q { \dot { ( } } y | x )$ is deterministic and the associated function is $f : \Sigma ^ { * } \longrightarrow Q$ , we assume that some example pairs $( x , f ( x ) )$ are known but no general algorithm for computing $f$ is known (if it were, synthesis would have already been performed). In practice synthesis starts with a sample $D _ { n } = \{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n }$ from $q ( x , y )$ with associated empirical Kullback-Leibler distance + +$$ +K _ { n } ( w ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \log \frac { q ( y _ { i } | x _ { i } ) } { p ( y _ { i } | x _ { i } , w ) } . +$$ + +If the synthesis problem is deterministic and $u \in W ^ { c o d e }$ then $K _ { n } ( u ) = 0$ if and only if $u$ explains the data in the sense that $\operatorname { s t e p } ^ { t } ( x _ { i } , u ) = y _ { i }$ for $1 \leq i \leq n$ . We now review two natural ways of finding such solutions in the context of machine learning. + +Synthesis by stochastic gradient descent (SGD). The first approach is to view the process of program synthesis as stochastic gradient descent for the function $K : W \longrightarrow \mathbb { R }$ . We view $D _ { n }$ as a large training set and further sample subsets $D _ { m }$ with $m \ll n$ and compute $\nabla K _ { m }$ to take gradient descent steps $w _ { i + 1 } = w _ { i } - \eta \nabla K _ { m } ( w _ { i } )$ for some learning rate $\eta$ . Stochastic gradient descent has the advantage (in principle) of scaling to high-dimensional parameter spaces $W$ , but in practice it is challenging to use gradient descent to find points of $W _ { 0 }$ (Gaunt et al., 2016). + +Synthesis by sampling. The second approach is to consider the Bayesian posterior associated to the synthesis problem, which can be viewed as an update on the prior distribution $\varphi$ after seeing $D _ { n }$ + +$$ +p ( w | D _ { n } ) = { \frac { p ( D _ { n } | w ) p ( w ) } { p ( D _ { n } ) } } = { \frac { 1 } { Z _ { n } } } \varphi ( w ) \prod _ { i = 1 } ^ { n } p ( y _ { i } | x _ { i } , w ) = { \frac { 1 } { Z _ { n } ^ { 0 } } } \exp \{ - n K _ { n } ( w ) + \log \varphi ( w ) \} +$$ + +where $\begin{array} { r } { Z _ { n } ^ { 0 } = \int \varphi ( w ) \exp ( - n K _ { n } ( w ) ) d w } \end{array}$ . If $n$ is large the posterior distribution concentrates around solutions $w \in W _ { 0 }$ and so sampling from the posterior will tend to produce machines that are (nearly) solutions. The gold standard sampling is Markov Chain Monte Carlo (MCMC). Scaling MCMC to where $W$ is high-dimensional is a challenging task with many attempts to bridge the gap with SGD (Welling & Teh, 2011; Chen et al., 2014; Ding et al., 2014; Zhang et al., 2020). Nonetheless in simple cases we demonstrate experimentally in Section 5 that machines may be synthesised by using MCMC to sample from the posterior. + +# 3 COMPLEXITY OF PROGRAMS + +Every Turing machine is the solution of a deterministic synthesis problem, so Section 2 associates to any Turing machine a singularity of a semi-analytic space $W _ { 0 }$ . To indicate that this connection is not vacuous, we sketch how the complexity of a program is related to the real log canonical threshold of a singularity. A more detailed discussion will appear elsewhere. + +Let $q ( x , y )$ be a deterministic synthesis problem for $\mathcal { U }$ which only involves input sequences in some restricted alphabet $\Sigma _ { i n p u t }$ , that is, $q ( x ) \bar { = } 0$ if $x \notin ( \Sigma _ { i n p u t } ) ^ { * }$ . Let $D _ { n }$ be sampled from $q ( x , y )$ and let $u , v \in W ^ { c o d e } \cap W _ { 0 }$ be two explanations for the sample in the sense that $K _ { n } ( u ) = K _ { n } ( v ) = 0$ . Which explanation for the data should we prefer? The classical answer based on Occam’s razor (Solomonoff, 1964) is that we should prefer the shorter program, that is, the one using the fewest states and symbols. + +Set $N = | \Sigma |$ and $M = | Q |$ . Any Turing machine $T$ using $N ^ { \prime } \leq N$ symbols and $M ^ { \prime } \leq M$ states has a code for $\mathcal { U }$ of length $c M ^ { \prime } N ^ { \prime }$ where $c$ is a constant. We assume that $\Sigma _ { i n p u t }$ is included in the tape alphabet of $T$ so that $N ^ { \prime } \geq | \Sigma _ { i n p u t } |$ and define the Kolmogorov complexity of $q$ with respect to $\mathcal { U }$ to be the infimum ${ \mathfrak { c } } ( q )$ of $M ^ { \prime } N ^ { \prime }$ over Turing machines $T$ that give classical solutions for $q$ . + +Let $\lambda$ be the RLCT of the triple $( p , q , \varphi )$ associated to the synthesis problem (Definition 2.4). + +Theorem 3.1. $\begin{array} { r } { \lambda \le \frac { 1 } { 2 } ( M + N ) \mathfrak { c } ( q ) } \end{array}$ . + +Proof. Let $u \in W ^ { c o d e } \cap W _ { 0 }$ be the code of a Turing machine realising the infimum in the definition of the Kolmogorov complexity and suppose that this machine only uses symbols in $\Sigma ^ { \prime }$ and states in $Q ^ { \prime }$ with $N ^ { \prime } = | \Sigma ^ { \prime } |$ and $\bar { M } ^ { \prime } = | Q ^ { \prime } |$ . The time evolution of the staged pseudo-UTM $\mathcal { U }$ simulating $u$ on $x \in \Sigma _ { i n p u t } ^ { * }$ is independent of the entries on the description tape that belong to tuples of the form $( \sigma , q , ? , ? , ? )$ with $( \sigma , q ) \notin \Sigma ^ { \prime } \times Q ^ { \prime }$ . Let $V \subseteq W$ be the submanifold of points which agree with $u$ on all tuples with $( \sigma , q ) \in \Sigma ^ { \prime } \times Q ^ { \prime }$ and are otherwise free. Then $u \in V \subseteq W _ { 0 }$ and $\operatorname { c o d i m } ( V ) =$ $M ^ { \prime } N ^ { \prime } ( \bar { M } + N )$ and by (Watanabe, 2009, Theorem 7.3) we have $\begin{array} { r } { \lambda \le \frac 1 2 \bmod { \mathrm { i m } } ( V ) } \end{array}$ . □ + +Remark 3.2. The Kolmogorov complexity depends only on the number of symbols and states used. The RLCT is a more refined invariant since it also depends on how each symbol and state is used (Clift & Murfet, 2018, Remark 7.8) as this affects the polynomials defining $W _ { 0 }$ (see Appendix D). + +# 4 PRACTICAL IMPLICATIONS + +Using singular learning theory we have explained how programs to be synthesised are singularities of analytic functions, and how the Kolmogorov complexity of a program bounds the RLCT of the associated singularity. We now sketch some practical insights that follow from this point of view. + +Synthesis minimises the free energy: the sampling-based approach to synthesis (Section 2.1) aims to approximate, via MCMC, sampling from the Bayesian posterior for the triple $( p , q , \varphi )$ associated to a synthesis problem. To understand the behaviour of these Markov chains we follow the asymptotic analysis of (Watanabe, 2009, Section 7.6). If we cover $W$ by small closed balls $V _ { \alpha }$ around points $w _ { \alpha }$ then we can compute the probability that a sample comes from $V _ { \alpha }$ by + +$$ +p _ { \alpha } = \frac { 1 } { Z _ { 0 } } \int _ { V _ { \alpha } } e ^ { - n K _ { n } ( w ) } \varphi ( w ) d w +$$ + +and if $n$ is sufficiently large this is proportional to $e ^ { - f _ { \alpha } }$ where the quantity + +$$ +f _ { \alpha } = K _ { \alpha } n + \lambda _ { \alpha } \log ( n ) +$$ + +is called the free energy. Here $K _ { \alpha }$ is the smallest value of the Kullback-Leibler divergence $K$ on $V _ { \alpha }$ and $\lambda _ { \alpha }$ is the RLCT of the set $W _ { K _ { \alpha } } \cap V _ { \alpha }$ where $W _ { c } = \{ w \in W | K ( w ) = c \}$ is a level set of $K$ . The Markov chains used to generate approximate samples from the posterior are attempting to minimise the free energy, which involves a tradeoff between the energy $K _ { \alpha } n$ and the entropy $\lambda _ { \alpha } \log ( n )$ . + +Why synthesis gets stuck: the kind of local minimum of the free energy that we want the synthesis process to find are solutions $w _ { \alpha } \in W _ { 0 }$ where $\lambda _ { \alpha }$ is minimal. By Section 3 one may think of these points as the “lowest complexity” solutions. However it is possible that there are other local minima of the free energy. Indeed, there may be local minima where the free energy is lower than the free energy at any solution since at finite $n$ it is possible to tradeoff an increase in $K _ { \alpha }$ against a decrease in the RLCT $\lambda _ { \alpha }$ . In practice, the existence of such “siren minima” of the free energy may manifest itself as regions where the synthesis process gets stuck and fails to converge to a solution. In such a region $\bar { K _ { \alpha } } n + \lambda _ { \alpha } \log ( n ) < \lambda \log ( \bar { n } )$ where $\lambda$ is the RLCT of the synthesis problem. In practice it has been observed that program synthesis by gradient descent often fails for complex problems in the sense that it fails to converge to a solution (Gaunt et al., 2016). While synthesis by SGD and sampling are different, it is a reasonable hypothesis that these siren minima are a significant contributing factor in both cases. + +Can we avoid siren minima? If we let $\lambda _ { c }$ denote the RLCT of the level set $W _ { c }$ then siren minima of the free energy will be impossible at a given value of n and c as long as λc ≥ λ−c nlog(n) . Recall that the more singular $W _ { c }$ is the lower the RLCT, so this lower bound says that the level sets should not become too singular too quickly as $c$ increases. At any given value of $n$ there is a “siren free” region in the range $c \geq { \frac { \lambda \log ( n ) } { n } }$ since the RLCT is non-negative (Figure 3). Thus the learning process will be more reliable the smaller $\frac { \lambda \log ( n ) } { n }$ is. This can arranged either by increasing $n$ (providing more examples) or decreasing $\lambda$ . + +While the RLCT is determined by the synthesis problem, it is possible to change its value by changing the structure of the UTM $\mathcal { U }$ . As we have defined it $\mathcal { U }$ is a “simulation type” UTM, but one could for example add special states such that if a code specifies a transition into that state a series of steps is executed by the UTM (i.e. a subroutine). This amounts to specifying codes in a higher level programming language. Hence one of the practical insights that can be derived from the geometric point of view on program synthesis is that varying this language is a natural way to engineer the singularities of the level sets of $K$ , which according to singular learning theory has direct implications for the learning process. + +![](images/e34800be3128393d0b71b422f4ec9e848e92967fd980b5ceb119d06d14b03a68.jpg) +Figure 3: Level sets above the cutoff cannot contain siren local minima of the free energy. + +# 5 EXPERIMENTS + +We estimate the RLCT for the triples $( p , q , \varphi )$ associated to the synthesis problems detectA (Example 2.5) and parityCheck. Hyperparameters of the various machines are contained in Table 3 of Appendix B. The true distribution $q ( x )$ is defined as follows: we fix a minimum and maximum sequence length $a \leq b$ and to sample $x \sim q ( x )$ we first sample a length $l$ uniformly from $[ a , b ]$ and then uniformly sample $x$ from $\{ A , { \cal B } \} ^ { l }$ . + +We perform MCMC on the weight vector for the model class $\{ p ( y | x , w ) : w \in W \}$ where $w$ is represented in our PyTorch implementation by three tensors of shape $\{ [ L , n _ { i } ] \} _ { 1 \leq i \leq 3 }$ where $L$ is the number of tuples in the description tape of the TM being simulated and $\{ n _ { i } \}$ are the number of symbols, states and directions respectively. A direct simulation of the UTM is used for all experiments to improve computational efficiency (Appendix G). We generate, for each inverse temperature $\beta$ and dataset $D _ { n }$ , a Markov chain via the No-U-turn sampler from Hoffman & Gelman (2014). We use the standard uniform distribution as our prior $\varphi$ . + +Table 1: RLCT estimates for detectA. + +
Max-lengthTemperatureRLCTStdR squared
7log(500)8.0892053.5247190.965384
7log(1000)6.5333622.0942780.966856
8log(500)4.6018001.1563250.974569
8log(1000)4.4316831.0690200.967847
9log(500)5.3025982.4156470.973016
9log(1000)4.0273241.8668020.958805
10log(500)3.2249101.1696990.963358
10log(1000)3.4336240.9999670.949972
+ +For the problem detectA given in Example 2.5 the dimension of parameter space is dim $W = 3 0$ . We use generalized least squares to fit the RLCT $\lambda$ (with goodness-of-fit measured by $R ^ { 2 }$ ), the algorithm of which is given in Appendix A. Our results are displayed in Table 1 and Figure 4. Our purpose in these experiments is not to provide high accuracy estimates of the RLCT, as these would require much longer Markov chains. Instead we demonstrate how rough estimates consistent with the theory can be obtained at low computational cost. If this model were regular the RLCT would be $\dim W / 2 = 1 5$ . + +![](images/9ace473833bc61d56cd4eaf090be6f773656e1bf4f188e396d12fd45dcd66217.jpg) +Figure 4: Plot of RLCT estimates for detectA. Shaded region shows one standard deviation. + +The deterministic synthesis problem parityCheck has + +$$ +\begin{array} { l } { \Sigma = \{ \Pi , A , B , X \} } \\ { Q = \{ \mathrm { r e j e c t , a c c e p t , g e t N e x t A B , g e t N e x t A , g e t N e x t B , g o t o S t a r t } \} . } \end{array} +$$ + +The distribution $q ( x )$ is as discussed in Section 5 and $q ( y | x )$ is determined by the function taking in a string of $A$ ’s and $B$ ’s, and terminating in state accept if the string contains the same number of $A$ ’s as $B$ ’s, and terminating in state reject otherwise. The string is assumed to contain no blank symbols. The true distribution is realisable because there is a Turing machine using $\Sigma$ and $Q$ which computes this function: the machine works by repeatedly overwriting pairs consisting of a single $A$ and $B$ with $X$ ’s; if there are any $A$ ’s without a matching $B$ left over (or vice versa), we reject, otherwise we accept. + +In more detail, the starting state getNextAB moves right on the tape until the first $A$ or $B$ is found, and overwrites it with an $X$ . If it’s an $A$ (resp. $B$ ) we enter state getNextB (resp. getNextA). If no $A$ or $B$ is found, we enter the state accept. The state getNextA (resp. getNextB) moves right until an $A$ (resp. $B$ ) is found, overwrites it with an $X$ and enters state gotoStart which moves left until a blank symbol is found (resetting the machine to the left end of the tape). If no $A$ ’s (resp. $B$ ’s) were left on the tape, we enter state reject. The dimension of the parameter space is $\dim W = 2 4 0$ . If this model were regular, the RLCT would be $\dim W / 2 = 1 2 { \bar { 0 } }$ . Our RLCT estimates are contained in Table 2. + +Table 2: RLCT estimates for parityCheck. + +
Max-lengthTemperatureRLCTStdR squared
5log(300)4.4117320.2524580.969500
6log(300)4.0056670.3658550.971619
7log(300)3.8876790.2763370.973716
+ +# 6 DISCUSSION + +We have developed a theoretical framework in which all programs can in principle be learnt from input-output examples via an existing optimisation procedure. This is done by associating to each program a smooth relaxation which, based on Clift & Murfet (2018), can be argued to be more canonical than existing approaches. This realization has important implications for the building of intelligent systems. + +In approaches to program synthesis based on gradient descent there is a tendency to think of solutions to the synthesis problem as isolated critical points of the loss function $K$ , but this is a false intuition based on regular models. Since neural networks, Bayesian networks, smooth relaxations of UTMs and all other extant approaches to smooth program synthesis are strictly singular models (the map from parameters to functions is not injective) the set $W _ { 0 }$ of parameters $w$ with $K ( w ) = 0$ is a complex extended object, whose geometry is shown by Watanabe’s singular learning theory to be deeply related to the learning process. We have examined this geometry in several specific examples and shown how to think about complexity of programs from a geometric perspective. It is our hope that algebraic geometry can assist in developing the next generation of synthesis machines. + +# REFERENCES + +Shun-ichi Amari, Tomoko Ozeki, and Hyeyoung Park. Learning and inference in hierarchical models with singularities. Systems and Computers in Japan, 34(7):34–42, 2003. + +Michael F Atiyah. Resolution of singularities and division of distributions. Communications on Pure and Applied Mathematics, 23(2):145–150, 1970. + +Alan W Biermann. On the inference of Turing machines from sample computations. Artificial Intelligence, 3:181–198, 1972. + +Rudy R Bunel, Alban Desmaison, Pawan K Mudigonda, Pushmeet Kohli, and Philip Torr. Adaptive neural compilation. In Advances in Neural Information Processing Systems, pp. 1444–1452, 2016. + +Tianqi Chen, Emily Fox, and Carlos Guestrin. Stochastic gradient Hamiltonian Monte Carlo. In International Conference on Machine Learning, pp. 1683–1691, 2014. + +Xinyun Chen, Chang Liu, and Dawn Song. Execution-guided neural program synthesis. In International Conference on Learning Representations, 2018. + +James Clift and Daniel Murfet. Derivatives of Turing machines in linear logic. arXiv preprint arXiv:1805.11813, 2018. + +Karel De Leeuw, Edward F Moore, Claude E Shannon, and Norman Shapiro. Computability by probabilistic machines. Automata studies, 34:183–198, 1956. + +Nan Ding, Youhan Fang, Ryan Babbush, Changyou Chen, Robert D Skeel, and Hartmut Neven. Bayesian sampling using stochastic gradient thermostats. In Advances in Neural Information Processing Systems, pp. 3203–3211, 2014. + +Richard Evans and Edward Grefenstette. Learning explanatory rules from noisy data. Journal of Artificial Intelligence Research, 61:1–64, 2018. + +Cameron E Freer, Daniel M Roy, and Joshua B Tenenbaum. Towards common-sense reasoning via conditional simulation: legacies of Turing in artificial intelligence. Turing’s Legacy, 42:195–252, 2014. + +Alexander L Gaunt, Marc Brockschmidt, Rishabh Singh, Nate Kushman, Pushmeet Kohli, Jonathan Taylor, and Daniel Tarlow. Terpret: A probabilistic programming language for program induction. arXiv preprint arXiv:1608.04428, 2016. + +Phillip Griffith and Joseph Harris. Principles of Algebraic Geometry. Wiley-Interscience, 1978. + +Sumit Gulwani, Oleksandr Polozov, and Rishabh Singh. Program synthesis. Foundations and Trends in Programming Languages, 4(1-2):1–119, 2017. + +Heisuke Hironaka. Resolution of singularities of an algebraic variety over a field of characteristic zero: I. Annals of Mathematics, 79(1):109–203, 1964. + +Matthew D Hoffman and Andrew Gelman. The No-U-Turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. J. Mach. Learn. Res., 15(1):1593–1623, 2014. + +Marcus Hutter. Universal artificial intelligence: Sequential decisions based on algorithmic probability. Springer Science & Business Media, 2004. + +Łukasz Kaiser and Ilya Sutskever. Neural GPUs learn algorithms. In International Conference on Learning Representations, 2016. + +Daniel Murfet, Susan Wei, Mingming Gong, Hui Li, Jesse Gell-Redman, and Thomas Quella. Deep learning is singular, and that’s good. arXiv preprint arXiv:2010.11560, 2020. + +Arvind Neelakantan, Quoc V. Le, and Ilya Sutskever. Neural programmer: Inducing latent programs with gradient descent. In International Conference on Learning Representations, ICLR 2016, 2016. + +Jurgen Schmidhuber. Discovering neural nets with low Kolmogorov complexity and high general- ¨ ization capability. Neural Networks, 10(5):857–873, 1997. + +Ray J Solomonoff. A formal theory of inductive inference. Part I. Information and control, 7(1): 1–22, 1964. + +Alan Turing. Intelligent machinery. NPL Mathematics Division, 1948. + +Sumio Watanabe. Almost all learning machines are singular. In 2007 IEEE Symposium on Foundations of Computational Intelligence, pp. 383–388. IEEE, 2007. + +Sumio Watanabe. Algebraic Geometry and Statistical Learning Theory, volume 25. Cambridge University Press, 2009. + +Sumio Watanabe. A widely applicable Bayesian information criterion. Journal of Machine Learning Research, 14:867–897, 2013. + +Max Welling and Yee W Teh. Bayesian learning via stochastic gradient Langevin dynamics. In Proceedings of the 28th International Conference on Machine Learning, pp. 681–688, 2011. + +Ruqi Zhang, Chunyuan Li, Jianyi Zhang, Changyou Chen, and Andrew Gordon Wilson. Cyclical stochastic gradient MCMC for Bayesian deep learning. In International Conference on Learning Representations, 2020. + +# APPENDIX + +# A ALGORITHM FOR ESTIMATING RLCTS + +Given a sample $D _ { n } = \{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n }$ from $q ( x , y )$ let $\begin{array} { r } { L _ { n } ( w ) : = - \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \log p ( y _ { i } | x _ { i } , w ) } \end{array}$ be the negative log likelihood. We would like to estimate + +$$ +\mathbb { E } _ { w } ^ { \beta } [ n L _ { n } ( w ) ] : = \frac { 1 } { Z _ { n } ^ { \beta } } \int n L _ { n } ( w ) \varphi ( w ) \prod _ { i = 1 } ^ { n } p ( y _ { i } | x _ { i } , w ) ^ { \beta } d w +$$ + +where $\begin{array} { r } { Z _ { n } ^ { \beta } = \int \varphi ( w ) \prod _ { i = 1 } ^ { n } p ( y _ { i } | x _ { i } , w ) ^ { \beta } d w } \end{array}$ for some inverse temperature $\beta$ . If $\begin{array} { r } { \beta = \frac { \beta _ { 0 } } { \log n } } \end{array}$ for some constant $\beta _ { 0 }$ , then by Theorem 4 of Watanabe (2013), + +$$ +\mathbb { E } _ { w } ^ { \beta } [ n L _ { n } ( w ) ] = n L _ { n } ( w _ { 0 } ) + \frac { \lambda \log n } { \beta _ { 0 } } + U _ { n } \sqrt { \frac { \lambda \log n } { 2 \beta _ { 0 } } } + O _ { p } ( 1 ) +$$ + +where $\{ U _ { n } \}$ is a sequence of random variables satisfying $\mathbb { E } [ U _ { n } ] = 0$ and $\lambda$ is the RLCT. In practice, the last two terms often vary negligibly with $1 / \beta$ and so $\mathbb { E } _ { w } ^ { \beta } [ n L _ { n } ( w ) ]$ approximates a linear function of $1 / \beta$ with slope $\lambda$ (Watanabe, 2013, Corollary 3). This is the foundation of the RLCT estimation procedure found in Algorithm 1 which is used in our experiments. + +# Algorithm 1 RLCT estimation + +
Input: range of β's, set of training sets T each of size n, approximate samples {w1,..,WR} from pβ(w|Dn) for each training set Dn and each β
for training set Dn ∈ T do
for β in range of β's do
ples from pβ(w|Dn)
end for
Perform generalised least squares to fit X in Equation (7),call result λ(Dn)
end for
Output: ∑Dn∈T λ(Dn)
+ +Each RLCT estimate $\hat { \lambda } ( \mathcal { D } _ { n } )$ in Algorithm 1 was performed by linear regression on the pairs $\{ ( 1 / \beta _ { i } , \mathbb { E } _ { w } ^ { \beta _ { i } } [ n L _ { n } ( w ) ] ) \} _ { i = 1 } ^ { 5 }$ where the five inverse temperatures $\beta _ { i }$ are centered on the inverse temperature $1 / T$ where $T$ is the temperature reported for each experiment in Table 1 and Table 2. + +From a Bayesian perspective, predictions about outputs $y$ should be made using the predictive distribution + +$$ +p ^ { * } ( y | x , D _ { n } ) = \int p ( y | x , w ) p ( w | D _ { n } ) d w . +$$ + +The Bayesian generalisation error associated to the Bayesian predictor is defined as the KullbackLeibler distance to the true conditional distribution + +$$ +B _ { g } ( n ) : = D _ { K L } ( q \| p ^ { * } ) = \int q ( y | x ) q ( x ) \log \left( { \frac { q ( y | x ) } { p ^ { * } ( y | x ) } } \right) d y d x . +$$ + +If some fundamental conditions are satisfied (Definition 6.1 and Definition 6.3 of Watanabe (2009)), then by Theorem 6.8 of loc.cit., there exists a random variable $B _ { g } ^ { * }$ such that as $n \to \infty$ , $\mathbb { E } [ n B _ { g } ( n ) ]$ converges to $\mathbb { E } [ B _ { g } ^ { * } ]$ . In particular, by Theorem 6.10 of Watanabe (2009), $\mathbb { E } [ B _ { g } ^ { * } ] = \lambda$ . + +# B HYPERPARAMETERS + +The hyperparameters for the various synthesis tasks are contained in Table 3. The number of samples is $R$ in Algorithm 1 and the number of datasets is $| \tau |$ . Samples are taken according to the Dirichlet distribution, a probability distribution over the simplex, which is controlled by the concentration. When the concentration is a constant across all dimensions, as is assumed here, this corresponds to a density which is symmetric about the uniform probability mass function occurring in the centre of the simplex. The value $\alpha = 1 . 0$ corresponds to the uniform distribution over the simplex. Finally, the chain temperature controls the default $\beta$ value, ie. all inverse temperature values are centered around $1 / T$ where $T$ is the chain temperature. + +Table 3: Hyperparameters for Datasets and MCMC. + +
HyperparameterdetectAparityCheck
Dataset size (n)200100
Minimum sequence length (a)41
Maximum sequence length (b)7/8/9/105/6/7
Number of samples (R)20.0002.000
Number of burn-in steps1,000500
Number of datasets (|T|)43
Target accept probability0.80.8
Concentration (α)1.01.0
Chain temperature (T)log(500)/log(1000)log(300)
Number of timesteps (t)1042
+ +# C THE SHIFT MACHINE + +The pseudo-UTM $\mathcal { U }$ is a complicated Turing machine, and the models $p ( \boldsymbol { y } | \boldsymbol { x } , \boldsymbol { w } )$ of Section 2 are therefore not easy to analyse by hand. To illustrate the kind of geometry that appears, we study the simple Turing machine shiftMachine of Clift & Murfet (2018) and formulate an associated statistical learning problem. The tape alphabet is $\Sigma = \{ \boxed { \begin{array} { r l } \end{array} } , A , B , 0 , 1 , 2 \}$ and the input to the machine will be a string of the form $\boxed { 1 } n a _ { 1 } a _ { 2 } a _ { 3 } \boxed { 2 }$ where $n$ is called the counter and $\bar { a _ { i } } \in \{ A , B \}$ . The transition function, given in loc.cit., will move the string of $A$ ’s and $B$ ’s leftwards by $n$ steps and fill the right hand end of the string with $A$ ’s, keeping the string length invariant. For example, if $\square 2 B A B \square$ is the input to $M$ , the output will be $\square 0 B A A \square$ . + +Set $W = \Delta \{ 0 , 2 \} \times \Delta \{ A , B \}$ and view $w = ( h , k ) \in W$ as representing a probability distribution $( 1 - h ) \cdot 0 + h \cdot 2$ for the counter and $( 1 - k ) \cdot B + k \cdot A$ for $a _ { 1 }$ . The model is + +$$ +p \big ( \boldsymbol { y } | \boldsymbol { x } = ( a _ { 2 } , a _ { 3 } ) , \boldsymbol { w } \big ) = ( 1 - h ) ^ { 2 } \boldsymbol { k } \cdot \boldsymbol { A } + ( 1 - h ) ^ { 2 } ( 1 - \boldsymbol { k } ) \cdot \boldsymbol { B } + \sum _ { i = 2 } ^ { 3 } \binom { 2 } { i - 1 } h ^ { i - 1 } ( 1 - h ) ^ { 3 - i } \cdot a _ { i } . +$$ + +This model is derived by propagating uncertainty through shiftMachine in the same way that $p ( \boldsymbol { y } | \boldsymbol { x } , \boldsymbol { w } )$ is derived from $\mathrm { \Delta } \mathrm { { s t e p } } ^ { t }$ in Section 2 by propagating uncertainty through $\mathcal { U }$ . We assume that some distribution $q ( x )$ over $\{ A , B \} ^ { 2 }$ is given. + +Example C.1. Suppose $q ( y | x ) = p ( y | x , w _ { 0 } )$ where $w _ { 0 } = ( 1 , 1 )$ . It is easy to see that + +$$ +K ( w ) = - { \frac { 1 } { 4 } } \sum _ { a _ { 2 } , a _ { 3 } } \log p \big ( y = a _ { 3 } | x = ( a _ { 2 } , a _ { 3 } ) , w \big ) = - { \frac { 1 } { 2 } } \log [ g ( h , k ) ] +$$ + +where $g ( h , k ) = \left( ( 1 - h ) ^ { 2 } k + h ^ { 2 } \right) \left( ( 1 - h ) ^ { 2 } ( 1 - k ) + h ^ { 2 } \right)$ is a polynomial in $w$ . Hence + +$$ +W _ { 0 } = \{ ( h , k ) \in W : g ( h , k ) = 1 \} = \mathbb { V } ( g - 1 ) \cap [ 0 , 1 ] ^ { 2 } +$$ + +is a semi-algebraic variety, that is, it is defined by polynomial equations and inequalities. Here $\mathbb { V } ( h )$ denotes the vanishing locus of a function $h$ . + +Example C.2. Suppose $q ( A B ) = 1$ and $\begin{array} { r } { q ( y | x = A B ) = \frac { 1 } { 2 } A + \frac { 1 } { 2 } B } \end{array}$ . Then the Kullback-Leibler divergence is $\begin{array} { r } { K ( h , k ) = - \frac { 1 } { 2 } \log ( 4 f ( 1 - f ) ) } \end{array}$ where $f = ( 1 - h ) ^ { 2 } k + 2 h ( 1 - h )$ . Hence $\nabla K =$ $\begin{array} { r } { ( f - \frac { 1 } { 2 } ) \frac { 1 } { f ( 1 - f ) } \nabla f . } \end{array}$ . Note that $f$ has no critical points, and so $\nabla K = 0$ at $( h , k ) \in ( 0 , 1 ) ^ { 2 }$ if and only if $\begin{array} { r } { f ( h , k ) = \frac { 1 } { 2 } } \end{array}$ . Since $K$ is non-negative, any $w \in W _ { 0 }$ satisfies $\nabla K ( w ) = 0$ and so + +$$ +W _ { 0 } = [ 0 , 1 ] ^ { 2 } \cap \mathbb { V } ( 4 f ( 1 - f ) - 1 ) = [ 0 , 1 ] ^ { 2 } \cap \mathbb { V } ( f - \frac { 1 } { 2 } ) +$$ + +is semi-algebraic. Note that the curve $\begin{array} { r } { f = \frac { 1 } { 2 } } \end{array}$ is regular while the curve $4 f ( 1 - f ) = 1$ is singular and it is the geometry of the singular curve that is related to the behaviour of $K$ . This curve is shown in Figure 5. It is straightforward to check that the determinant of the Hessian of $K$ is identically zero on $W _ { 0 }$ , so that every point on $W _ { 0 }$ is a degenerate critical point of $K$ . + +![](images/6e526ca2a8223c4c58975329dac1b2911fc337af16f18add2a573fa4cdb23bdf.jpg) +Figure 5: Values of $K ( h , k )$ on $[ 0 , 1 ] ^ { 2 }$ are shown by colour, ranging from blue (zero) to red (0.01). The singular analytic space $K = 0$ (white) and the regular analytic level set $K = 0 . 0 0 1$ (black). + +# D GENERAL SOLUTION FOR DETERMINISTIC SYNTHESIS PROBLEMS + +In this section we consider the case of a deterministic synthesis problem $q ( x , y )$ which is finitely supported in the sense that there exists a finite set $\mathcal { X } \subseteq \Sigma ^ { * }$ such that $q ( x ) = c$ for all $x \in \mathcal { X }$ and $q ( x ) = 0$ for all $x \notin \mathcal { X }$ . We first need to discuss the coordinates on the parameter space $W$ of (3). To specify a point on $W$ is to specify for each pair $( \sigma , q ) \in \Sigma \times Q$ (that is, for each tuple on the description tape) a triple of probability distributions + +$$ +\begin{array} { r l } & { \displaystyle \sum _ { \sigma ^ { \prime } \in Q } x _ { \sigma ^ { \prime } } ^ { \sigma , q } \cdot \sigma ^ { \prime } \in \Delta \Sigma , } \\ & { \displaystyle \sum _ { q ^ { \prime } \in Q } y _ { q ^ { \prime } } ^ { \sigma , q } \cdot q ^ { \prime } \in \Delta Q , } \\ & { \displaystyle \sum _ { d \in \{ L , S , R \} } z _ { d } ^ { \sigma , q } \cdot d \in \Delta \{ L , S , R \} . } \end{array} +$$ + +The space $W$ of distributions is therefore contained in the affine space with coordinate ring + +$$ +R _ { W } = \mathbb { R } \big [ \big \{ x _ { \sigma ^ { \prime } } ^ { \sigma , q } \big \} _ { \sigma , q , \sigma ^ { \prime } } , \big \{ y _ { q ^ { \prime } } ^ { \sigma , q } \big \} _ { \sigma , q , q ^ { \prime } } , \big \{ z _ { d } ^ { \sigma , q } \big \} _ { \sigma , q , d } \big ] . +$$ + +The function $F ^ { x } = \Delta \mathrm { s t e p } ^ { t } ( x , - ) : W \longrightarrow \Delta Q$ is polynomial (Clift & Murfet, 2018, Proposition 4.2) and we denote for $s \in Q$ by $F _ { s } ^ { x } \in R _ { W }$ the polynomial computing the associated component of the function $F ^ { x }$ . Let $\partial W$ denote the boundary of the manifold with corners $W$ , that is, the set of all points on $W$ where at least one of the coordinate functions given above vanishes + +$$ +\partial W = \mathbb { V } \big ( \prod _ { \sigma , q } \Big [ \prod _ { \sigma ^ { \prime } \in Q } x _ { \sigma ^ { \prime } } ^ { \sigma , q } \prod _ { q ^ { \prime } \in Q } y _ { q ^ { \prime } } ^ { \sigma , q } \prod _ { \substack { d \in \{ L , S , R \} } } z _ { d } ^ { \sigma , q } \Big ] \big ) +$$ + +where $\mathbb { V } ( h )$ denotes the vanishing locus of $h$ . + +Lemma D.1. $W _ { 0 } \neq W$ + +Proof. Choose $x \in \mathcal { X }$ with $q ( x ) > 0$ and let $y$ be such that $q ( y | x ) = 1$ . Let $w \in W ^ { c o d e }$ be the code for the Turing machine which ignores the symbol under the head and current state, transitions to some fixed state $s \neq y$ and stays. Then $w \not \in W _ { 0 }$ . □ + +Lemma D.2. The set $W _ { 0 }$ is semi-algebraic and $W _ { 0 } \subseteq \partial W$ . + +Proof. Given $x \in \Sigma ^ { * }$ with $q ( x ) > 0$ we write $y = y ( x )$ for the unique state with $q ( x , y ) \neq 0$ . In this notation the Kullback-Leibler divergence is + +$$ +K ( w ) = \sum _ { x \in \mathcal { X } } c D _ { K L } ( y | | F ^ { x } ( w ) ) = - c \sum _ { x \in \mathcal { X } } \log F _ { y } ^ { x } ( w ) = - c \log \prod _ { x \in \mathcal { X } } F _ { y } ^ { x } ( w ) . +$$ + +Hence + +$$ +W _ { 0 } = W \cap \bigcap _ { x \in \mathcal { X } } \mathbb { V } ( 1 - F _ { y } ^ { x } ( w ) ) +$$ + +is semi-algebraic. + +Recall that the function $\Delta \mathrm { s t e p } ^ { t }$ is associated to an encoding of the UTM in linear logic by the Sweedler semantics (Clift & Murfet, 2018) and the particular polynomials involved have a form that is determined by the details of that encoding (Clift & Murfet, 2018, Proposition 4.3). From the design of our UTM we obtain positive integers $l _ { \sigma } , m _ { q } , n _ { d }$ for $\sigma \in \Sigma , q \in \bar { Q } , d \in \{ L , S , R \}$ and a function $\pi : \Theta \longrightarrow Q$ where + +$$ +\Theta = \prod _ { \sigma , q } \Sigma ^ { l _ { \sigma } } \times Q ^ { m _ { q } } \times \{ L , S , R \} ^ { n _ { d } } . +$$ + +We represent elements of $\Theta$ by tuples $( \mu , \zeta , \xi ) \in \Theta$ where $\mu ( \sigma , q , i ) \in \Sigma$ for $\sigma \in \Sigma , q \in Q$ and $1 \leq i \leq l _ { \sigma }$ and similarly $\zeta ( \sigma , q , j ) \in Q$ and $\xi ( \sigma , q , k ) \in \{ L , S , R \}$ . The polynomial $F _ { s } ^ { x }$ is + +$$ +F _ { s } ^ { x } = \sum _ { ( \mu , \zeta , \xi ) \in \Theta } \delta ( s = \pi ( \mu , \zeta , \xi ) ) \prod _ { \sigma , q } \Big [ \prod _ { i = 1 } ^ { l _ { \sigma } } x _ { \mu ( \sigma , q , i ) } ^ { \sigma , q } \prod _ { j = 1 } ^ { m _ { q } } y _ { \zeta ( \sigma , q , j ) } ^ { \sigma , q } \prod _ { k = 1 } ^ { n _ { d } } z _ { \xi ( \sigma , q , k ) } ^ { \sigma , q } \Big ] +$$ + +where $\delta$ is a Kronecker delta. With this in hand we may compute + +$$ +\begin{array} { r } { W _ { 0 } = W \cap \displaystyle \bigcap _ { x \in \mathcal { X } } \mathbb { V } ( 1 - F _ { y } ^ { x } ( w ) ) } \\ { = W \cap \displaystyle \bigcap _ { x \in \mathcal { X } } \bigcap _ { s \neq y } \mathbb { V } ( F _ { s } ^ { x } ( w ) ) . } \end{array} +$$ + +But $F _ { s } ^ { x }$ is a polynomial with non-negative integer coefficients, which takes values in $[ 0 , 1 ]$ for $w \in$ $W$ . Hence it vanishes on $w$ if and only if for each triple $\mu , \zeta , \xi$ with $s = \pi ( \mu , \zeta , \xi )$ one or more of the coordinate functions xσ,qµ(σ,q,i), yσ,qζ(σ,q,j), zσ,qξ(σ,q,k) vanishes on $w$ . + +The desired conclusion follows unless for every $x \in \mathcal { X }$ and $( \mu , \zeta , \xi ) \in \Theta$ we have $\pi ( \mu , \zeta , \xi ) = y$ so that $F _ { s } ^ { x } = 0$ for all $s \neq y$ . But in this case case $W _ { 0 } = W$ which contradicts Lemma D.1. □ + +# E STAGED PSEUDO-UTM + +Simulating a Turing machine $M$ with tape alphabet $\Sigma$ and set of states $Q$ on a standard UTM requires the specification of an encoding of $\Sigma$ and $Q$ in the tape alphabet of the UTM. From the point of view of exploring the geometry of program synthesis, this additional complexity is uninteresting and so here we consider a staged pseudo-UTM whose alphabet is + +$$ +\Sigma _ { \mathrm { U T M } } = \Sigma \cup Q \cup \{ L , R , S \} \cup \{ X , \sqsubseteq \} +$$ + +where the union is disjoint where $\boxed { \begin{array} { r l } \end{array} }$ is the blank symbol (which is distinct from the blank symbol of $M$ ). Such a machine is capable of simulating any machine with tape alphabet $\Sigma$ and set of states $Q$ but cannot simulate arbitrary machines and is not a UTM in the standard sense. The adjective staged refers to the design of the UTM, which we now explain. The set of states is + +$$ +\begin{array} { r } { Q _ { \mathrm { U T M } } = \{ \mathrm { c o m p S y m b o l , c o m p S t a t e , c o p y S y m b o l , c o p y S t a t e , c o p y } \mathrm { ~ } \forall \mathrm { t o p y ~ } \mathrm { ~ c o p y ~ } \mathrm { ~ c o p y ~ } \mathrm { ~ c o p y ~ } \mathrm { ~ c o p y ~ } \mathrm { ~ c o p y ~ } \mathrm { ~ c o p y ~ } \mathrm { ~ c o m p ~ } } \\ \mathrm { ~ \ " c o m p S t a t e , ~ } \mathrm { \ " { c o p y S y m b o l , } \mathrm { ~ \ " { c o p y S t a t e , } \mathrm { ~ - c o p y S t a t e , } \mathrm { ~ - c o p y D i r , } \mathrm { ~ \ ~ } } } \\ \mathrm { ~ \ " { u p d a t e S y m b o l , u p d a t e S t a t e , u p d a t e D i r , r e s e t D e s c r ~ } \} . } \end{array} +$$ + +The UTM has four tapes numbered from 0 to 3, which we refer to as the description tape, the staging tape, the state tape and the working tape respectively. Initially the description tape contains a string of the form + +$$ +X s _ { 0 } q _ { 0 } s _ { 0 } ^ { \prime } q _ { 0 } ^ { \prime } d _ { 0 } s _ { 1 } q _ { 1 } s _ { 1 } ^ { \prime } q _ { 1 } ^ { \prime } d _ { 1 } \dots s _ { N } q _ { N } s _ { N } ^ { \prime } q _ { N } ^ { \prime } d _ { N } X , +$$ + +corresponding to the tuples which define $M$ , with the tape head initially on $s _ { 0 }$ . The staging tape is initially a string $X X X$ with the tape head over the second $X$ . The state tape has a single square containing some distribution in $\Delta Q$ , corresponding to the initial state of the simulated machine $M$ , with the tape head over that square. Each square on the the working tape is some distribution in $\Delta \Sigma$ with only finitely many distributions different from $\boxed { \begin{array} { r l } \end{array} }$ . The UTM is initialized in state compSymbol. + +The operation of the UTM is outlined in Figure 6. It consists of two phases; the scan phase (middle and right path), and the update phase (left path). During the scan phase, the description tape is scanned from left to right, and the first two squares of each tuple are compared to the contents of the working tape and state tape respectively. If both agree, then the last three symbols of the tuple are written to the staging tape (middle path), otherwise the tuple is ignored (right path). Once the $X$ at the end of the description tape is reached, the UTM begins the update phase, wherein the three symbols on the staging tape are then used to print the new symbol on the working tape, to update the simulated state on the state tape, and to move the working tape head in the appropriate direction. The tape head on the description tape is then reset to the initial $X$ . + +Remark E.1. One could imagine a variant of the UTM which did not include a staging tape, instead performing the actions on the work and state tape directly upon reading the appropriate tuple on the description tape. However, this is problematic when the contents of the state or working tape are distributions, as the exact time-step of the simulated machine can become unsynchronised, increasing entropy. As a simple example, suppose that the contents of the state tape were $0 . 5 q + 0 . 5 p$ , and the symbol under the working tape head was $s$ . Upon encountering the tuple $s q s { ' } q { ' } R$ , the machine would enter a superposition of states corresponding to the tape head having both moved right and not moved, complicating the future behaviour. + +We define the period of the UTM to be the smallest nonzero time interval taken for the tape head on the description tape to return to the initial $X$ , and the machine to reenter the state compSymbol. If the number of tuples on the description tape is $N$ , then the period of the UTM is $T = 1 0 N + 5$ . Moreover, other than the working tape, the position of the tape heads are $T$ -periodic. + +# F SMOOTH TURING MACHINES + +Let $\mathcal { U }$ be the staged pseudo-UTM of Appendix E. In defining the model $p ( \boldsymbol { y } | \boldsymbol { x } , \boldsymbol { w } )$ associated to a synthesis problem in Section 2 we use a smooth relaxation $\Delta \mathrm { { s t e p } } ^ { t }$ of the step function of $\mathcal { U }$ . In this appendix we define the smooth relaxation of any Turing machine following Clift & Murfet (2018). + +Let $M = ( \Sigma , Q , \delta )$ be a Turing machine with a finite set of symbols $\Sigma$ , a finite set of states $Q$ and transition function $\delta : \Sigma \times Q \bar { \to } \Sigma \times Q \times \{ - 1 , 0 , 1 \}$ . We write $\delta _ { i } = \mathsf { p r o j } _ { i } \circ \delta$ for the $i$ th component of $\delta$ for $i \in \{ 1 , 2 , 3 \}$ . For $\sqsubseteq \Sigma$ , let + +$$ +\Sigma ^ { \mathbb { Z } , \sqcap } = \{ f : \mathbb { Z } \to \Sigma | f ( i ) = \bigsqcup \mathrm { e x c e p t ~ f o r ~ f i n i t e l y ~ m a n y ~ } i \} . +$$ + +![](images/8439277769c907e73a890f5d22819ba3dcc4dc5cc4f53963fd296742f6e34336.jpg) +Figure 6: The UTM. Each of the rectangles are states, and an arrow $q q ^ { \prime }$ has the following interpretation: if the UTM is in state $q$ and sees the tape symbols (on the four tapes) as indicated by the source of the arrow, then the UTM transitions to state $q ^ { \prime }$ , writes the indicated symbols (or if there is no write instruction, simply rewrites the same symbols back onto the tapes), and performs the indicated movements of each of the tape heads. The symbols $a , b , c , d$ stand for generic symbols which are not $X$ . + +We can associate to $M$ a discrete dynamical system ${ \widehat { M } } = ( \Sigma ^ { \mathbb { Z } , \sqcup } \times Q , { \mathrm { s t e } } ]$ p) where + +$$ +{ \mathrm { s t e p } } : \Sigma ^ { \mathbb { Z } , \sqcap } \times Q \Sigma ^ { \mathbb { Z } , \sqcap } \times Q +$$ + +is the step function defined by + +$$ +\mathrm { s t e p } ( \sigma , q ) = \Bigl ( \alpha ^ { \delta _ { 3 } ( \sigma _ { 0 } , q ) } \bigl ( \ldots , \sigma _ { - 2 } , \sigma _ { - 1 } , \delta _ { 1 } ( \sigma _ { 0 } , q ) , \sigma _ { 1 } , \sigma _ { 2 } , \ldots \bigr ) , \delta _ { 2 } ( \sigma _ { 0 } , q ) \Bigr ) . +$$ + +with shift map $\alpha ^ { \delta _ { 3 } ( \sigma _ { 0 } , q ) } ( \sigma ) _ { u } = \sigma _ { u + \delta _ { 3 } ( \sigma _ { 0 } , q ) } .$ + +Let $X$ be a finite set. The standard $X$ -simplex is defined as + +$$ +\Delta X = \{ \sum _ { x \in X } \lambda _ { x } x \in \mathbb { R } X | \sum _ { x } \lambda _ { x } = 1 { \mathrm { a n d } } \lambda _ { x } \geq 0 { \mathrm { f o r ~ a l l } } x \in X \} +$$ + +where $\mathbb { R } X$ is the free vector space on $X$ . We often identify $X$ with the vertices of $\Delta X$ under the canonical inclusion $i : X \to \Delta X$ given by $\begin{array} { r } { i ( x ) = \sum _ { x ^ { \prime } \in X } \delta _ { x = x ^ { \prime } } x ^ { \prime } } \end{array}$ . For example $\{ 0 , 1 \} \subset$ $\Delta ( \{ 0 , 1 \} ) \simeq [ 0 , 1 ]$ . + +A tape square is said to be at relative position $u \in \mathbb { Z }$ if it is labelled $u$ after enumerating all squares in increasing order from left to right such that the square currently under the head is assigned zero. Consider the following random variables at times $t \geq 0$ : + +• $Y _ { u , t } \in \Sigma$ : the content of the tape square at relative position $u$ at time $t$ . +• $S _ { t } \in Q$ : the internal state at time $t$ . +· $W r _ { t } \in \Sigma$ : the symbol to be written, in the transition from time $t$ to $t + 1$ . +· $M v _ { t } \in \{ L , S , R \}$ : the direction to move, in the transition from time $t$ to $t + 1$ . + +We call a smooth dynamical system a pair $( A , \phi )$ consisting of a smooth manifold $A$ with corners together with a smooth transformation $\phi : A A$ . + +Definition F.1. Let $M = ( \Sigma , Q , \delta )$ be a Turing machine. The smooth relaxation of $M$ is the smooth dynamical system $( ( \Delta \Sigma ) ^ { \mathbb { Z } , \sqsupset } \times \Delta Q , \Delta \mathrm { s t e p } )$ where + +$$ +\Delta \mathrm { s t e p } : ( \Delta \Sigma ) ^ { \mathbb { Z } , \square } \times \Delta Q ( \Delta \Sigma ) ^ { \mathbb { Z } , \square } \times \Delta Q +$$ + +is a smooth transformation sending a state $( \{ P ( Y _ { u , t } ) \} _ { u \in \mathbb { Z } } , P ( S _ { t } ) )$ to $( \{ P ( Y _ { u , t + 1 } ) \} _ { u \in \mathbb { Z } } , P ( S _ { t + 1 } ) )$ determined by the equations + +$$ +\begin{array} { r } { P ( M v _ { t } = d | C ) = \sum _ { \sigma , q } \delta _ { \delta _ { 3 } ( \sigma , q ) = d } P ( Y _ { 0 , t } = \sigma | C ) P ( S _ { t } = q | C ) , } \end{array} +$$ + +$$ +\begin{array} { r } { P ( W r _ { t } = \sigma | C ) = \sum _ { \sigma ^ { \prime } , q } \delta _ { \delta _ { 1 } ( \sigma ^ { \prime } , q ) = \sigma } P ( Y _ { 0 , t } = \sigma ^ { \prime } | C ) P ( S _ { t } = q | C ) , } \end{array} +$$ + +$$ +\begin{array} { r l } & { P ( Y _ { u , t + 1 } = \sigma | C ) = P ( M v _ { t } = L | C ) \Big ( \delta _ { u \neq 1 } P ( Y _ { u - 1 , t } = \sigma | C ) + \delta _ { u = 1 } P ( W r _ { t } = \sigma | C ) \Big ) } \\ & { \qquad + P ( M v _ { t } = S | C ) \Big ( \delta _ { u \neq 0 } P ( Y _ { u , t } = \sigma | C ) + \delta _ { u = 0 } P ( W r _ { t } = \sigma | C ) \Big ) } \\ & { \qquad + P ( M v _ { t } = R | C ) \Big ( \delta _ { u \neq - 1 } P ( Y _ { u + 1 , t } = \sigma | C ) + \delta _ { u = - 1 } P ( W r _ { t } = \sigma | C ) \Big ) , } \end{array} +$$ + +where $C \in ( \Delta \Sigma ) ^ { \mathbb { Z } , \square } \times \Delta Q$ is an initial state. + +We will call the smooth relaxation of a Turing machine a smooth Turing machine. A smooth Turing machine encodes uncertainty in the initial configuration of a Turing machine together with an update rule for how to propagate this uncertainty over time. We interpret the smooth step function as updating the state of belief of a “naive” Bayesian observer. This nomenclature comes from the assumption of conditional independence between random variables in our probability functions. + +Remark F.2. Propagating uncertainty using standard probability leads to a smooth dynamical system which encodes the state evolution of an “ordinary” Bayesian observer of the Turing machine. This requires the calculation of various joint distributions which makes such an extension computationally difficult to work with. Computation aside, the naive probabilistic extension is justified from the point of view of derivatives of algorithms according to the denotational semantics of differential linear logic. See Clift & Murfet (2018) for further details. + +We call the smooth extension of a universal Turing machine a smooth universal Turing machine. Recall that the staged pseudo-UTM $\mathcal { U }$ has four tapes: the description tape, the staging tape, the state tape and working tape. The smooth relaxation of $\mathcal { U }$ is a smooth dynamical system + +$$ +\Delta \mathrm { s t e p } _ { \mathcal { U } } : [ ( \Delta \Sigma _ { \mathrm { U T M } } ) ^ { \mathbb { Z } , \bigtriangledown } ] ^ { 4 } \times \Delta Q _ { \mathrm { U T M } } \to [ ( \Delta \Sigma _ { \mathrm { U T M } } ) ^ { \mathbb { Z } , \bigtriangledown } ] ^ { 4 } \times \Delta Q _ { \mathrm { U T M } } . +$$ + +If we use the staged pseudo-UTM to simulate a Turing machine with tape alphabet $\Sigma \subseteq \Sigma _ { \mathrm { U T M } }$ and states $Q \subseteq \Sigma _ { \mathrm { U T M } }$ then with some determined initial state the function $\Delta$ step restricts to + +$$ +\Delta \mathrm { s t e p } _ { \mathscr { U } } : ( \Delta \Sigma ) ^ { \mathbb { Z } , \sharp } \times { \mathscr { W } } \times \Delta Q \times \mathscr { X } \longrightarrow ( \Delta \Sigma ) ^ { \mathbb { Z } , \sharp } \times { \mathscr { W } } \times \Delta Q \times \mathscr { X } +$$ + +where the first factor is the configuration of the work tape, $W$ is as in (3) and + +$$ +\mathcal { X } = [ ( \Delta \Sigma _ { \mathrm { U T M } } ) ^ { \mathbb { Z } , \sqcap } ] \times \Delta Q _ { \mathrm { U T M } } +$$ + +where the first factor is the configuration of the staging tape. Since $\mathcal { U }$ is periodic of period $T =$ $1 0 N + 5$ (Appendix E) the iterated function $( \Delta \mathrm { s t e p } _ { \mathscr { U } } ) ^ { T }$ takes an input with staging tape in its + +default state $X X X$ and UTM state compSymbol and returns a configuration with the same staging tape and state, but with the configuration of the work tape, description tape and state tape updated by one complete simulation step. That is, + +$$ +( \Delta \operatorname { s t e p } _ { \mathcal { U } } ) ^ { T } ( x , w , q , X X X , \mathrm { c o m p S y m b o l } ) = ( F ( x , w , q ) , X X X , \mathrm { c o m p S y m b o l } ) +$$ + +for some smooth function + +$$ +F : ( \Delta \Sigma ) ^ { \mathbb { Z } , \sharp } \times W \times \Delta Q \longrightarrow ( \Delta \Sigma ) ^ { \mathbb { Z } , \sharp } \times W \times \Delta Q . +$$ + +Finally we can define the function $\Delta \mathrm { s t e p } ^ { t }$ of (4). We assume all Turing machines are initialised in some common state init $\in Q$ . + +Definition F.3. Given $t \geq 0$ we define $\Delta \operatorname { s t e p } ^ { t } : \Sigma ^ { * } \times W \longrightarrow \Delta Q$ by + +$$ +\Delta \operatorname { s t e p } ^ { t } ( x , w ) = \Pi _ { Q } F ^ { t } ( x , w , \operatorname { i n i t } ) +$$ + +where $\Pi _ { Q }$ is the projection onto $\Delta Q$ . + +# G DIRECT SIMULATION + +For computational efficiency in our PyTorch implementation of the staged pseudo-UTM we implement $F$ of (9) rather than $\Delta \mathrm { s t e p } _ { \mathcal { U } }$ . We refer to this as direction simulation since it means that we update in one step the state and working tape of the UTM for a full cycle where a cycle consists of $T = 1 0 N + 5$ steps of the UTM. + +Let $S ( t )$ and $Y _ { u } ( t )$ be random variables describing the contents of state tape and working tape in relative positions $0 , u$ respectively after $t \geq 0$ time steps of the UTM. We define ${ \widetilde { S } } ( t ) : = S ( 4 + T t )$ and $\widetilde { Y } _ { u } ( t ) : = Y _ { u } ( 4 + T t )$ where $t \geq 0$ and $u \in \mathbb { Z }$ . The task then is to define functions $f , g$ such that + +$$ +\widetilde { S } ( t + 1 ) = f ( \widetilde { S } ( t ) ) +$$ + +$$ +\widetilde Y _ { u } ( t + 1 ) = g ( \widetilde Y _ { u } ( t ) ) . +$$ + +The functional relationship is given as follows: for $1 \leq i \leq N$ indexing tuples on the description tape, while processing that tuple, the UTM is in a state distribution $\lambda _ { i } \cdot \bar { q } + ( 1 - \lambda _ { i } ) \cdot \neg \bar { q }$ where $\bar { q } \in$ {copySymbol, copyState, $\mathrm { c o p y D i r } \}$ . Given the initial state of the description tape, we assume uncertainty about $s ^ { \prime } , q ^ { \prime } , d$ only. This determines a map + +$$ +\theta : \{ 1 , \dots , N \} \to \Sigma \times Q +$$ + +where the description tape at tuple number $i$ is given by $\theta ( i ) _ { 1 } \theta ( i ) _ { 2 } P ( s _ { i } ^ { \prime } ) P ( q _ { i } ^ { \prime } ) P ( d _ { i } )$ . We define the conditionally independent joint distribution between $\{ \widetilde { Y } _ { 0 , t - 1 } , \widetilde { S } _ { t - 1 } \}$ by + +$$ +\begin{array} { l } { { \lambda _ { i } = \displaystyle \sum _ { \sigma \in \Sigma } \delta _ { \theta ( i ) _ { 1 } = \sigma } P ( \widetilde { Y } _ { 0 , t - 1 } = \sigma ) \cdot \sum _ { q \in Q } \delta _ { \theta ( i ) _ { 2 } = q } P ( \widetilde { S } _ { t - 1 } = q ) \hfill } } \\ { { \quad = P ( \widetilde { Y } _ { 0 , t - 1 } = \theta ( i ) _ { 1 } ) \cdot P ( \widetilde { S } _ { t - 1 } = \theta ( i ) _ { 2 } ) . } } \end{array} +$$ + +We then calculate a recursive set of equations for $0 \leq j \leq N$ describing distributions $P ( \hat { s } _ { j } ) , P ( \hat { q } _ { j } )$ and $P ( \hat { d } _ { j } )$ on the staging tape after processing all tuples up to and including tuple $j$ . These are given by $P ( \hat { s } _ { 0 } ) = P ( \hat { q } _ { 0 } ) = P ( \hat { d } _ { 0 } ) = 1 \cdot X$ and + +$$ +\begin{array} { r l } & { \displaystyle { P ( \hat { s } _ { i } ) = \sum _ { \sigma \in \Sigma } \{ \lambda _ { i } \cdot P ( s _ { i } ^ { \prime } = \sigma ) + ( 1 - \lambda _ { i } ) \cdot P ( \hat { s } _ { i - 1 } = \sigma ) \} \cdot \sigma + ( 1 - \lambda _ { i } ) \cdot P ( \hat { s } _ { i - 1 } = X ) \cdot X } } \\ & { \displaystyle { P ( \hat { q } _ { i } ) = \sum _ { \ q \in Q } \{ \lambda _ { i } \cdot P ( q _ { i } ^ { \prime } = q ) + ( 1 - \lambda _ { i } ) \cdot P ( \hat { q } _ { i - 1 } = q ) \} \cdot q + ( 1 - \lambda _ { i } ) \cdot P ( \hat { q } _ { i - 1 } = X ) \cdot X } } \\ & { \displaystyle { \hat { l } _ { i } ) = \sum _ { \alpha \in \{ L , R , S \} } \{ \lambda _ { i } \cdot P ( d _ { i } = a ) + ( 1 - \lambda _ { i } ) \cdot P ( \hat { d } _ { i - 1 } = a ) \} \cdot a + ( 1 - \lambda _ { i } ) \cdot P ( \hat { d } _ { i - 1 } = X ) \cdot X . } } \end{array} +$$ + +Let $A _ { \sigma } = P ( \widehat { s } _ { N } = X ) \cdot P ( \widetilde { Y } _ { 0 , t - 1 } = \sigma ) + P ( \widehat { s } _ { N } = \sigma ) .$ . In terms of the above distributions + +$$ +P ( \widetilde { S } _ { t } ) = \sum _ { q \in Q } \Big ( P ( \hat { q } _ { N } = X ) \cdot P ( \widetilde { S } _ { t - 1 } = q ) + P ( \hat { q } _ { N } = q ) \Big ) \cdot q +$$ + +and + +$$ +\begin{array} { r l } & { P ( \widetilde { Y } _ { u , t } = \sigma ) = P ( \hat { d } _ { N } = L ) \left( \delta _ { u \neq 1 } P ( \widetilde { Y } _ { u - 1 , t - 1 } = \sigma ) + \delta _ { u = 1 } A _ { \sigma } \right) } \\ & { \quad \quad \quad \quad \quad + P ( \hat { d } _ { N } = R ) \left( \delta _ { u \neq - 1 } P ( \widetilde { Y } _ { u + 1 , t - 1 } = \sigma ) + \delta _ { u = - 1 } A _ { \sigma } \right) } \\ & { \quad \quad \quad \quad \quad + P ( \hat { d } _ { N } = S ) \left( \delta _ { u \neq 0 } P ( \widetilde { Y } _ { u , t - 1 } = \sigma ) + \delta _ { u = 0 } A _ { \sigma } \right) } \\ & { \quad \quad \quad \quad \quad + P ( \hat { d } _ { N } = X ) \left( \delta _ { u \neq 0 } P ( \widetilde { Y } _ { u , t - 1 } = \sigma ) + \delta _ { u = 0 } A _ { \sigma } \right) . } \end{array} +$$ + +Using these equations, we can state efficient update rules for the staging tape. We have + +$$ +\begin{array} { l l l } { { \displaystyle P ( \hat { s } _ { N } = X ) = \prod _ { j = 1 } ^ { N } ( 1 - \lambda _ { j } ) , \quad } } & { { \displaystyle P ( \hat { s } _ { N } = \sigma ) = \sum _ { j = 1 } ^ { N } \lambda _ { j } \cdot P ( s _ { j } ^ { \prime } = \sigma ) \prod _ { l = j + 1 } ^ { N } ( 1 - \lambda _ { l } ) } } \\ { { \displaystyle P ( \hat { q } _ { N } = X ) = \prod _ { j = 1 } ^ { N } ( 1 - \lambda _ { j } ) , \quad } } & { { \displaystyle P ( \hat { q } _ { N } = q ) = \sum _ { j = 1 } ^ { N } \lambda _ { j } \cdot P ( q _ { j } ^ { \prime } = q ) \prod _ { l = j + 1 } ^ { N } ( 1 - \lambda _ { l } ) } } \\ { { \displaystyle P ( \hat { d } _ { N } = X ) = \prod _ { j = 1 } ^ { N } ( 1 - \lambda _ { j } ) , \quad } } & { { \displaystyle P ( \hat { d } _ { N } = a ) = \sum _ { j = 1 } ^ { N } \lambda _ { j } \cdot P ( d _ { j } = a ) \prod _ { l = j + 1 } ^ { N } ( 1 - \lambda _ { l } ) . } } \end{array} +$$ + +To enable efficient computation, we can express these equations using tensor calculus. Let $\lambda =$ $( \lambda _ { 1 } , \dots , \lambda _ { N } ) \in \mathbb { R } ^ { N }$ . We view + +$$ +\theta : \mathbb { R } ^ { N } \xrightarrow { } \mathbb { R } \Sigma \otimes \mathbb { R } Q +$$ + +as a tensor and so $\begin{array} { r } { \theta = \sum _ { i = 1 } ^ { N } i \otimes \theta ( i ) _ { 1 } \otimes \theta ( i ) _ { 2 } \in \mathbb { R } ^ { N } \otimes \mathbb { R } \Sigma \otimes \mathbb { R } Q . } \end{array}$ . Then + +$$ +\theta _ { - } \left( P ( \widetilde { Y } _ { 0 , t - 1 } ) \otimes P ( \widetilde { S } _ { t - 1 } ) \right) = \sum _ { i = 1 } ^ { N } i \cdot P ( \widetilde { Y } _ { 0 , t - 1 } = \theta ( i ) _ { 1 } ) \cdot P ( \widetilde { S } _ { t - 1 } = \theta ( i ) _ { 2 } ) = \lambda . +$$ + +If we view $P ( s _ { * } ^ { \prime } = \bullet ) \in \mathbb { R } ^ { N } \otimes \mathbb { R } ^ { \Sigma }$ as a tensor, then + +$$ +{ \mathcal { S } } ( { \widehat { \mathfrak { s } } } _ { N } ) = \sum _ { j = 1 } ^ { N } P ( s _ { j } ^ { \prime } = \bullet ) \cdot \left( \lambda _ { j } \prod _ { l = j + 1 } ^ { N } ( 1 - \lambda _ { l } ) \right) = \lambda \cdot \left( \prod _ { l = 2 } ^ { N } ( 1 - \lambda _ { l } ) , \prod _ { l = 3 } ^ { N } ( 1 - \lambda _ { l } ) , \ldots , ( 1 - \lambda _ { N } ) , 1 \right) +$$ + +can be expressed in terms on the vector $\lambda$ only. Similarly, $P ( q _ { * } ^ { \prime } = \bullet ) \in \mathbb { R } ^ { N } \otimes \mathbb { R } ^ { Q }$ with + +$$ +{ \cal P } ( \hat { q } _ { N } ) = \sum _ { j = 1 } ^ { N } P ( q _ { j } ^ { \prime } = \bullet ) \cdot \left( \lambda _ { j } \prod _ { l = j + 1 } ^ { N } ( 1 - \lambda _ { l } ) \right) = \lambda \cdot \left( \prod _ { l = 2 } ^ { N } ( 1 - \lambda _ { l } ) , \prod _ { l = 3 } ^ { N } ( 1 - \lambda _ { l } ) , \ldots , ( 1 - \lambda _ { N } ) , 1 \right) +$$ + +and $P ( d _ { * } = \bullet ) \in \mathbb { R } ^ { N } \otimes \mathbb { R } ^ { 3 }$ with + +$$ +^ { > } ( \hat { d } _ { N } ) = \sum _ { j = 1 } ^ { N } P ( d _ { j } = \bullet ) \cdot \left( \lambda _ { j } \prod _ { l = j + 1 } ^ { N } ( 1 - \lambda _ { l } ) \right) = \lambda \cdot \left( \prod _ { l = 2 } ^ { N } ( 1 - \lambda _ { l } ) , \prod _ { l = 3 } ^ { N } ( 1 - \lambda _ { l } ) , \ldots , ( 1 - \lambda _ { N } ) , 1 \right) . +$$ \ No newline at end of file diff --git a/parse/train/qiydAcw6Re/qiydAcw6Re_content_list.json b/parse/train/qiydAcw6Re/qiydAcw6Re_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..d42612782f8bfe62409ae3086823b7c1fd5aa786 --- /dev/null +++ b/parse/train/qiydAcw6Re/qiydAcw6Re_content_list.json @@ -0,0 +1,2983 @@ +[ + { + "type": "text", + "text": "GEOMETRY OF PROGRAM SYNTHESIS ", + "text_level": 1, + "bbox": [ + 174, + 98, + 627, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 145, + 398, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 210, + 544, + 226 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We present a new perspective on program synthesis in which programs may be identified with singularities of analytic functions. As an example, Turing machines are synthesised from input-output examples by propagating uncertainty through a smooth relaxation of a universal Turing machine. The posterior distribution over weights is approximated using Markov chain Monte Carlo and bounds on the generalisation error of these models is estimated using the real log canonical threshold, a geometric invariant from singular learning theory. ", + "bbox": [ + 232, + 242, + 764, + 340 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 371, + 334, + 386 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The idea of program synthesis dates back to the birth of modern computation itself (Turing, 1948) and is recognised as one of the most important open problems in computer science (Gulwani et al., 2017). However, there appear to be serious obstacles to synthesising programs by gradient descent at scale (Neelakantan et al., 2016; Kaiser & Sutskever, 2016; Bunel et al., 2016; Gaunt et al., 2016; Evans & Grefenstette, 2018; Chen et al., 2018) and these problems suggest that it would be appropriate to make a fundamental study of the geometry of loss surfaces in program synthesis, since this geometry determines the learning process. To that end, in this paper we explain a new point of view on program synthesis using the singular learning theory of Watanabe (2009) and the smooth relaxation of Turing machines from Clift & Murfet (2018). ", + "bbox": [ + 174, + 404, + 825, + 529 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In broad strokes this new geometric point of view on program synthesis says: ", + "bbox": [ + 176, + 535, + 678, + 550 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "• Programs to be synthesised are singularities of analytic functions. If $U \\subseteq \\mathbb { R } ^ { d }$ is open and $K : U \\longrightarrow \\mathbb { R }$ is analytic, then $x \\in U$ is a critical point of $K$ if $\\nabla K ( x ) = 0$ and a singularity of the function $K$ if it is a critical point where $K ( x ) = 0$ . The Kolmogorov complexity of a program is related to a geometric invariant of the associated singularity called the Real Log Canonical Threshold (RLCT). This invariant controls both the generalisation error and the learning process, and is therefore an appropriate measure of “complexity” in continuous program synthesis. See Section 3. The geometry has concrete practical implications. For example, a MCMC-based approach to program synthesis will find, with high probability, a solution that is of low complexity (if it finds a solution at all). We sketch a novel point of view on the problem of “bad local minima” (Gaunt et al., 2016) based on these ideas. See Section 4. ", + "bbox": [ + 215, + 563, + 825, + 728 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We demonstrate all of these principles in experiments with toy examples of synthesis problems. ", + "bbox": [ + 178, + 742, + 797, + 756 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Program synthesis as inference. We use Turing machines, but mutatis mutandis everything applies to other programming languages. Let $T$ be a Turing machine with tape alphabet $\\Sigma$ and set of states $Q$ and assume that on any input $x \\in \\Sigma ^ { * }$ the machine eventually halts with output $T ( x ) \\in \\Sigma ^ { * }$ . Then to the machine $T$ we may associate the set $\\{ ( x , T ( x ) ) \\} _ { x \\in \\Sigma ^ { * } } \\subseteq \\Sigma ^ { * } \\times \\Sigma ^ { * }$ . Program synthesis is the study of the inverse problem: given a subset of $\\Sigma ^ { * } \\times \\Sigma ^ { * }$ we would like to determine (if possible) a Turing machine which computes the given outputs on the given inputs. ", + "bbox": [ + 174, + 762, + 825, + 847 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "If we presume given a probability distribution $q ( x )$ on $\\Sigma ^ { * }$ then we can formulate this as a problem of statistical inference: given a probability distribution $q ( x , y )$ on $\\Sigma ^ { * } \\times \\Sigma ^ { * }$ determine the most likely machine producing the observed distribution $q ( x , y ) = q ( y | x ) q ( x )$ . If we fix a universal Turing machine $\\mathcal { U }$ then Turing machines can be parametrised by codes $\\dot { w } \\in W ^ { c o d e }$ with $\\mathcal { U } ( x , w ) = T ( x )$ for all $x \\in \\Sigma ^ { * }$ . We let ${ \\bar { p } } ( y | x , w )$ denote the probability of $\\mathcal { U } ( x , w ) = y$ (which is either zero or one) ", + "bbox": [ + 176, + 854, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "so that solutions to the synthesis problem are in bijection with the zeros of the Kullback-Leibler divergence between the true distribution and the model ", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/d3da2bd582ae2575ffc9fe4975a17ea33e2e7e5d6822aa86726e2a8859d775b8.jpg", + "text": "$$\nK ( w ) = \\int \\int q ( y | x ) q ( x ) \\log \\frac { q ( y | x ) } { p ( y | x , w ) } d x d y .\n$$", + "text_format": "latex", + "bbox": [ + 343, + 138, + 653, + 172 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "So far this is just a trivial rephrasing of the combinatorial optimisation problem of finding a Turing machine $T$ with $T ( x ) = y$ for all $( x , y )$ with $q ( x , y ) > 0$ . ", + "bbox": [ + 173, + 179, + 823, + 209 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Smooth relaxation. One approach is to seek a smooth relaxation of the synthesis problem consisting of an analytic manifold $W \\bar { \\supseteq } W ^ { c o d e }$ and an extension of $K$ to an analytic function $K : W \\longrightarrow \\mathbb { R }$ so that we can search for the zeros of $K$ using gradient descent. Perhaps the most natural way to construct such a smooth relaxation is to take $W$ to be a space of probability distributions over $W ^ { \\dot { c } o d e }$ and prescribe a model $p ( \\boldsymbol { y } | \\boldsymbol { x } , \\boldsymbol { w } )$ for propagating uncertainty about codes to uncertainty about outputs (Gaunt et al., 2016; Evans & Grefenstette, 2018). The particular model we choose is based on the semantics of linear logic (Clift & Murfet, 2018). Supposing that such a smooth relaxation has been chosen together with a prior $\\varphi ( w )$ over $W$ , smooth program synthesis becomes the study of the statistical learning theory of the triple $( p , q , \\varphi )$ . ", + "bbox": [ + 173, + 214, + 825, + 340 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "There are perhaps two primary reasons to consider the smooth relaxation. Firstly, one might hope that stochastic gradient descent or techniques like Markov chain Monte Carlo will be effective means of solving the original combinatorial optimisation problem. This is not a new idea (Gulwani et al., 2017, §6) but so far its effectiveness for large programs has not been proven. Independently, one might hope to find powerful new mathematical ideas that apply to the relaxed problem and shed light on the nature of program synthesis. This is the purpose of the present paper. ", + "bbox": [ + 173, + 347, + 825, + 431 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Singular learning theory. We denote by $W _ { 0 } = \\{ w \\in W | K ( w ) = 0 \\}$ so that ", + "bbox": [ + 174, + 435, + 686, + 453 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/86ade0437cc02ec74cf772862fcd0f85b59d0d58456410aeb8aa4fceae3052a3.jpg", + "text": "$$\nW _ { 0 } \\cap W ^ { c o d e } \\subseteq W _ { 0 } \\subseteq W\n$$", + "text_format": "latex", + "bbox": [ + 413, + 459, + 584, + 477 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $W _ { 0 } \\cap W ^ { c o d e }$ is the discrete set of solutions to the original synthesis problem. We refer to these as the classical solutions. As the vanishing locus of an analytic function, $W _ { 0 }$ is an analytic space over $\\mathbb { R }$ (Hironaka, 1964, $\\ S 0 . 1 )$ , (Griffith & Harris, 1978) and it is interesting to study the geometry of this space near the classical solutions. Since $K$ is a Kullback-Leibler divergence it is non-negative and so it not only vanishes on $W _ { 0 }$ but $\\nabla K$ also vanishes, hence every point of $W _ { 0 }$ is a singular point. ", + "bbox": [ + 173, + 484, + 825, + 555 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Beyond this the geometry of $W _ { 0 }$ depends on the particular model $p ( \\boldsymbol { y } | \\boldsymbol { x } , \\boldsymbol { w } )$ that has been chosen, but some aspects are universal: the nature of program synthesis means that typically $W _ { 0 }$ is an extended object (i.e. it contains points other than the classical solutions) and the Hessian matrix of second order partial derivatives of $K$ at a classical solution is not invertible - that is, the classical solutions are degenerate critical points of $K$ . This means that singularity theory is the appropriate branch of mathematics for studying the geometry of $W _ { 0 }$ near a classical solution. It also means that the Fisher information matrix ", + "bbox": [ + 173, + 561, + 825, + 659 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/b8dfea439a1747d3611410930a42886e436c6fc67f00c4da375cd2997dad9780.jpg", + "text": "$$\nI ( w ) _ { i j } = \\int \\int \\frac { \\partial } { \\partial w _ { i } } \\big [ \\log p ( y | x , w ) \\big ] \\frac { \\partial } { \\partial w _ { j } } \\big [ \\log p ( y | x , w ) \\big ] q ( y | x ) q ( x ) d x d y ,\n$$", + "text_format": "latex", + "bbox": [ + 254, + 662, + 740, + 698 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "is degenerate at a classical solution, so that the appropriate branch of statistical learning theory is singular learning theory (Watanabe, 2007; 2009). For an introduction to singular learning theory in the context of deep learning see (Murfet et al., 2020). ", + "bbox": [ + 174, + 703, + 823, + 746 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Broadly speaking the contribution of this paper is to realise program synthesis within the framework of singular learning theory, at both a theoretical and an experimental level. In more detail the contents of the paper are: ", + "bbox": [ + 176, + 752, + 821, + 795 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We define a staged pseudo-UTM (Appendix E) which is well-suited to experiments with the ideas discussed above. Propagating uncertainty about the code through this UTM using the ideas of (Clift & Murfet, 2018) defines a triple $( p , q , \\varphi )$ associated to a synthesis problem. This formally embeds program synthesis within singular learning theory. We realise this embedding in code by providing an implementation in PyTorch of this propagation of uncertainty through a UTM. Using the No-U-Turn variant of MCMC (Hoffman & Gelman, 2014) we can approximate the Bayesian posterior of any program synthesis problem (of course in practice we are limited by computational constraints in doing so). ", + "bbox": [ + 215, + 806, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We explain how the real log canonical threshold (a geometric invariant) is related to Kolmogorov complexity (Section 3). • We give a simple example (Appendix C) in which $W _ { 0 }$ contains the set of classical solutions as a proper subset and every point of $W _ { 0 }$ is a degenerate critical point of $K$ . For two simple synthesis problems detectA and parityCheck we demonstrate all of the above, using MCMC to approximate the Bayesian posterior and theorems from Watanabe (2013) to estimate the RLCT (Section 5). We discuss how $W _ { 0 }$ is an extended object and how the RLCT relates to the local dimension of $W _ { 0 }$ near a classical solution. ", + "bbox": [ + 212, + 103, + 825, + 227 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "RELATED WORK ", + "text_level": 1, + "bbox": [ + 174, + 250, + 310, + 265 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The idea of synthesising Turing machines can be traced back to the work of Solomonoff on inductive inference (Solomonoff, 1964). A more explicit form of the problem was given in Biermann (1972) who proposed an algorithmic method. Machine learning based approaches appear in Schmidhuber (1997) and Hutter (2004), which pay particular attention to model complexity, and Gaunt et al. (2016) and Freer et al. (2014), the latter using the notion of “universal probabilistic Turing machine” (De Leeuw et al., 1956). A different probabilistic extension of a universal Turing machine was introduced in Clift & Murfet (2018) via linear logic. Studies of the singular geometry of learning models go back to Amari et al. (2003) and notably, the extensive work of Watanabe (2007; 2009). ", + "bbox": [ + 174, + 281, + 825, + 393 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 TURING MACHINE SYNTHESIS AS SINGULAR LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 415, + 676, + 431 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "All known approaches to program synthesis can be formulated in terms of a singular learning problem. Singular learning theory is the extension of statistical learning theory to account for the fact that the set of learned parameters $W _ { 0 }$ has the structure of an analytic space as opposed to an analytic manifold (Watanabe, 2007; 2009). It is organised around triples $( p , q , \\varphi )$ consisting of a class of models $\\{ p ( y | x , w ) : w \\in W \\}$ , a true distribution $q ( y | x )$ and a prior $\\varphi$ on $W$ . ", + "bbox": [ + 174, + 446, + 825, + 517 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In our approach we fix a Universal Turing Machine (UTM), denoted $\\mathcal { U }$ , with a description tape (which specifies the code of the Turing machine to be executed), a work tape (simulating the tape of that Turing machine during its operation) and a state tape (simulating the state of that Turing machine). The general statistical learning problem that can be formulated using $\\mathcal { U }$ is the following: given some initial string $x$ on the work tape, predict the state of the simulated machine and the contents of the work tape after some specified number of steps (Clift & Murfet, 2018, $\\ S 7 . 1 )$ . For simplicity, in this paper we consider models that only predict the final state; the necessary modifications in the general case are routine. We also assume that $W$ parametrises Turing machines whose tape alphabet $\\Sigma$ and set of states $Q$ have been encoded by individual symbols in the tape alphabet of $\\mathcal { U }$ . Hence $\\mathcal { U }$ is actually what we call a pseudo-UTM (see Appendix E). Again, treating the general case is routine and for the present purposes only introduces uninteresting complexity. ", + "bbox": [ + 173, + 523, + 825, + 678 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Let $\\Sigma$ denote the tape alphabet of the simulated machine, $Q$ the set of states and let $L , S , R$ stand for left, stay and right, the possible motions of the Turing machine head. We assume that $| Q | > 1$ since otherwise the synthesis problem is trivial. The set of ordinary codes $W ^ { c o d e }$ for a Turing machine sits inside a compact space of probability distributions $W$ over codes ", + "bbox": [ + 174, + 684, + 825, + 739 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/cd50a701fdf44997a4a77d7267e011542a930c3a306e5f11d1b0e6f45f9825b0.jpg", + "text": "$$\nW ^ { c o d e } : = \\prod _ { \\sigma , q } \\Sigma \\times Q \\times \\{ L , S , R \\} \\subseteq \\prod _ { \\sigma , q } \\Delta \\Sigma \\times \\Delta Q \\times \\Delta \\{ L , S , R \\} = : W\n$$", + "text_format": "latex", + "bbox": [ + 254, + 747, + 743, + 782 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\Delta X$ denotes the set of probability distributions over a set $X$ , see (8), and the product is over pairs $( \\sigma , q ) \\in \\Sigma \\times Q$ .1 For example the point $\\{ ( \\sigma ^ { \\prime } , q ^ { \\prime } , d ) \\} _ { \\sigma , q } \\in W ^ { c o d e }$ encodes the machine which when it reads $\\sigma$ under the head in state $q$ writes $\\sigma ^ { \\prime }$ , transitions into state $q ^ { \\prime }$ and moves in direction $d$ . Given $w \\in W ^ { c o d e }$ let $\\operatorname { s t e p } ^ { t } ( x , w ) \\in Q$ denote the contents of the state tape of $\\mathcal { U }$ after $t$ timesteps (of the simulated machine) when the work tape is initialised with $x$ and the description tape with $w$ . ", + "bbox": [ + 173, + 789, + 825, + 859 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "There is a principled extension of this operation of $\\mathcal { U }$ to a smooth function ", + "bbox": [ + 174, + 103, + 660, + 118 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/5dea214bbab5e854e2798a1539cbdc71b497e31002b3bffddc549555d436dd43.jpg", + "text": "$$\n\\Delta \\operatorname { s t e p } ^ { t } : { \\Sigma } ^ { * } \\times W \\longrightarrow \\Delta Q\n$$", + "text_format": "latex", + "bbox": [ + 405, + 126, + 593, + 143 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "which propagates uncertainty about the symbols on the description tape to uncertainty about the final state and we refer to this extension as the smooth relaxation of $\\mathcal { U }$ . The details are given in Appendix F but at an informal level the idea behind the relaxation is easy to understand: to sample from $\\Delta \\operatorname { s t e p } ^ { t } ( x , w )$ we run $\\mathcal { U }$ to simulate $t$ timesteps in such a way that whenever the UTM needs to “look at” an entry on the description tape we sample from the corresponding distribution specified by $w$ .2 The significance of the particular smooth relaxation that we use is that its derivatives have a logical interpretation (Clift & Murfet, 2018, $\\ S 7 . 1 \\ r _ { . }$ ). ", + "bbox": [ + 173, + 151, + 825, + 251 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The class of models that we consider is ", + "bbox": [ + 176, + 257, + 431, + 271 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c3fa021dbc6482a3b52d76502708359b0457132f60210ddd4277aecba15a00c3.jpg", + "text": "$$\np ( y | x , w ) = \\Delta \\mathrm { s t e p } ^ { t } ( x , w )\n$$", + "text_format": "latex", + "bbox": [ + 408, + 279, + 589, + 297 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $t$ is fixed for simplicity in this paper. More generally we could also view $x$ as consisting of a sequence and a timeout, as is done in (Clift & Murfet, 2018, $\\ S 7 . 1 )$ . The construction of this model is summarised in Figure 1. ", + "bbox": [ + 174, + 305, + 825, + 348 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/4a6eaec4d0956e25ce7728bf29fb6ddeee2cacf5f806fd63f0d36ae211bc6fec.jpg", + "image_caption": [ + "Figure 1: The state of $\\mathcal { U }$ is represented by the state of the work tape, state tape and description (code) tape. The work tape is initialised with a sequence $x \\in \\Sigma ^ { * }$ , the code tape with $w \\in W$ and the state tape with some standard initial state, the smooth relaxation $\\Delta$ step of the pseudo-UTM is run for $t$ steps and the final probability distribution over states is $y$ . " + ], + "image_footnote": [], + "bbox": [ + 240, + 366, + 756, + 512 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Definition 2.1 (Synthesis problem). A synthesis problem for $\\mathcal { U }$ consists of a probability distribution $q ( x , y )$ over $\\Sigma ^ { * } \\times Q$ . We say that the synthesis problem is deterministic if there is $f : \\Sigma ^ { * } \\longrightarrow Q$ such that $q ( y = f ( x ) | x ) = 1$ for all $x \\in \\Sigma ^ { * }$ . ", + "bbox": [ + 174, + 606, + 825, + 648 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Definition 2.2. The triple $( p , q , \\varphi )$ associated to a synthesis problem is the model $p$ of (5) together with the true distribution $q$ and uniform prior $\\varphi$ on the parameter space $W$ . The Kullback-Leibler function $K ( w )$ of the synthesis problem is defined by (1) and a solution to the synthesis problem is a point of $W _ { 0 }$ . A classical solution is a point of $W _ { 0 } \\cap W ^ { c o d e }$ . ", + "bbox": [ + 174, + 654, + 825, + 709 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As $\\Delta \\mathrm { s t e p } ^ { t }$ is a polynomial function, $K$ is analytic and so $W _ { 0 }$ is a semi-analytic space (it is cut out of the semi-analytic space $W$ by the vanishing of $K$ ). If the synthesis problem is deterministic and $q ( x )$ is uniform on some finite subset of $\\Sigma ^ { * }$ then $W _ { 0 }$ is semi-algebraic (it is cut out of $W$ by polynomial equations) and all solutions lie at the boundary of the parameter space $W$ (Appendix D). However in general $W _ { 0 }$ is only semi-analytic and intersects the interior of $W$ (Example C.2). We assume that ${ \\bar { q } } ( y | x )$ is realisable that is, there exists $w _ { 0 } \\in W$ with $q ( y | x ) = p ( y | x , w _ { 0 } )$ . ", + "bbox": [ + 173, + 722, + 825, + 806 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "A triple $( p , q , \\varphi )$ is regular if the model is identifiable, ie. for all inputs $x \\in \\mathbb { R } ^ { n }$ , the map sending $w$ to the conditional probability distribution $p ( \\boldsymbol { y } | \\boldsymbol { x } , \\boldsymbol { w } )$ is one-to-one, and the Fisher information matrix is non-degenerate. Otherwise, the learning machine is strictly singular (Watanabe, 2009, $\\ S 1 . 2 . 1 $ . Triples arising from synthesis problems are typically singular: in Example 2.5 below we show an explicit example where multiple parameters $w$ determine the same model, and in Example C.2 we give an example where the Hessian of $K$ is degenerate everywhere on $W _ { 0 }$ (Watanabe, 2009, §1.1.3). ", + "bbox": [ + 173, + 813, + 825, + 896 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Remark 2.3. Non-deterministic synthesis problems arise naturally in various contexts, for example in the fitting of algorithms to the behaviour of deep reinforcement learning agents. Suppose an agent is acting in an environment with starting states encoded by $x \\in \\Sigma ^ { * }$ and possible episode end states by $y \\in Q$ . Even if the optimal policy is known to determine a computable function $\\Sigma ^ { * } \\longrightarrow Q$ the statistics of the observed behaviour after finite training time will only provide a function $\\Sigma ^ { * } \\longrightarrow \\Delta Q$ and if we wish to fit algorithms to behaviour it makes sense to deal with this uncertainty directly. ", + "bbox": [ + 173, + 103, + 825, + 188 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Definition 2.4. Let $( p , q , \\varphi )$ be the triple associated to a synthesis problem. The Real Log Canonical Threshold (RLCT) $\\lambda$ of the synthesis problem is defined so that $- \\lambda$ is the largest pole of the meromorphic extension (Atiyah, 1970) of the zeta function $\\begin{array} { r } { \\zeta ( z ) = \\int K ( w ) ^ { z } \\varphi ( w ) \\hat { d w } } \\end{array}$ . ", + "bbox": [ + 173, + 191, + 825, + 234 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The more singular the analytic space $W _ { 0 }$ of solutions is, the smaller the RLCT. One way to think of the RLCT is as a count of the effective number of parameters near $W _ { 0 }$ (Murfet et al., 2020, $\\ S 4$ ). In Section 3 we relate the RLCT to Kolmogorov complexity and in Section 5 we estimate the RLCT of the synthesis problem detectA given below, using the method explained in Appendix A. ", + "bbox": [ + 173, + 244, + 825, + 303 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Example 2.5 (detectA). The deterministic synthesis problem detectA has $\\Sigma = \\{ \\boxed \\} , A , B \\}$ , $Q = \\{ { \\mathrm { r e j e c t } } , { \\mathrm { a c c e p t } } \\}$ and $q ( y | x )$ is determined by the function taking in a string $x$ of $A$ ’s and $B$ ’s and returning the state accept if the string contains an $A$ and state reject otherwise. The conditional true distribution $q ( y | x )$ is realisable because this function is computed by a Turing machine. ", + "bbox": [ + 173, + 306, + 825, + 363 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Two solutions are shown in Figure 2. On the left is a parameter $w _ { l } \\in \\ b { W } _ { 0 } \\setminus \\ b { W } ^ { c o d e }$ and on the right is $w _ { r } \\in W _ { 0 } \\cap W ^ { c o d e }$ . Varying the distributions in $w _ { l }$ that have nonzero entropy we obtain a submanifold $V \\subseteq W _ { 0 }$ containing $w _ { l }$ of dimension 14. This leads by (Watanabe, 2009, Remark 7.3) to a bound on the RLCT of $\\lambda \\le \\frac { 1 } { 2 } ( 3 0 - 1 4 ) = 8$ which is consistent with the experimental results in Table 1. This highlights that solutions need not lie at vertices of the probability simplex, and $W _ { 0 }$ may contain a high-dimensional submanifold around a given classical solution. ", + "bbox": [ + 173, + 369, + 825, + 454 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/94e2cb3be00f1414e5f172ce0b09cdea4b00fe6fbc48f36b360a03838c63cdbb.jpg", + "image_caption": [ + "Figure 2: Visualisation of two solutions for the synthesis problem detectA . " + ], + "image_footnote": [], + "bbox": [ + 191, + 468, + 803, + 584 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.1 THE SYNTHESIS PROCESS ", + "text_level": 1, + "bbox": [ + 174, + 637, + 397, + 652 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Synthesis is a problem because we do not assume that the true distribution is known: for example, if $q { \\dot { ( } } y | x )$ is deterministic and the associated function is $f : \\Sigma ^ { * } \\longrightarrow Q$ , we assume that some example pairs $( x , f ( x ) )$ are known but no general algorithm for computing $f$ is known (if it were, synthesis would have already been performed). In practice synthesis starts with a sample $D _ { n } = \\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { n }$ from $q ( x , y )$ with associated empirical Kullback-Leibler distance ", + "bbox": [ + 173, + 664, + 825, + 734 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/7d14e8fbfe73c032c6de880f843e24e7d2034b028aec0b312f8aaa04795f064c.jpg", + "text": "$$\nK _ { n } ( w ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\log \\frac { q ( y _ { i } | x _ { i } ) } { p ( y _ { i } | x _ { i } , w ) } .\n$$", + "text_format": "latex", + "bbox": [ + 387, + 741, + 611, + 782 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "If the synthesis problem is deterministic and $u \\in W ^ { c o d e }$ then $K _ { n } ( u ) = 0$ if and only if $u$ explains the data in the sense that $\\operatorname { s t e p } ^ { t } ( x _ { i } , u ) = y _ { i }$ for $1 \\leq i \\leq n$ . We now review two natural ways of finding such solutions in the context of machine learning. ", + "bbox": [ + 176, + 790, + 825, + 833 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Synthesis by stochastic gradient descent (SGD). The first approach is to view the process of program synthesis as stochastic gradient descent for the function $K : W \\longrightarrow \\mathbb { R }$ . We view $D _ { n }$ as a large training set and further sample subsets $D _ { m }$ with $m \\ll n$ and compute $\\nabla K _ { m }$ to take gradient descent steps $w _ { i + 1 } = w _ { i } - \\eta \\nabla K _ { m } ( w _ { i } )$ for some learning rate $\\eta$ . Stochastic gradient descent has the advantage (in principle) of scaling to high-dimensional parameter spaces $W$ , but in practice it is challenging to use gradient descent to find points of $W _ { 0 }$ (Gaunt et al., 2016). ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Synthesis by sampling. The second approach is to consider the Bayesian posterior associated to the synthesis problem, which can be viewed as an update on the prior distribution $\\varphi$ after seeing $D _ { n }$ ", + "bbox": [ + 171, + 103, + 825, + 132 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/85dfd95a7a7c20074844aa594f0db4f5329f3fcaec6759bced1b5a0c8373d2d2.jpg", + "text": "$$\np ( w | D _ { n } ) = { \\frac { p ( D _ { n } | w ) p ( w ) } { p ( D _ { n } ) } } = { \\frac { 1 } { Z _ { n } } } \\varphi ( w ) \\prod _ { i = 1 } ^ { n } p ( y _ { i } | x _ { i } , w ) = { \\frac { 1 } { Z _ { n } ^ { 0 } } } \\exp \\{ - n K _ { n } ( w ) + \\log \\varphi ( w ) \\}\n$$", + "text_format": "latex", + "bbox": [ + 196, + 138, + 800, + 181 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { Z _ { n } ^ { 0 } = \\int \\varphi ( w ) \\exp ( - n K _ { n } ( w ) ) d w } \\end{array}$ . If $n$ is large the posterior distribution concentrates around solutions $w \\in W _ { 0 }$ and so sampling from the posterior will tend to produce machines that are (nearly) solutions. The gold standard sampling is Markov Chain Monte Carlo (MCMC). Scaling MCMC to where $W$ is high-dimensional is a challenging task with many attempts to bridge the gap with SGD (Welling & Teh, 2011; Chen et al., 2014; Ding et al., 2014; Zhang et al., 2020). Nonetheless in simple cases we demonstrate experimentally in Section 5 that machines may be synthesised by using MCMC to sample from the posterior. ", + "bbox": [ + 174, + 188, + 825, + 287 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3 COMPLEXITY OF PROGRAMS ", + "text_level": 1, + "bbox": [ + 176, + 308, + 446, + 324 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Every Turing machine is the solution of a deterministic synthesis problem, so Section 2 associates to any Turing machine a singularity of a semi-analytic space $W _ { 0 }$ . To indicate that this connection is not vacuous, we sketch how the complexity of a program is related to the real log canonical threshold of a singularity. A more detailed discussion will appear elsewhere. ", + "bbox": [ + 174, + 339, + 825, + 396 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Let $q ( x , y )$ be a deterministic synthesis problem for $\\mathcal { U }$ which only involves input sequences in some restricted alphabet $\\Sigma _ { i n p u t }$ , that is, $q ( x ) \\bar { = } 0$ if $x \\notin ( \\Sigma _ { i n p u t } ) ^ { * }$ . Let $D _ { n }$ be sampled from $q ( x , y )$ and let $u , v \\in W ^ { c o d e } \\cap W _ { 0 }$ be two explanations for the sample in the sense that $K _ { n } ( u ) = K _ { n } ( v ) = 0$ . Which explanation for the data should we prefer? The classical answer based on Occam’s razor (Solomonoff, 1964) is that we should prefer the shorter program, that is, the one using the fewest states and symbols. ", + "bbox": [ + 173, + 402, + 825, + 488 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Set $N = | \\Sigma |$ and $M = | Q |$ . Any Turing machine $T$ using $N ^ { \\prime } \\leq N$ symbols and $M ^ { \\prime } \\leq M$ states has a code for $\\mathcal { U }$ of length $c M ^ { \\prime } N ^ { \\prime }$ where $c$ is a constant. We assume that $\\Sigma _ { i n p u t }$ is included in the tape alphabet of $T$ so that $N ^ { \\prime } \\geq | \\Sigma _ { i n p u t } |$ and define the Kolmogorov complexity of $q$ with respect to $\\mathcal { U }$ to be the infimum ${ \\mathfrak { c } } ( q )$ of $M ^ { \\prime } N ^ { \\prime }$ over Turing machines $T$ that give classical solutions for $q$ . ", + "bbox": [ + 173, + 494, + 825, + 551 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Let $\\lambda$ be the RLCT of the triple $( p , q , \\varphi )$ associated to the synthesis problem (Definition 2.4). ", + "bbox": [ + 169, + 558, + 781, + 573 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 3.1. $\\begin{array} { r } { \\lambda \\le \\frac { 1 } { 2 } ( M + N ) \\mathfrak { c } ( q ) } \\end{array}$ . ", + "bbox": [ + 176, + 575, + 413, + 593 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proof. Let $u \\in W ^ { c o d e } \\cap W _ { 0 }$ be the code of a Turing machine realising the infimum in the definition of the Kolmogorov complexity and suppose that this machine only uses symbols in $\\Sigma ^ { \\prime }$ and states in $Q ^ { \\prime }$ with $N ^ { \\prime } = | \\Sigma ^ { \\prime } |$ and $\\bar { M } ^ { \\prime } = | Q ^ { \\prime } |$ . The time evolution of the staged pseudo-UTM $\\mathcal { U }$ simulating $u$ on $x \\in \\Sigma _ { i n p u t } ^ { * }$ is independent of the entries on the description tape that belong to tuples of the form $( \\sigma , q , ? , ? , ? )$ with $( \\sigma , q ) \\notin \\Sigma ^ { \\prime } \\times Q ^ { \\prime }$ . Let $V \\subseteq W$ be the submanifold of points which agree with $u$ on all tuples with $( \\sigma , q ) \\in \\Sigma ^ { \\prime } \\times Q ^ { \\prime }$ and are otherwise free. Then $u \\in V \\subseteq W _ { 0 }$ and $\\operatorname { c o d i m } ( V ) =$ $M ^ { \\prime } N ^ { \\prime } ( \\bar { M } + N )$ and by (Watanabe, 2009, Theorem 7.3) we have $\\begin{array} { r } { \\lambda \\le \\frac 1 2 \\bmod { \\mathrm { i m } } ( V ) } \\end{array}$ . □ ", + "bbox": [ + 173, + 611, + 825, + 713 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Remark 3.2. The Kolmogorov complexity depends only on the number of symbols and states used. The RLCT is a more refined invariant since it also depends on how each symbol and state is used (Clift & Murfet, 2018, Remark 7.8) as this affects the polynomials defining $W _ { 0 }$ (see Appendix D). ", + "bbox": [ + 174, + 723, + 825, + 765 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 PRACTICAL IMPLICATIONS ", + "text_level": 1, + "bbox": [ + 176, + 786, + 428, + 803 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Using singular learning theory we have explained how programs to be synthesised are singularities of analytic functions, and how the Kolmogorov complexity of a program bounds the RLCT of the associated singularity. We now sketch some practical insights that follow from this point of view. ", + "bbox": [ + 174, + 818, + 825, + 861 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Synthesis minimises the free energy: the sampling-based approach to synthesis (Section 2.1) aims to approximate, via MCMC, sampling from the Bayesian posterior for the triple $( p , q , \\varphi )$ associated to a synthesis problem. To understand the behaviour of these Markov chains we follow the asymptotic analysis of (Watanabe, 2009, Section 7.6). If we cover $W$ by small closed balls $V _ { \\alpha }$ around points $w _ { \\alpha }$ then we can compute the probability that a sample comes from $V _ { \\alpha }$ by ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 697, + 119 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/913ad1faf9c2102a1bb88c71f0e8806d6d34d68a21500305eec785971a3adc2d.jpg", + "text": "$$\np _ { \\alpha } = \\frac { 1 } { Z _ { 0 } } \\int _ { V _ { \\alpha } } e ^ { - n K _ { n } ( w ) } \\varphi ( w ) d w\n$$", + "text_format": "latex", + "bbox": [ + 388, + 126, + 607, + 161 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "and if $n$ is sufficiently large this is proportional to $e ^ { - f _ { \\alpha } }$ where the quantity ", + "bbox": [ + 173, + 170, + 661, + 185 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/177356d3ae0be560077e8d6a24d0651047330b1c13c5e35f3ed01828ee833782.jpg", + "text": "$$\nf _ { \\alpha } = K _ { \\alpha } n + \\lambda _ { \\alpha } \\log ( n )\n$$", + "text_format": "latex", + "bbox": [ + 418, + 194, + 580, + 210 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "is called the free energy. Here $K _ { \\alpha }$ is the smallest value of the Kullback-Leibler divergence $K$ on $V _ { \\alpha }$ and $\\lambda _ { \\alpha }$ is the RLCT of the set $W _ { K _ { \\alpha } } \\cap V _ { \\alpha }$ where $W _ { c } = \\{ w \\in W | K ( w ) = c \\}$ is a level set of $K$ . The Markov chains used to generate approximate samples from the posterior are attempting to minimise the free energy, which involves a tradeoff between the energy $K _ { \\alpha } n$ and the entropy $\\lambda _ { \\alpha } \\log ( n )$ . ", + "bbox": [ + 173, + 218, + 825, + 276 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Why synthesis gets stuck: the kind of local minimum of the free energy that we want the synthesis process to find are solutions $w _ { \\alpha } \\in W _ { 0 }$ where $\\lambda _ { \\alpha }$ is minimal. By Section 3 one may think of these points as the “lowest complexity” solutions. However it is possible that there are other local minima of the free energy. Indeed, there may be local minima where the free energy is lower than the free energy at any solution since at finite $n$ it is possible to tradeoff an increase in $K _ { \\alpha }$ against a decrease in the RLCT $\\lambda _ { \\alpha }$ . In practice, the existence of such “siren minima” of the free energy may manifest itself as regions where the synthesis process gets stuck and fails to converge to a solution. In such a region $\\bar { K _ { \\alpha } } n + \\lambda _ { \\alpha } \\log ( n ) < \\lambda \\log ( \\bar { n } )$ where $\\lambda$ is the RLCT of the synthesis problem. In practice it has been observed that program synthesis by gradient descent often fails for complex problems in the sense that it fails to converge to a solution (Gaunt et al., 2016). While synthesis by SGD and sampling are different, it is a reasonable hypothesis that these siren minima are a significant contributing factor in both cases. ", + "bbox": [ + 173, + 281, + 825, + 449 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Can we avoid siren minima? If we let $\\lambda _ { c }$ denote the RLCT of the level set $W _ { c }$ then siren minima of the free energy will be impossible at a given value of n and c as long as λc ≥ λ−c nlog(n) . Recall that the more singular $W _ { c }$ is the lower the RLCT, so this lower bound says that the level sets should not become too singular too quickly as $c$ increases. At any given value of $n$ there is a “siren free” region in the range $c \\geq { \\frac { \\lambda \\log ( n ) } { n } }$ since the RLCT is non-negative (Figure 3). Thus the learning process will be more reliable the smaller $\\frac { \\lambda \\log ( n ) } { n }$ is. This can arranged either by increasing $n$ (providing more examples) or decreasing $\\lambda$ . ", + "bbox": [ + 173, + 455, + 825, + 564 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "While the RLCT is determined by the synthesis problem, it is possible to change its value by changing the structure of the UTM $\\mathcal { U }$ . As we have defined it $\\mathcal { U }$ is a “simulation type” UTM, but one could for example add special states such that if a code specifies a transition into that state a series of steps is executed by the UTM (i.e. a subroutine). This amounts to specifying codes in a higher level programming language. Hence one of the practical insights that can be derived from the geometric point of view on program synthesis is that varying this language is a natural way to engineer the singularities of the level sets of $K$ , which according to singular learning theory has direct implications for the learning process. ", + "bbox": [ + 173, + 569, + 825, + 681 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/e34800be3128393d0b71b422f4ec9e848e92967fd980b5ceb119d06d14b03a68.jpg", + "image_caption": [ + "Figure 3: Level sets above the cutoff cannot contain siren local minima of the free energy. " + ], + "image_footnote": [], + "bbox": [ + 387, + 704, + 604, + 875 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 326, + 118 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We estimate the RLCT for the triples $( p , q , \\varphi )$ associated to the synthesis problems detectA (Example 2.5) and parityCheck. Hyperparameters of the various machines are contained in Table 3 of Appendix B. The true distribution $q ( x )$ is defined as follows: we fix a minimum and maximum sequence length $a \\leq b$ and to sample $x \\sim q ( x )$ we first sample a length $l$ uniformly from $[ a , b ]$ and then uniformly sample $x$ from $\\{ A , { \\cal B } \\} ^ { l }$ . ", + "bbox": [ + 173, + 132, + 825, + 204 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We perform MCMC on the weight vector for the model class $\\{ p ( y | x , w ) : w \\in W \\}$ where $w$ is represented in our PyTorch implementation by three tensors of shape $\\{ [ L , n _ { i } ] \\} _ { 1 \\leq i \\leq 3 }$ where $L$ is the number of tuples in the description tape of the TM being simulated and $\\{ n _ { i } \\}$ are the number of symbols, states and directions respectively. A direct simulation of the UTM is used for all experiments to improve computational efficiency (Appendix G). We generate, for each inverse temperature $\\beta$ and dataset $D _ { n }$ , a Markov chain via the No-U-turn sampler from Hoffman & Gelman (2014). We use the standard uniform distribution as our prior $\\varphi$ . ", + "bbox": [ + 173, + 209, + 825, + 308 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/5d4afa355e7944efc656d6fc95adefe04ce2d1b8b5d4c0468d69412c54084812.jpg", + "table_caption": [ + "Table 1: RLCT estimates for detectA. " + ], + "table_footnote": [], + "table_body": "
Max-lengthTemperatureRLCTStdR squared
7log(500)8.0892053.5247190.965384
7log(1000)6.5333622.0942780.966856
8log(500)4.6018001.1563250.974569
8log(1000)4.4316831.0690200.967847
9log(500)5.3025982.4156470.973016
9log(1000)4.0273241.8668020.958805
10log(500)3.2249101.1696990.963358
10log(1000)3.4336240.9999670.949972
", + "bbox": [ + 284, + 319, + 714, + 449 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For the problem detectA given in Example 2.5 the dimension of parameter space is dim $W = 3 0$ . We use generalized least squares to fit the RLCT $\\lambda$ (with goodness-of-fit measured by $R ^ { 2 }$ ), the algorithm of which is given in Appendix A. Our results are displayed in Table 1 and Figure 4. Our purpose in these experiments is not to provide high accuracy estimates of the RLCT, as these would require much longer Markov chains. Instead we demonstrate how rough estimates consistent with the theory can be obtained at low computational cost. If this model were regular the RLCT would be $\\dim W / 2 = 1 5$ . ", + "bbox": [ + 173, + 494, + 826, + 593 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/9ace473833bc61d56cd4eaf090be6f773656e1bf4f188e396d12fd45dcd66217.jpg", + "image_caption": [ + "Figure 4: Plot of RLCT estimates for detectA. Shaded region shows one standard deviation. " + ], + "image_footnote": [], + "bbox": [ + 302, + 608, + 691, + 818 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The deterministic synthesis problem parityCheck has ", + "bbox": [ + 176, + 867, + 553, + 882 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/586a4e6297eddaed76a38f30a3312cc424089c43682afd35d9bcb0fedb06db8a.jpg", + "text": "$$\n\\begin{array} { l } { \\Sigma = \\{ \\Pi , A , B , X \\} } \\\\ { Q = \\{ \\mathrm { r e j e c t , a c c e p t , g e t N e x t A B , g e t N e x t A , g e t N e x t B , g o t o S t a r t } \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 264, + 886, + 728, + 921 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The distribution $q ( x )$ is as discussed in Section 5 and $q ( y | x )$ is determined by the function taking in a string of $A$ ’s and $B$ ’s, and terminating in state accept if the string contains the same number of $A$ ’s as $B$ ’s, and terminating in state reject otherwise. The string is assumed to contain no blank symbols. The true distribution is realisable because there is a Turing machine using $\\Sigma$ and $Q$ which computes this function: the machine works by repeatedly overwriting pairs consisting of a single $A$ and $B$ with $X$ ’s; if there are any $A$ ’s without a matching $B$ left over (or vice versa), we reject, otherwise we accept. ", + "bbox": [ + 173, + 103, + 825, + 202 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In more detail, the starting state getNextAB moves right on the tape until the first $A$ or $B$ is found, and overwrites it with an $X$ . If it’s an $A$ (resp. $B$ ) we enter state getNextB (resp. getNextA). If no $A$ or $B$ is found, we enter the state accept. The state getNextA (resp. getNextB) moves right until an $A$ (resp. $B$ ) is found, overwrites it with an $X$ and enters state gotoStart which moves left until a blank symbol is found (resetting the machine to the left end of the tape). If no $A$ ’s (resp. $B$ ’s) were left on the tape, we enter state reject. The dimension of the parameter space is $\\dim W = 2 4 0$ . If this model were regular, the RLCT would be $\\dim W / 2 = 1 2 { \\bar { 0 } }$ . Our RLCT estimates are contained in Table 2. ", + "bbox": [ + 173, + 208, + 825, + 319 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/509b50040e6f76ba80e26f4d6f57fd36e0f2da05f22a5bcf5029ea5b32f517d6.jpg", + "table_caption": [ + "Table 2: RLCT estimates for parityCheck. " + ], + "table_footnote": [], + "table_body": "
Max-lengthTemperatureRLCTStdR squared
5log(300)4.4117320.2524580.969500
6log(300)4.0056670.3658550.971619
7log(300)3.8876790.2763370.973716
", + "bbox": [ + 282, + 329, + 714, + 397 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 DISCUSSION ", + "text_level": 1, + "bbox": [ + 174, + 459, + 310, + 476 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We have developed a theoretical framework in which all programs can in principle be learnt from input-output examples via an existing optimisation procedure. This is done by associating to each program a smooth relaxation which, based on Clift & Murfet (2018), can be argued to be more canonical than existing approaches. This realization has important implications for the building of intelligent systems. ", + "bbox": [ + 174, + 491, + 825, + 560 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In approaches to program synthesis based on gradient descent there is a tendency to think of solutions to the synthesis problem as isolated critical points of the loss function $K$ , but this is a false intuition based on regular models. Since neural networks, Bayesian networks, smooth relaxations of UTMs and all other extant approaches to smooth program synthesis are strictly singular models (the map from parameters to functions is not injective) the set $W _ { 0 }$ of parameters $w$ with $K ( w ) = 0$ is a complex extended object, whose geometry is shown by Watanabe’s singular learning theory to be deeply related to the learning process. We have examined this geometry in several specific examples and shown how to think about complexity of programs from a geometric perspective. It is our hope that algebraic geometry can assist in developing the next generation of synthesis machines. ", + "bbox": [ + 173, + 568, + 825, + 693 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 103, + 287, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Shun-ichi Amari, Tomoko Ozeki, and Hyeyoung Park. 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", + "bbox": [ + 173, + 363, + 823, + 406 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "APPENDIX ", + "text_level": 1, + "bbox": [ + 174, + 435, + 264, + 450 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A ALGORITHM FOR ESTIMATING RLCTS ", + "text_level": 1, + "bbox": [ + 174, + 468, + 534, + 486 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Given a sample $D _ { n } = \\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { n }$ from $q ( x , y )$ let $\\begin{array} { r } { L _ { n } ( w ) : = - \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\log p ( y _ { i } | x _ { i } , w ) } \\end{array}$ be the negative log likelihood. We would like to estimate ", + "bbox": [ + 173, + 498, + 823, + 530 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/8d8e755703fb165a62e5c1ade202f2a6560bac00263a0d9806155d8da28ae7d4.jpg", + "text": "$$\n\\mathbb { E } _ { w } ^ { \\beta } [ n L _ { n } ( w ) ] : = \\frac { 1 } { Z _ { n } ^ { \\beta } } \\int n L _ { n } ( w ) \\varphi ( w ) \\prod _ { i = 1 } ^ { n } p ( y _ { i } | x _ { i } , w ) ^ { \\beta } d w\n$$", + "text_format": "latex", + "bbox": [ + 307, + 536, + 691, + 578 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { Z _ { n } ^ { \\beta } = \\int \\varphi ( w ) \\prod _ { i = 1 } ^ { n } p ( y _ { i } | x _ { i } , w ) ^ { \\beta } d w } \\end{array}$ for some inverse temperature $\\beta$ . If $\\begin{array} { r } { \\beta = \\frac { \\beta _ { 0 } } { \\log n } } \\end{array}$ for some constant $\\beta _ { 0 }$ , then by Theorem 4 of Watanabe (2013), ", + "bbox": [ + 173, + 585, + 825, + 619 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/9ac51b44fb76b345f8e352ce537dd93abee705ef95c94315ed301762c547332a.jpg", + "text": "$$\n\\mathbb { E } _ { w } ^ { \\beta } [ n L _ { n } ( w ) ] = n L _ { n } ( w _ { 0 } ) + \\frac { \\lambda \\log n } { \\beta _ { 0 } } + U _ { n } \\sqrt { \\frac { \\lambda \\log n } { 2 \\beta _ { 0 } } } + O _ { p } ( 1 )\n$$", + "text_format": "latex", + "bbox": [ + 294, + 626, + 704, + 667 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "where $\\{ U _ { n } \\}$ is a sequence of random variables satisfying $\\mathbb { E } [ U _ { n } ] = 0$ and $\\lambda$ is the RLCT. In practice, the last two terms often vary negligibly with $1 / \\beta$ and so $\\mathbb { E } _ { w } ^ { \\beta } [ n L _ { n } ( w ) ]$ approximates a linear function of $1 / \\beta$ with slope $\\lambda$ (Watanabe, 2013, Corollary 3). This is the foundation of the RLCT estimation procedure found in Algorithm 1 which is used in our experiments. ", + "bbox": [ + 173, + 674, + 825, + 732 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Algorithm 1 RLCT estimation ", + "text_level": 1, + "bbox": [ + 173, + 747, + 379, + 762 + ], + "page_idx": 10 + }, + { + "type": "table", + "img_path": "images/ee0ac6996d3e6c803b6a9bed0ca9815b2389ef7b2a1ac8ca1f2d1035dde64725.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Input: range of β's, set of training sets T each of size n, approximate samples {w1,..,WR} from pβ(w|Dn) for each training set Dn and each β
for training set Dn ∈ T do
for β in range of β's do
ples from pβ(w|Dn)
end for
Perform generalised least squares to fit X in Equation (7),call result λ(Dn)
end for
Output: ∑Dn∈T λ(Dn)
", + "bbox": [ + 184, + 766, + 825, + 920 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Each RLCT estimate $\\hat { \\lambda } ( \\mathcal { D } _ { n } )$ in Algorithm 1 was performed by linear regression on the pairs $\\{ ( 1 / \\beta _ { i } , \\mathbb { E } _ { w } ^ { \\beta _ { i } } [ n L _ { n } ( w ) ] ) \\} _ { i = 1 } ^ { 5 }$ where the five inverse temperatures $\\beta _ { i }$ are centered on the inverse temperature $1 / T$ where $T$ is the temperature reported for each experiment in Table 1 and Table 2. ", + "bbox": [ + 174, + 102, + 825, + 146 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "From a Bayesian perspective, predictions about outputs $y$ should be made using the predictive distribution ", + "bbox": [ + 171, + 152, + 823, + 179 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/e6ef740e0ca909d79bde3a4cbe0efecca74a6c9a2e0f2c9a5a1a39b435ba7bd5.jpg", + "text": "$$\np ^ { * } ( y | x , D _ { n } ) = \\int p ( y | x , w ) p ( w | D _ { n } ) d w .\n$$", + "text_format": "latex", + "bbox": [ + 361, + 176, + 635, + 208 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The Bayesian generalisation error associated to the Bayesian predictor is defined as the KullbackLeibler distance to the true conditional distribution ", + "bbox": [ + 176, + 208, + 820, + 236 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/d9361ae57a73f5e53a4489d10f0d66d21fc543139e815e7f64104eb404905b0d.jpg", + "text": "$$\nB _ { g } ( n ) : = D _ { K L } ( q \\| p ^ { * } ) = \\int q ( y | x ) q ( x ) \\log \\left( { \\frac { q ( y | x ) } { p ^ { * } ( y | x ) } } \\right) d y d x .\n$$", + "text_format": "latex", + "bbox": [ + 292, + 236, + 705, + 270 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "If some fundamental conditions are satisfied (Definition 6.1 and Definition 6.3 of Watanabe (2009)), then by Theorem 6.8 of loc.cit., there exists a random variable $B _ { g } ^ { * }$ such that as $n \\to \\infty$ , $\\mathbb { E } [ n B _ { g } ( n ) ]$ converges to $\\mathbb { E } [ B _ { g } ^ { * } ]$ . In particular, by Theorem 6.10 of Watanabe (2009), $\\mathbb { E } [ B _ { g } ^ { * } ] = \\lambda$ . ", + "bbox": [ + 173, + 277, + 825, + 323 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B HYPERPARAMETERS ", + "text_level": 1, + "bbox": [ + 176, + 340, + 380, + 357 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The hyperparameters for the various synthesis tasks are contained in Table 3. The number of samples is $R$ in Algorithm 1 and the number of datasets is $| \\tau |$ . Samples are taken according to the Dirichlet distribution, a probability distribution over the simplex, which is controlled by the concentration. When the concentration is a constant across all dimensions, as is assumed here, this corresponds to a density which is symmetric about the uniform probability mass function occurring in the centre of the simplex. The value $\\alpha = 1 . 0$ corresponds to the uniform distribution over the simplex. Finally, the chain temperature controls the default $\\beta$ value, ie. all inverse temperature values are centered around $1 / T$ where $T$ is the chain temperature. ", + "bbox": [ + 173, + 371, + 826, + 483 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/043b4a973caea9a964f76b8b4143b500dd1b30d16ecb6a1c76a44bbfb3876c39.jpg", + "table_caption": [ + "Table 3: Hyperparameters for Datasets and MCMC. " + ], + "table_footnote": [], + "table_body": "
HyperparameterdetectAparityCheck
Dataset size (n)200100
Minimum sequence length (a)41
Maximum sequence length (b)7/8/9/105/6/7
Number of samples (R)20.0002.000
Number of burn-in steps1,000500
Number of datasets (|T|)43
Target accept probability0.80.8
Concentration (α)1.01.0
Chain temperature (T)log(500)/log(1000)log(300)
Number of timesteps (t)1042
", + "bbox": [ + 267, + 494, + 728, + 648 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "C THE SHIFT MACHINE ", + "text_level": 1, + "bbox": [ + 174, + 705, + 388, + 722 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The pseudo-UTM $\\mathcal { U }$ is a complicated Turing machine, and the models $p ( \\boldsymbol { y } | \\boldsymbol { x } , \\boldsymbol { w } )$ of Section 2 are therefore not easy to analyse by hand. To illustrate the kind of geometry that appears, we study the simple Turing machine shiftMachine of Clift & Murfet (2018) and formulate an associated statistical learning problem. The tape alphabet is $\\Sigma = \\{ \\boxed { \\begin{array} { r l } \\end{array} } , A , B , 0 , 1 , 2 \\}$ and the input to the machine will be a string of the form $\\boxed { 1 } n a _ { 1 } a _ { 2 } a _ { 3 } \\boxed { 2 }$ where $n$ is called the counter and $\\bar { a _ { i } } \\in \\{ A , B \\}$ . The transition function, given in loc.cit., will move the string of $A$ ’s and $B$ ’s leftwards by $n$ steps and fill the right hand end of the string with $A$ ’s, keeping the string length invariant. For example, if $\\square 2 B A B \\square$ is the input to $M$ , the output will be $\\square 0 B A A \\square$ . ", + "bbox": [ + 173, + 736, + 825, + 849 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Set $W = \\Delta \\{ 0 , 2 \\} \\times \\Delta \\{ A , B \\}$ and view $w = ( h , k ) \\in W$ as representing a probability distribution $( 1 - h ) \\cdot 0 + h \\cdot 2$ for the counter and $( 1 - k ) \\cdot B + k \\cdot A$ for $a _ { 1 }$ . The model is ", + "bbox": [ + 171, + 854, + 823, + 883 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/655d3d051a0208c4d5adc9a91bb8bf32c20a7f77e5857efbc71e978138af1849.jpg", + "text": "$$\np \\big ( \\boldsymbol { y } | \\boldsymbol { x } = ( a _ { 2 } , a _ { 3 } ) , \\boldsymbol { w } \\big ) = ( 1 - h ) ^ { 2 } \\boldsymbol { k } \\cdot \\boldsymbol { A } + ( 1 - h ) ^ { 2 } ( 1 - \\boldsymbol { k } ) \\cdot \\boldsymbol { B } + \\sum _ { i = 2 } ^ { 3 } \\binom { 2 } { i - 1 } h ^ { i - 1 } ( 1 - h ) ^ { 3 - i } \\cdot a _ { i } .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 886, + 820, + 928 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "This model is derived by propagating uncertainty through shiftMachine in the same way that $p ( \\boldsymbol { y } | \\boldsymbol { x } , \\boldsymbol { w } )$ is derived from $\\mathrm { \\Delta } \\mathrm { { s t e p } } ^ { t }$ in Section 2 by propagating uncertainty through $\\mathcal { U }$ . We assume that some distribution $q ( x )$ over $\\{ A , B \\} ^ { 2 }$ is given. ", + "bbox": [ + 174, + 103, + 825, + 146 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Example C.1. Suppose $q ( y | x ) = p ( y | x , w _ { 0 } )$ where $w _ { 0 } = ( 1 , 1 )$ . It is easy to see that ", + "bbox": [ + 174, + 148, + 730, + 166 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/b29a76eb679442a67367c43a891a13bfa9eff320e573767248836bc96aa07018.jpg", + "text": "$$\nK ( w ) = - { \\frac { 1 } { 4 } } \\sum _ { a _ { 2 } , a _ { 3 } } \\log p \\big ( y = a _ { 3 } | x = ( a _ { 2 } , a _ { 3 } ) , w \\big ) = - { \\frac { 1 } { 2 } } \\log [ g ( h , k ) ]\n$$", + "text_format": "latex", + "bbox": [ + 272, + 170, + 725, + 208 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where $g ( h , k ) = \\left( ( 1 - h ) ^ { 2 } k + h ^ { 2 } \\right) \\left( ( 1 - h ) ^ { 2 } ( 1 - k ) + h ^ { 2 } \\right)$ is a polynomial in $w$ . Hence ", + "bbox": [ + 169, + 213, + 754, + 233 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/908fcb4fd665de455f62bb90a7952c7ac7c4614cac4247c979470c2625d932ef.jpg", + "text": "$$\nW _ { 0 } = \\{ ( h , k ) \\in W : g ( h , k ) = 1 \\} = \\mathbb { V } ( g - 1 ) \\cap [ 0 , 1 ] ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 312, + 238, + 684, + 257 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "is a semi-algebraic variety, that is, it is defined by polynomial equations and inequalities. Here $\\mathbb { V } ( h )$ denotes the vanishing locus of a function $h$ . ", + "bbox": [ + 173, + 262, + 821, + 291 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Example C.2. Suppose $q ( A B ) = 1$ and $\\begin{array} { r } { q ( y | x = A B ) = \\frac { 1 } { 2 } A + \\frac { 1 } { 2 } B } \\end{array}$ . Then the Kullback-Leibler divergence is $\\begin{array} { r } { K ( h , k ) = - \\frac { 1 } { 2 } \\log ( 4 f ( 1 - f ) ) } \\end{array}$ where $f = ( 1 - h ) ^ { 2 } k + 2 h ( 1 - h )$ . Hence $\\nabla K =$ $\\begin{array} { r } { ( f - \\frac { 1 } { 2 } ) \\frac { 1 } { f ( 1 - f ) } \\nabla f . } \\end{array}$ . Note that $f$ has no critical points, and so $\\nabla K = 0$ at $( h , k ) \\in ( 0 , 1 ) ^ { 2 }$ if and only if $\\begin{array} { r } { f ( h , k ) = \\frac { 1 } { 2 } } \\end{array}$ . Since $K$ is non-negative, any $w \\in W _ { 0 }$ satisfies $\\nabla K ( w ) = 0$ and so ", + "bbox": [ + 173, + 292, + 825, + 361 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/aa5a8ee57b6abc3254fa01e12c98c50d16b3d6e995ffe42d683b1e144830794e.jpg", + "text": "$$\nW _ { 0 } = [ 0 , 1 ] ^ { 2 } \\cap \\mathbb { V } ( 4 f ( 1 - f ) - 1 ) = [ 0 , 1 ] ^ { 2 } \\cap \\mathbb { V } ( f - \\frac { 1 } { 2 } )\n$$", + "text_format": "latex", + "bbox": [ + 313, + 367, + 684, + 387 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "is semi-algebraic. Note that the curve $\\begin{array} { r } { f = \\frac { 1 } { 2 } } \\end{array}$ is regular while the curve $4 f ( 1 - f ) = 1$ is singular and it is the geometry of the singular curve that is related to the behaviour of $K$ . This curve is shown in Figure 5. It is straightforward to check that the determinant of the Hessian of $K$ is identically zero on $W _ { 0 }$ , so that every point on $W _ { 0 }$ is a degenerate critical point of $K$ . ", + "bbox": [ + 173, + 393, + 825, + 450 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/6e526ca2a8223c4c58975329dac1b2911fc337af16f18add2a573fa4cdb23bdf.jpg", + "image_caption": [ + "Figure 5: Values of $K ( h , k )$ on $[ 0 , 1 ] ^ { 2 }$ are shown by colour, ranging from blue (zero) to red (0.01). The singular analytic space $K = 0$ (white) and the regular analytic level set $K = 0 . 0 0 1$ (black). " + ], + "image_footnote": [], + "bbox": [ + 369, + 469, + 627, + 630 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "D GENERAL SOLUTION FOR DETERMINISTIC SYNTHESIS PROBLEMS ", + "text_level": 1, + "bbox": [ + 169, + 705, + 753, + 722 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In this section we consider the case of a deterministic synthesis problem $q ( x , y )$ which is finitely supported in the sense that there exists a finite set $\\mathcal { X } \\subseteq \\Sigma ^ { * }$ such that $q ( x ) = c$ for all $x \\in \\mathcal { X }$ and $q ( x ) = 0$ for all $x \\notin \\mathcal { X }$ . We first need to discuss the coordinates on the parameter space $W$ of (3). To specify a point on $W$ is to specify for each pair $( \\sigma , q ) \\in \\Sigma \\times Q$ (that is, for each tuple on the description tape) a triple of probability distributions ", + "bbox": [ + 173, + 736, + 825, + 808 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/ceb6d53c80c07c8e14f9f68787c1b51e746104e01a46b307617df70297373cda.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\sum _ { \\sigma ^ { \\prime } \\in Q } x _ { \\sigma ^ { \\prime } } ^ { \\sigma , q } \\cdot \\sigma ^ { \\prime } \\in \\Delta \\Sigma , } \\\\ & { \\displaystyle \\sum _ { q ^ { \\prime } \\in Q } y _ { q ^ { \\prime } } ^ { \\sigma , q } \\cdot q ^ { \\prime } \\in \\Delta Q , } \\\\ & { \\displaystyle \\sum _ { d \\in \\{ L , S , R \\} } z _ { d } ^ { \\sigma , q } \\cdot d \\in \\Delta \\{ L , S , R \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 385, + 814, + 611, + 921 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The space $W$ of distributions is therefore contained in the affine space with coordinate ring ", + "bbox": [ + 171, + 102, + 771, + 118 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/fbbe5794dd1a62728877fbddd627af9ffe4e67284551ae2834cdd6d641da9325.jpg", + "text": "$$\nR _ { W } = \\mathbb { R } \\big [ \\big \\{ x _ { \\sigma ^ { \\prime } } ^ { \\sigma , q } \\big \\} _ { \\sigma , q , \\sigma ^ { \\prime } } , \\big \\{ y _ { q ^ { \\prime } } ^ { \\sigma , q } \\big \\} _ { \\sigma , q , q ^ { \\prime } } , \\big \\{ z _ { d } ^ { \\sigma , q } \\big \\} _ { \\sigma , q , d } \\big ] .\n$$", + "text_format": "latex", + "bbox": [ + 331, + 122, + 665, + 145 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The function $F ^ { x } = \\Delta \\mathrm { s t e p } ^ { t } ( x , - ) : W \\longrightarrow \\Delta Q$ is polynomial (Clift & Murfet, 2018, Proposition 4.2) and we denote for $s \\in Q$ by $F _ { s } ^ { x } \\in R _ { W }$ the polynomial computing the associated component of the function $F ^ { x }$ . Let $\\partial W$ denote the boundary of the manifold with corners $W$ , that is, the set of all points on $W$ where at least one of the coordinate functions given above vanishes ", + "bbox": [ + 173, + 151, + 826, + 208 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/b5c6037fbacc5bc1bbefefebfe01f4d2a4b1883c4c987bdfc8e6ce7dbda9fe9b.jpg", + "text": "$$\n\\partial W = \\mathbb { V } \\big ( \\prod _ { \\sigma , q } \\Big [ \\prod _ { \\sigma ^ { \\prime } \\in Q } x _ { \\sigma ^ { \\prime } } ^ { \\sigma , q } \\prod _ { q ^ { \\prime } \\in Q } y _ { q ^ { \\prime } } ^ { \\sigma , q } \\prod _ { \\substack { d \\in \\{ L , S , R \\} } } z _ { d } ^ { \\sigma , q } \\Big ] \\big )\n$$", + "text_format": "latex", + "bbox": [ + 326, + 213, + 671, + 252 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $\\mathbb { V } ( h )$ denotes the vanishing locus of $h$ . ", + "bbox": [ + 176, + 258, + 472, + 273 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Lemma D.1. $W _ { 0 } \\neq W$ ", + "bbox": [ + 174, + 276, + 334, + 292 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof. Choose $x \\in \\mathcal { X }$ with $q ( x ) > 0$ and let $y$ be such that $q ( y | x ) = 1$ . Let $w \\in W ^ { c o d e }$ be the code for the Turing machine which ignores the symbol under the head and current state, transitions to some fixed state $s \\neq y$ and stays. Then $w \\not \\in W _ { 0 }$ . □ ", + "bbox": [ + 173, + 306, + 825, + 351 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Lemma D.2. The set $W _ { 0 }$ is semi-algebraic and $W _ { 0 } \\subseteq \\partial W$ . ", + "bbox": [ + 173, + 357, + 566, + 372 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof. Given $x \\in \\Sigma ^ { * }$ with $q ( x ) > 0$ we write $y = y ( x )$ for the unique state with $q ( x , y ) \\neq 0$ . In this notation the Kullback-Leibler divergence is ", + "bbox": [ + 173, + 387, + 825, + 416 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/0c762935a362819fdbd7538cdfac2d59d015b87dc485620b7b2c813c4c593257.jpg", + "text": "$$\nK ( w ) = \\sum _ { x \\in \\mathcal { X } } c D _ { K L } ( y | | F ^ { x } ( w ) ) = - c \\sum _ { x \\in \\mathcal { X } } \\log F _ { y } ^ { x } ( w ) = - c \\log \\prod _ { x \\in \\mathcal { X } } F _ { y } ^ { x } ( w ) .\n$$", + "text_format": "latex", + "bbox": [ + 243, + 422, + 754, + 457 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Hence ", + "bbox": [ + 173, + 462, + 218, + 477 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/2c5fd9c205eb37e6c4fab6c9fb80af46c80b3cb65c7707540728e36317f80812.jpg", + "text": "$$\nW _ { 0 } = W \\cap \\bigcap _ { x \\in \\mathcal { X } } \\mathbb { V } ( 1 - F _ { y } ^ { x } ( w ) )\n$$", + "text_format": "latex", + "bbox": [ + 392, + 479, + 606, + 513 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "is semi-algebraic. ", + "bbox": [ + 173, + 520, + 290, + 534 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Recall that the function $\\Delta \\mathrm { s t e p } ^ { t }$ is associated to an encoding of the UTM in linear logic by the Sweedler semantics (Clift & Murfet, 2018) and the particular polynomials involved have a form that is determined by the details of that encoding (Clift & Murfet, 2018, Proposition 4.3). From the design of our UTM we obtain positive integers $l _ { \\sigma } , m _ { q } , n _ { d }$ for $\\sigma \\in \\Sigma , q \\in \\bar { Q } , d \\in \\{ L , S , R \\}$ and a function $\\pi : \\Theta \\longrightarrow Q$ where ", + "bbox": [ + 173, + 540, + 826, + 611 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/0bfcf940add0b17859f0e5c8051c2ad0b5850a532acaaffd78bec37b94873ffe.jpg", + "text": "$$\n\\Theta = \\prod _ { \\sigma , q } \\Sigma ^ { l _ { \\sigma } } \\times Q ^ { m _ { q } } \\times \\{ L , S , R \\} ^ { n _ { d } } .\n$$", + "text_format": "latex", + "bbox": [ + 375, + 616, + 620, + 651 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We represent elements of $\\Theta$ by tuples $( \\mu , \\zeta , \\xi ) \\in \\Theta$ where $\\mu ( \\sigma , q , i ) \\in \\Sigma$ for $\\sigma \\in \\Sigma , q \\in Q$ and $1 \\leq i \\leq l _ { \\sigma }$ and similarly $\\zeta ( \\sigma , q , j ) \\in Q$ and $\\xi ( \\sigma , q , k ) \\in \\{ L , S , R \\}$ . The polynomial $F _ { s } ^ { x }$ is ", + "bbox": [ + 173, + 656, + 823, + 688 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/ec7f164df26b9d18dca289720af09fe60bfe4d716f62bb0524650935eca9bf35.jpg", + "text": "$$\nF _ { s } ^ { x } = \\sum _ { ( \\mu , \\zeta , \\xi ) \\in \\Theta } \\delta ( s = \\pi ( \\mu , \\zeta , \\xi ) ) \\prod _ { \\sigma , q } \\Big [ \\prod _ { i = 1 } ^ { l _ { \\sigma } } x _ { \\mu ( \\sigma , q , i ) } ^ { \\sigma , q } \\prod _ { j = 1 } ^ { m _ { q } } y _ { \\zeta ( \\sigma , q , j ) } ^ { \\sigma , q } \\prod _ { k = 1 } ^ { n _ { d } } z _ { \\xi ( \\sigma , q , k ) } ^ { \\sigma , q } \\Big ]\n$$", + "text_format": "latex", + "bbox": [ + 245, + 693, + 753, + 739 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $\\delta$ is a Kronecker delta. With this in hand we may compute ", + "bbox": [ + 174, + 744, + 601, + 760 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/ac771f11c67d3891be5cece10a275aa657d6e1044f6dc0a9a7feef0fd986cedf.jpg", + "text": "$$\n\\begin{array} { r } { W _ { 0 } = W \\cap \\displaystyle \\bigcap _ { x \\in \\mathcal { X } } \\mathbb { V } ( 1 - F _ { y } ^ { x } ( w ) ) } \\\\ { = W \\cap \\displaystyle \\bigcap _ { x \\in \\mathcal { X } } \\bigcap _ { s \\neq y } \\mathbb { V } ( F _ { s } ^ { x } ( w ) ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 390, + 766, + 607, + 837 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "But $F _ { s } ^ { x }$ is a polynomial with non-negative integer coefficients, which takes values in $[ 0 , 1 ]$ for $w \\in$ $W$ . Hence it vanishes on $w$ if and only if for each triple $\\mu , \\zeta , \\xi$ with $s = \\pi ( \\mu , \\zeta , \\xi )$ one or more of the coordinate functions xσ,qµ(σ,q,i), yσ,qζ(σ,q,j), zσ,qξ(σ,q,k) vanishes on $w$ . ", + "bbox": [ + 173, + 843, + 826, + 890 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The desired conclusion follows unless for every $x \\in \\mathcal { X }$ and $( \\mu , \\zeta , \\xi ) \\in \\Theta$ we have $\\pi ( \\mu , \\zeta , \\xi ) = y$ so that $F _ { s } ^ { x } = 0$ for all $s \\neq y$ . But in this case case $W _ { 0 } = W$ which contradicts Lemma D.1. □ ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "E STAGED PSEUDO-UTM ", + "text_level": 1, + "bbox": [ + 176, + 101, + 403, + 118 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Simulating a Turing machine $M$ with tape alphabet $\\Sigma$ and set of states $Q$ on a standard UTM requires the specification of an encoding of $\\Sigma$ and $Q$ in the tape alphabet of the UTM. From the point of view of exploring the geometry of program synthesis, this additional complexity is uninteresting and so here we consider a staged pseudo-UTM whose alphabet is ", + "bbox": [ + 173, + 132, + 825, + 188 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/47f784d767f57b579cb9a41c8e56f298fa8eea94a0fa855856aa2e9b613efbd7.jpg", + "text": "$$\n\\Sigma _ { \\mathrm { U T M } } = \\Sigma \\cup Q \\cup \\{ L , R , S \\} \\cup \\{ X , \\sqsubseteq \\}\n$$", + "text_format": "latex", + "bbox": [ + 369, + 191, + 627, + 208 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where the union is disjoint where $\\boxed { \\begin{array} { r l } \\end{array} }$ is the blank symbol (which is distinct from the blank symbol of $M$ ). Such a machine is capable of simulating any machine with tape alphabet $\\Sigma$ and set of states $Q$ but cannot simulate arbitrary machines and is not a UTM in the standard sense. The adjective staged refers to the design of the UTM, which we now explain. The set of states is ", + "bbox": [ + 176, + 210, + 825, + 266 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/b3a2fa940172cc778c9b8e4ee80141b2b78103b5513ade8d54750f3baf6c4d72.jpg", + "text": "$$\n\\begin{array} { r } { Q _ { \\mathrm { U T M } } = \\{ \\mathrm { c o m p S y m b o l , c o m p S t a t e , c o p y S y m b o l , c o p y S t a t e , c o p y } \\mathrm { ~ } \\forall \\mathrm { t o p y ~ } \\mathrm { ~ c o p y ~ } \\mathrm { ~ c o p y ~ } \\mathrm { ~ c o p y ~ } \\mathrm { ~ c o p y ~ } \\mathrm { ~ c o p y ~ } \\mathrm { ~ c o p y ~ } \\mathrm { ~ c o m p ~ } } \\\\ \\mathrm { ~ \\ \" c o m p S t a t e , ~ } \\mathrm { \\ \" { c o p y S y m b o l , } \\mathrm { ~ \\ \" { c o p y S t a t e , } \\mathrm { ~ - c o p y S t a t e , } \\mathrm { ~ - c o p y D i r , } \\mathrm { ~ \\ ~ } } } \\\\ \\mathrm { ~ \\ \" { u p d a t e S y m b o l , u p d a t e S t a t e , u p d a t e D i r , r e s e t D e s c r ~ } \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 264, + 267, + 694, + 323 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The UTM has four tapes numbered from 0 to 3, which we refer to as the description tape, the staging tape, the state tape and the working tape respectively. Initially the description tape contains a string of the form ", + "bbox": [ + 174, + 323, + 821, + 363 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/e20513688d9121bee8af67f0f13a1b91614ecd3c7df2358faec9690fbf851037.jpg", + "text": "$$\nX s _ { 0 } q _ { 0 } s _ { 0 } ^ { \\prime } q _ { 0 } ^ { \\prime } d _ { 0 } s _ { 1 } q _ { 1 } s _ { 1 } ^ { \\prime } q _ { 1 } ^ { \\prime } d _ { 1 } \\dots s _ { N } q _ { N } s _ { N } ^ { \\prime } q _ { N } ^ { \\prime } d _ { N } X ,\n$$", + "text_format": "latex", + "bbox": [ + 343, + 362, + 651, + 380 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "corresponding to the tuples which define $M$ , with the tape head initially on $s _ { 0 }$ . The staging tape is initially a string $X X X$ with the tape head over the second $X$ . The state tape has a single square containing some distribution in $\\Delta Q$ , corresponding to the initial state of the simulated machine $M$ , with the tape head over that square. Each square on the the working tape is some distribution in $\\Delta \\Sigma$ with only finitely many distributions different from $\\boxed { \\begin{array} { r l } \\end{array} }$ . The UTM is initialized in state compSymbol. ", + "bbox": [ + 174, + 381, + 825, + 450 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The operation of the UTM is outlined in Figure 6. It consists of two phases; the scan phase (middle and right path), and the update phase (left path). During the scan phase, the description tape is scanned from left to right, and the first two squares of each tuple are compared to the contents of the working tape and state tape respectively. If both agree, then the last three symbols of the tuple are written to the staging tape (middle path), otherwise the tuple is ignored (right path). Once the $X$ at the end of the description tape is reached, the UTM begins the update phase, wherein the three symbols on the staging tape are then used to print the new symbol on the working tape, to update the simulated state on the state tape, and to move the working tape head in the appropriate direction. The tape head on the description tape is then reset to the initial $X$ . ", + "bbox": [ + 173, + 455, + 825, + 583 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Remark E.1. One could imagine a variant of the UTM which did not include a staging tape, instead performing the actions on the work and state tape directly upon reading the appropriate tuple on the description tape. However, this is problematic when the contents of the state or working tape are distributions, as the exact time-step of the simulated machine can become unsynchronised, increasing entropy. As a simple example, suppose that the contents of the state tape were $0 . 5 q + 0 . 5 p$ , and the symbol under the working tape head was $s$ . Upon encountering the tuple $s q s { ' } q { ' } R$ , the machine would enter a superposition of states corresponding to the tape head having both moved right and not moved, complicating the future behaviour. ", + "bbox": [ + 173, + 584, + 825, + 696 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We define the period of the UTM to be the smallest nonzero time interval taken for the tape head on the description tape to return to the initial $X$ , and the machine to reenter the state compSymbol. If the number of tuples on the description tape is $N$ , then the period of the UTM is $T = 1 0 N + 5$ . Moreover, other than the working tape, the position of the tape heads are $T$ -periodic. ", + "bbox": [ + 174, + 705, + 825, + 762 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "F SMOOTH TURING MACHINES ", + "text_level": 1, + "bbox": [ + 176, + 781, + 449, + 797 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Let $\\mathcal { U }$ be the staged pseudo-UTM of Appendix E. In defining the model $p ( \\boldsymbol { y } | \\boldsymbol { x } , \\boldsymbol { w } )$ associated to a synthesis problem in Section 2 we use a smooth relaxation $\\Delta \\mathrm { { s t e p } } ^ { t }$ of the step function of $\\mathcal { U }$ . In this appendix we define the smooth relaxation of any Turing machine following Clift & Murfet (2018). ", + "bbox": [ + 174, + 811, + 825, + 854 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Let $M = ( \\Sigma , Q , \\delta )$ be a Turing machine with a finite set of symbols $\\Sigma$ , a finite set of states $Q$ and transition function $\\delta : \\Sigma \\times Q \\bar { \\to } \\Sigma \\times Q \\times \\{ - 1 , 0 , 1 \\}$ . We write $\\delta _ { i } = \\mathsf { p r o j } _ { i } \\circ \\delta$ for the $i$ th component of $\\delta$ for $i \\in \\{ 1 , 2 , 3 \\}$ . For $\\sqsubseteq \\Sigma$ , let ", + "bbox": [ + 174, + 861, + 825, + 904 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/f3487b835791ef405aca7831e8d5f0b285bc6d2fd3410dde01e3985b09ed852f.jpg", + "text": "$$\n\\Sigma ^ { \\mathbb { Z } , \\sqcap } = \\{ f : \\mathbb { Z } \\to \\Sigma | f ( i ) = \\bigsqcup \\mathrm { e x c e p t ~ f o r ~ f i n i t e l y ~ m a n y ~ } i \\} .\n$$", + "text_format": "latex", + "bbox": [ + 305, + 907, + 692, + 926 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/8439277769c907e73a890f5d22819ba3dcc4dc5cc4f53963fd296742f6e34336.jpg", + "image_caption": [ + "Figure 6: The UTM. Each of the rectangles are states, and an arrow $q q ^ { \\prime }$ has the following interpretation: if the UTM is in state $q$ and sees the tape symbols (on the four tapes) as indicated by the source of the arrow, then the UTM transitions to state $q ^ { \\prime }$ , writes the indicated symbols (or if there is no write instruction, simply rewrites the same symbols back onto the tapes), and performs the indicated movements of each of the tape heads. The symbols $a , b , c , d$ stand for generic symbols which are not $X$ . " + ], + "image_footnote": [], + "bbox": [ + 245, + 113, + 727, + 510 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We can associate to $M$ a discrete dynamical system ${ \\widehat { M } } = ( \\Sigma ^ { \\mathbb { Z } , \\sqcup } \\times Q , { \\mathrm { s t e } } ]$ p) where ", + "bbox": [ + 173, + 637, + 714, + 656 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/f9a01a1b4dad6f8e9e15d64bf32f27b15e1db14b68a0e6e3eadcfef197c2583b.jpg", + "text": "$$\n{ \\mathrm { s t e p } } : \\Sigma ^ { \\mathbb { Z } , \\sqcap } \\times Q \\Sigma ^ { \\mathbb { Z } , \\sqcap } \\times Q\n$$", + "text_format": "latex", + "bbox": [ + 397, + 661, + 601, + 681 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "is the step function defined by ", + "bbox": [ + 173, + 689, + 370, + 704 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/8266cb583c44e5283a73e1c246870b772e20f8b416a9fea405703fa70848f576.jpg", + "text": "$$\n\\mathrm { s t e p } ( \\sigma , q ) = \\Bigl ( \\alpha ^ { \\delta _ { 3 } ( \\sigma _ { 0 } , q ) } \\bigl ( \\ldots , \\sigma _ { - 2 } , \\sigma _ { - 1 } , \\delta _ { 1 } ( \\sigma _ { 0 } , q ) , \\sigma _ { 1 } , \\sigma _ { 2 } , \\ldots \\bigr ) , \\delta _ { 2 } ( \\sigma _ { 0 } , q ) \\Bigr ) .\n$$", + "text_format": "latex", + "bbox": [ + 254, + 710, + 740, + 738 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "with shift map $\\alpha ^ { \\delta _ { 3 } ( \\sigma _ { 0 } , q ) } ( \\sigma ) _ { u } = \\sigma _ { u + \\delta _ { 3 } ( \\sigma _ { 0 } , q ) } .$ ", + "bbox": [ + 173, + 746, + 464, + 763 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Let $X$ be a finite set. The standard $X$ -simplex is defined as ", + "bbox": [ + 173, + 768, + 562, + 784 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/fb04d8564aef14526f7ee251a3ac6b1a31381e38c4a965b7257ff3b29a6987bc.jpg", + "text": "$$\n\\Delta X = \\{ \\sum _ { x \\in X } \\lambda _ { x } x \\in \\mathbb { R } X | \\sum _ { x } \\lambda _ { x } = 1 { \\mathrm { a n d } } \\lambda _ { x } \\geq 0 { \\mathrm { f o r ~ a l l } } x \\in X \\}\n$$", + "text_format": "latex", + "bbox": [ + 287, + 790, + 710, + 824 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $\\mathbb { R } X$ is the free vector space on $X$ . We often identify $X$ with the vertices of $\\Delta X$ under the canonical inclusion $i : X \\to \\Delta X$ given by $\\begin{array} { r } { i ( x ) = \\sum _ { x ^ { \\prime } \\in X } \\delta _ { x = x ^ { \\prime } } x ^ { \\prime } } \\end{array}$ . For example $\\{ 0 , 1 \\} \\subset$ $\\Delta ( \\{ 0 , 1 \\} ) \\simeq [ 0 , 1 ]$ . ", + "bbox": [ + 174, + 830, + 825, + 876 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A tape square is said to be at relative position $u \\in \\mathbb { Z }$ if it is labelled $u$ after enumerating all squares in increasing order from left to right such that the square currently under the head is assigned zero. Consider the following random variables at times $t \\geq 0$ : ", + "bbox": [ + 173, + 881, + 825, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "• $Y _ { u , t } \\in \\Sigma$ : the content of the tape square at relative position $u$ at time $t$ . \n• $S _ { t } \\in Q$ : the internal state at time $t$ . \n· $W r _ { t } \\in \\Sigma$ : the symbol to be written, in the transition from time $t$ to $t + 1$ . \n· $M v _ { t } \\in \\{ L , S , R \\}$ : the direction to move, in the transition from time $t$ to $t + 1$ . ", + "bbox": [ + 214, + 103, + 746, + 176 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We call a smooth dynamical system a pair $( A , \\phi )$ consisting of a smooth manifold $A$ with corners together with a smooth transformation $\\phi : A A$ . ", + "bbox": [ + 173, + 185, + 823, + 215 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Definition F.1. Let $M = ( \\Sigma , Q , \\delta )$ be a Turing machine. The smooth relaxation of $M$ is the smooth dynamical system $( ( \\Delta \\Sigma ) ^ { \\mathbb { Z } , \\sqsupset } \\times \\Delta Q , \\Delta \\mathrm { s t e p } )$ where ", + "bbox": [ + 174, + 218, + 821, + 250 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/72b1aa44f2de884700842f66e48f63028a81392f430e1671c249d7cc87b59b8d.jpg", + "text": "$$\n\\Delta \\mathrm { s t e p } : ( \\Delta \\Sigma ) ^ { \\mathbb { Z } , \\square } \\times \\Delta Q ( \\Delta \\Sigma ) ^ { \\mathbb { Z } , \\square } \\times \\Delta Q\n$$", + "text_format": "latex", + "bbox": [ + 349, + 255, + 647, + 275 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "is a smooth transformation sending a state $( \\{ P ( Y _ { u , t } ) \\} _ { u \\in \\mathbb { Z } } , P ( S _ { t } ) )$ to $( \\{ P ( Y _ { u , t + 1 } ) \\} _ { u \\in \\mathbb { Z } } , P ( S _ { t + 1 } ) )$ determined by the equations ", + "bbox": [ + 174, + 280, + 820, + 310 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/1b4007c46825393aef4aa4521ba4b324339f93f3df64105c7d175487d591569c.jpg", + "text": "$$\n\\begin{array} { r } { P ( M v _ { t } = d | C ) = \\sum _ { \\sigma , q } \\delta _ { \\delta _ { 3 } ( \\sigma , q ) = d } P ( Y _ { 0 , t } = \\sigma | C ) P ( S _ { t } = q | C ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 212, + 320, + 643, + 340 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/919c45ad9588dbcda6b701e326612cb29b59ce549ab3de2174aa479d62c58d1d.jpg", + "text": "$$\n\\begin{array} { r } { P ( W r _ { t } = \\sigma | C ) = \\sum _ { \\sigma ^ { \\prime } , q } \\delta _ { \\delta _ { 1 } ( \\sigma ^ { \\prime } , q ) = \\sigma } P ( Y _ { 0 , t } = \\sigma ^ { \\prime } | C ) P ( S _ { t } = q | C ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 358, + 661, + 377 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/ac6ffd9db28543066c5647c7c715ab7e342982d3a5b07f74916f12368ec08bbc.jpg", + "text": "$$\n\\begin{array} { r l } & { P ( Y _ { u , t + 1 } = \\sigma | C ) = P ( M v _ { t } = L | C ) \\Big ( \\delta _ { u \\neq 1 } P ( Y _ { u - 1 , t } = \\sigma | C ) + \\delta _ { u = 1 } P ( W r _ { t } = \\sigma | C ) \\Big ) } \\\\ & { \\qquad + P ( M v _ { t } = S | C ) \\Big ( \\delta _ { u \\neq 0 } P ( Y _ { u , t } = \\sigma | C ) + \\delta _ { u = 0 } P ( W r _ { t } = \\sigma | C ) \\Big ) } \\\\ & { \\qquad + P ( M v _ { t } = R | C ) \\Big ( \\delta _ { u \\neq - 1 } P ( Y _ { u + 1 , t } = \\sigma | C ) + \\delta _ { u = - 1 } P ( W r _ { t } = \\sigma | C ) \\Big ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 430, + 823, + 507 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where $C \\in ( \\Delta \\Sigma ) ^ { \\mathbb { Z } , \\square } \\times \\Delta Q$ is an initial state. ", + "bbox": [ + 176, + 516, + 473, + 534 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We will call the smooth relaxation of a Turing machine a smooth Turing machine. A smooth Turing machine encodes uncertainty in the initial configuration of a Turing machine together with an update rule for how to propagate this uncertainty over time. We interpret the smooth step function as updating the state of belief of a “naive” Bayesian observer. This nomenclature comes from the assumption of conditional independence between random variables in our probability functions. ", + "bbox": [ + 173, + 542, + 825, + 614 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Remark F.2. Propagating uncertainty using standard probability leads to a smooth dynamical system which encodes the state evolution of an “ordinary” Bayesian observer of the Turing machine. This requires the calculation of various joint distributions which makes such an extension computationally difficult to work with. Computation aside, the naive probabilistic extension is justified from the point of view of derivatives of algorithms according to the denotational semantics of differential linear logic. See Clift & Murfet (2018) for further details. ", + "bbox": [ + 173, + 617, + 825, + 702 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We call the smooth extension of a universal Turing machine a smooth universal Turing machine. Recall that the staged pseudo-UTM $\\mathcal { U }$ has four tapes: the description tape, the staging tape, the state tape and working tape. The smooth relaxation of $\\mathcal { U }$ is a smooth dynamical system ", + "bbox": [ + 173, + 713, + 825, + 756 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/d20fb7f204d6eccdb2ee56ba113b01bd24c7b142123b231a5c77b06cc916b5ad.jpg", + "text": "$$\n\\Delta \\mathrm { s t e p } _ { \\mathcal { U } } : [ ( \\Delta \\Sigma _ { \\mathrm { U T M } } ) ^ { \\mathbb { Z } , \\bigtriangledown } ] ^ { 4 } \\times \\Delta Q _ { \\mathrm { U T M } } \\to [ ( \\Delta \\Sigma _ { \\mathrm { U T M } } ) ^ { \\mathbb { Z } , \\bigtriangledown } ] ^ { 4 } \\times \\Delta Q _ { \\mathrm { U T M } } .\n$$", + "text_format": "latex", + "bbox": [ + 272, + 762, + 723, + 782 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "If we use the staged pseudo-UTM to simulate a Turing machine with tape alphabet $\\Sigma \\subseteq \\Sigma _ { \\mathrm { U T M } }$ and states $Q \\subseteq \\Sigma _ { \\mathrm { U T M } }$ then with some determined initial state the function $\\Delta$ step restricts to ", + "bbox": [ + 174, + 787, + 825, + 816 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/0be3c84ebbb22ca055bd21d7b51bbcc2b8693a203027e1c3f7d90462b695d95d.jpg", + "text": "$$\n\\Delta \\mathrm { s t e p } _ { \\mathscr { U } } : ( \\Delta \\Sigma ) ^ { \\mathbb { Z } , \\sharp } \\times { \\mathscr { W } } \\times \\Delta Q \\times \\mathscr { X } \\longrightarrow ( \\Delta \\Sigma ) ^ { \\mathbb { Z } , \\sharp } \\times { \\mathscr { W } } \\times \\Delta Q \\times \\mathscr { X }\n$$", + "text_format": "latex", + "bbox": [ + 269, + 821, + 728, + 842 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where the first factor is the configuration of the work tape, $W$ is as in (3) and ", + "bbox": [ + 173, + 848, + 679, + 863 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/d2cf55518ec2b7772068c8a57ed2ca16ca8c1d6d2e87e22a848749a59f378ba9.jpg", + "text": "$$\n\\mathcal { X } = [ ( \\Delta \\Sigma _ { \\mathrm { U T M } } ) ^ { \\mathbb { Z } , \\sqcap } ] \\times \\Delta Q _ { \\mathrm { U T M } }\n$$", + "text_format": "latex", + "bbox": [ + 395, + 869, + 602, + 890 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where the first factor is the configuration of the staging tape. Since $\\mathcal { U }$ is periodic of period $T =$ $1 0 N + 5$ (Appendix E) the iterated function $( \\Delta \\mathrm { s t e p } _ { \\mathscr { U } } ) ^ { T }$ takes an input with staging tape in its ", + "bbox": [ + 174, + 895, + 825, + 925 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "default state $X X X$ and UTM state compSymbol and returns a configuration with the same staging tape and state, but with the configuration of the work tape, description tape and state tape updated by one complete simulation step. That is, ", + "bbox": [ + 173, + 103, + 826, + 146 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/271d548bcfb3c3cd71521abb4bde6826466f54ed2b06dede0612062692d3e8c2.jpg", + "text": "$$\n( \\Delta \\operatorname { s t e p } _ { \\mathcal { U } } ) ^ { T } ( x , w , q , X X X , \\mathrm { c o m p S y m b o l } ) = ( F ( x , w , q ) , X X X , \\mathrm { c o m p S y m b o l } )\n$$", + "text_format": "latex", + "bbox": [ + 232, + 151, + 764, + 170 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "for some smooth function ", + "bbox": [ + 174, + 174, + 346, + 189 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/52bd475d96829f650a9fe6f4cda0ffd2a8880933ce6771e40742cf756c7a5a6e.jpg", + "text": "$$\nF : ( \\Delta \\Sigma ) ^ { \\mathbb { Z } , \\sharp } \\times W \\times \\Delta Q \\longrightarrow ( \\Delta \\Sigma ) ^ { \\mathbb { Z } , \\sharp } \\times W \\times \\Delta Q .\n$$", + "text_format": "latex", + "bbox": [ + 318, + 190, + 679, + 210 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Finally we can define the function $\\Delta \\mathrm { s t e p } ^ { t }$ of (4). We assume all Turing machines are initialised in some common state init $\\in Q$ . ", + "bbox": [ + 176, + 214, + 823, + 244 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Definition F.3. Given $t \\geq 0$ we define $\\Delta \\operatorname { s t e p } ^ { t } : \\Sigma ^ { * } \\times W \\longrightarrow \\Delta Q$ by ", + "bbox": [ + 174, + 246, + 632, + 262 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/74696f823bfc89d626700cc4f9a1a8bb8d8f406ee6306dab64b37d909b2ce88c.jpg", + "text": "$$\n\\Delta \\operatorname { s t e p } ^ { t } ( x , w ) = \\Pi _ { Q } F ^ { t } ( x , w , \\operatorname { i n i t } )\n$$", + "text_format": "latex", + "bbox": [ + 383, + 265, + 614, + 284 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where $\\Pi _ { Q }$ is the projection onto $\\Delta Q$ . ", + "bbox": [ + 176, + 287, + 419, + 304 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "G DIRECT SIMULATION ", + "text_level": 1, + "bbox": [ + 174, + 321, + 388, + 338 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "For computational efficiency in our PyTorch implementation of the staged pseudo-UTM we implement $F$ of (9) rather than $\\Delta \\mathrm { s t e p } _ { \\mathcal { U } }$ . We refer to this as direction simulation since it means that we update in one step the state and working tape of the UTM for a full cycle where a cycle consists of $T = 1 0 N + 5$ steps of the UTM. ", + "bbox": [ + 173, + 353, + 825, + 410 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Let $S ( t )$ and $Y _ { u } ( t )$ be random variables describing the contents of state tape and working tape in relative positions $0 , u$ respectively after $t \\geq 0$ time steps of the UTM. We define ${ \\widetilde { S } } ( t ) : = S ( 4 + T t )$ and $\\widetilde { Y } _ { u } ( t ) : = Y _ { u } ( 4 + T t )$ where $t \\geq 0$ and $u \\in \\mathbb { Z }$ . The task then is to define functions $f , g$ such that ", + "bbox": [ + 173, + 416, + 825, + 465 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/01589a065d21d81fafaf4c122c7537746528ccc3172c3a93611a75f1ce056d6a.jpg", + "text": "$$\n\\widetilde { S } ( t + 1 ) = f ( \\widetilde { S } ( t ) )\n$$", + "text_format": "latex", + "bbox": [ + 429, + 469, + 566, + 489 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/ce3e5f30df241bc7588b4371d14629e7c41e23165272c2b7dae8f3ee5418d8d8.jpg", + "text": "$$\n\\widetilde Y _ { u } ( t + 1 ) = g ( \\widetilde Y _ { u } ( t ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 423, + 492, + 573, + 512 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The functional relationship is given as follows: for $1 \\leq i \\leq N$ indexing tuples on the description tape, while processing that tuple, the UTM is in a state distribution $\\lambda _ { i } \\cdot \\bar { q } + ( 1 - \\lambda _ { i } ) \\cdot \\neg \\bar { q }$ where $\\bar { q } \\in$ {copySymbol, copyState, $\\mathrm { c o p y D i r } \\}$ . Given the initial state of the description tape, we assume uncertainty about $s ^ { \\prime } , q ^ { \\prime } , d$ only. This determines a map ", + "bbox": [ + 173, + 513, + 825, + 570 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/7aa90452bd127888ac878a82b20c4859c54dac1f3fb343d910a8fcaec6de7504.jpg", + "text": "$$\n\\theta : \\{ 1 , \\dots , N \\} \\to \\Sigma \\times Q\n$$", + "text_format": "latex", + "bbox": [ + 413, + 575, + 584, + 593 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where the description tape at tuple number $i$ is given by $\\theta ( i ) _ { 1 } \\theta ( i ) _ { 2 } P ( s _ { i } ^ { \\prime } ) P ( q _ { i } ^ { \\prime } ) P ( d _ { i } )$ . We define the conditionally independent joint distribution between $\\{ \\widetilde { Y } _ { 0 , t - 1 } , \\widetilde { S } _ { t - 1 } \\}$ by ", + "bbox": [ + 174, + 597, + 821, + 630 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/00f2cdcbaf17862a9524c183d3cd6929539574a482f723be628aae9bcc2006d0.jpg", + "text": "$$\n\\begin{array} { l } { { \\lambda _ { i } = \\displaystyle \\sum _ { \\sigma \\in \\Sigma } \\delta _ { \\theta ( i ) _ { 1 } = \\sigma } P ( \\widetilde { Y } _ { 0 , t - 1 } = \\sigma ) \\cdot \\sum _ { q \\in Q } \\delta _ { \\theta ( i ) _ { 2 } = q } P ( \\widetilde { S } _ { t - 1 } = q ) \\hfill } } \\\\ { { \\quad = P ( \\widetilde { Y } _ { 0 , t - 1 } = \\theta ( i ) _ { 1 } ) \\cdot P ( \\widetilde { S } _ { t - 1 } = \\theta ( i ) _ { 2 } ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 297, + 633, + 700, + 693 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We then calculate a recursive set of equations for $0 \\leq j \\leq N$ describing distributions $P ( \\hat { s } _ { j } ) , P ( \\hat { q } _ { j } )$ and $P ( \\hat { d } _ { j } )$ on the staging tape after processing all tuples up to and including tuple $j$ . These are given by $P ( \\hat { s } _ { 0 } ) = P ( \\hat { q } _ { 0 } ) = P ( \\hat { d } _ { 0 } ) = 1 \\cdot X$ and ", + "bbox": [ + 173, + 702, + 826, + 752 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/1c4ab3562612c52181d5fb39c49de516678335b4023913e0cae6179ec06de62f.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle { P ( \\hat { s } _ { i } ) = \\sum _ { \\sigma \\in \\Sigma } \\{ \\lambda _ { i } \\cdot P ( s _ { i } ^ { \\prime } = \\sigma ) + ( 1 - \\lambda _ { i } ) \\cdot P ( \\hat { s } _ { i - 1 } = \\sigma ) \\} \\cdot \\sigma + ( 1 - \\lambda _ { i } ) \\cdot P ( \\hat { s } _ { i - 1 } = X ) \\cdot X } } \\\\ & { \\displaystyle { P ( \\hat { q } _ { i } ) = \\sum _ { \\ q \\in Q } \\{ \\lambda _ { i } \\cdot P ( q _ { i } ^ { \\prime } = q ) + ( 1 - \\lambda _ { i } ) \\cdot P ( \\hat { q } _ { i - 1 } = q ) \\} \\cdot q + ( 1 - \\lambda _ { i } ) \\cdot P ( \\hat { q } _ { i - 1 } = X ) \\cdot X } } \\\\ & { \\displaystyle { \\hat { l } _ { i } ) = \\sum _ { \\alpha \\in \\{ L , R , S \\} } \\{ \\lambda _ { i } \\cdot P ( d _ { i } = a ) + ( 1 - \\lambda _ { i } ) \\cdot P ( \\hat { d } _ { i - 1 } = a ) \\} \\cdot a + ( 1 - \\lambda _ { i } ) \\cdot P ( \\hat { d } _ { i - 1 } = X ) \\cdot X . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 176, + 757, + 823, + 868 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Let $A _ { \\sigma } = P ( \\widehat { s } _ { N } = X ) \\cdot P ( \\widetilde { Y } _ { 0 , t - 1 } = \\sigma ) + P ( \\widehat { s } _ { N } = \\sigma ) .$ . In terms of the above distributions ", + "bbox": [ + 169, + 868, + 767, + 887 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/870a01861cc402a8d5f1573c80cdb5933997da086a036400c1fa3874d947fcda.jpg", + "text": "$$\nP ( \\widetilde { S } _ { t } ) = \\sum _ { q \\in Q } \\Big ( P ( \\hat { q } _ { N } = X ) \\cdot P ( \\widetilde { S } _ { t - 1 } = q ) + P ( \\hat { q } _ { N } = q ) \\Big ) \\cdot q\n$$", + "text_format": "latex", + "bbox": [ + 294, + 892, + 702, + 929 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 104, + 202, + 118 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/48870491c119bf3fb60dd1692af389768f4ed2e7428746fb34f022b5de1c0c77.jpg", + "text": "$$\n\\begin{array} { r l } & { P ( \\widetilde { Y } _ { u , t } = \\sigma ) = P ( \\hat { d } _ { N } = L ) \\left( \\delta _ { u \\neq 1 } P ( \\widetilde { Y } _ { u - 1 , t - 1 } = \\sigma ) + \\delta _ { u = 1 } A _ { \\sigma } \\right) } \\\\ & { \\quad \\quad \\quad \\quad \\quad + P ( \\hat { d } _ { N } = R ) \\left( \\delta _ { u \\neq - 1 } P ( \\widetilde { Y } _ { u + 1 , t - 1 } = \\sigma ) + \\delta _ { u = - 1 } A _ { \\sigma } \\right) } \\\\ & { \\quad \\quad \\quad \\quad \\quad + P ( \\hat { d } _ { N } = S ) \\left( \\delta _ { u \\neq 0 } P ( \\widetilde { Y } _ { u , t - 1 } = \\sigma ) + \\delta _ { u = 0 } A _ { \\sigma } \\right) } \\\\ & { \\quad \\quad \\quad \\quad \\quad + P ( \\hat { d } _ { N } = X ) \\left( \\delta _ { u \\neq 0 } P ( \\widetilde { Y } _ { u , t - 1 } = \\sigma ) + \\delta _ { u = 0 } A _ { \\sigma } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 271, + 122, + 725, + 234 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Using these equations, we can state efficient update rules for the staging tape. We have ", + "bbox": [ + 168, + 244, + 743, + 261 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/34a718ae3fe055dbad0ae6ff9d6580def17a8c81f53748b3f5050351d5edd06e.jpg", + "text": "$$\n\\begin{array} { l l l } { { \\displaystyle P ( \\hat { s } _ { N } = X ) = \\prod _ { j = 1 } ^ { N } ( 1 - \\lambda _ { j } ) , \\quad } } & { { \\displaystyle P ( \\hat { s } _ { N } = \\sigma ) = \\sum _ { j = 1 } ^ { N } \\lambda _ { j } \\cdot P ( s _ { j } ^ { \\prime } = \\sigma ) \\prod _ { l = j + 1 } ^ { N } ( 1 - \\lambda _ { l } ) } } \\\\ { { \\displaystyle P ( \\hat { q } _ { N } = X ) = \\prod _ { j = 1 } ^ { N } ( 1 - \\lambda _ { j } ) , \\quad } } & { { \\displaystyle P ( \\hat { q } _ { N } = q ) = \\sum _ { j = 1 } ^ { N } \\lambda _ { j } \\cdot P ( q _ { j } ^ { \\prime } = q ) \\prod _ { l = j + 1 } ^ { N } ( 1 - \\lambda _ { l } ) } } \\\\ { { \\displaystyle P ( \\hat { d } _ { N } = X ) = \\prod _ { j = 1 } ^ { N } ( 1 - \\lambda _ { j } ) , \\quad } } & { { \\displaystyle P ( \\hat { d } _ { N } = a ) = \\sum _ { j = 1 } ^ { N } \\lambda _ { j } \\cdot P ( d _ { j } = a ) \\prod _ { l = j + 1 } ^ { N } ( 1 - \\lambda _ { l } ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 223, + 266, + 774, + 412 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "To enable efficient computation, we can express these equations using tensor calculus. Let $\\lambda =$ $( \\lambda _ { 1 } , \\dots , \\lambda _ { N } ) \\in \\mathbb { R } ^ { N }$ . We view ", + "bbox": [ + 171, + 420, + 821, + 448 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/922bc650c8af1c587aa886ee81dcdcd1e3e0f61ef0354eb95e31030c0c55f9ca.jpg", + "text": "$$\n\\theta : \\mathbb { R } ^ { N } \\xrightarrow { } \\mathbb { R } \\Sigma \\otimes \\mathbb { R } Q\n$$", + "text_format": "latex", + "bbox": [ + 426, + 446, + 570, + 464 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "as a tensor and so $\\begin{array} { r } { \\theta = \\sum _ { i = 1 } ^ { N } i \\otimes \\theta ( i ) _ { 1 } \\otimes \\theta ( i ) _ { 2 } \\in \\mathbb { R } ^ { N } \\otimes \\mathbb { R } \\Sigma \\otimes \\mathbb { R } Q . } \\end{array}$ . Then ", + "bbox": [ + 173, + 468, + 656, + 488 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/9407a8ae5aa4a54ff3c7e57aacca083315889ddf658dad1de7198f7fec34fd3f.jpg", + "text": "$$\n\\theta _ { - } \\left( P ( \\widetilde { Y } _ { 0 , t - 1 } ) \\otimes P ( \\widetilde { S } _ { t - 1 } ) \\right) = \\sum _ { i = 1 } ^ { N } i \\cdot P ( \\widetilde { Y } _ { 0 , t - 1 } = \\theta ( i ) _ { 1 } ) \\cdot P ( \\widetilde { S } _ { t - 1 } = \\theta ( i ) _ { 2 } ) = \\lambda .\n$$", + "text_format": "latex", + "bbox": [ + 230, + 493, + 766, + 536 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "If we view $P ( s _ { * } ^ { \\prime } = \\bullet ) \\in \\mathbb { R } ^ { N } \\otimes \\mathbb { R } ^ { \\Sigma }$ as a tensor, then ", + "bbox": [ + 173, + 542, + 516, + 560 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/8bb785fffae36c9f98d84eb10a135961729c3af486d3f03168861cfc947c8a20.jpg", + "text": "$$\n{ \\mathcal { S } } ( { \\widehat { \\mathfrak { s } } } _ { N } ) = \\sum _ { j = 1 } ^ { N } P ( s _ { j } ^ { \\prime } = \\bullet ) \\cdot \\left( \\lambda _ { j } \\prod _ { l = j + 1 } ^ { N } ( 1 - \\lambda _ { l } ) \\right) = \\lambda \\cdot \\left( \\prod _ { l = 2 } ^ { N } ( 1 - \\lambda _ { l } ) , \\prod _ { l = 3 } ^ { N } ( 1 - \\lambda _ { l } ) , \\ldots , ( 1 - \\lambda _ { N } ) , 1 \\right)\n$$", + "text_format": "latex", + "bbox": [ + 181, + 565, + 826, + 617 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "can be expressed in terms on the vector $\\lambda$ only. Similarly, $P ( q _ { * } ^ { \\prime } = \\bullet ) \\in \\mathbb { R } ^ { N } \\otimes \\mathbb { R } ^ { Q }$ with ", + "bbox": [ + 171, + 622, + 746, + 640 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/dd24ee215a5f81118a4f1bab317e8808b056df708f3b7065937e57068bcbfc20.jpg", + "text": "$$\n{ \\cal P } ( \\hat { q } _ { N } ) = \\sum _ { j = 1 } ^ { N } P ( q _ { j } ^ { \\prime } = \\bullet ) \\cdot \\left( \\lambda _ { j } \\prod _ { l = j + 1 } ^ { N } ( 1 - \\lambda _ { l } ) \\right) = \\lambda \\cdot \\left( \\prod _ { l = 2 } ^ { N } ( 1 - \\lambda _ { l } ) , \\prod _ { l = 3 } ^ { N } ( 1 - \\lambda _ { l } ) , \\ldots , ( 1 - \\lambda _ { N } ) , 1 \\right)\n$$", + "text_format": "latex", + "bbox": [ + 181, + 645, + 826, + 695 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "and $P ( d _ { * } = \\bullet ) \\in \\mathbb { R } ^ { N } \\otimes \\mathbb { R } ^ { 3 }$ with ", + "bbox": [ + 173, + 702, + 393, + 718 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/357054ab85b8301fc14c8cf543d7f246fd09a1a1fdd0a8f982875d736e0c534a.jpg", + "text": "$$\n^ { > } ( \\hat { d } _ { N } ) = \\sum _ { j = 1 } ^ { N } P ( d _ { j } = \\bullet ) \\cdot \\left( \\lambda _ { j } \\prod _ { l = j + 1 } ^ { N } ( 1 - \\lambda _ { l } ) \\right) = \\lambda \\cdot \\left( \\prod _ { l = 2 } ^ { N } ( 1 - \\lambda _ { l } ) , \\prod _ { l = 3 } ^ { N } ( 1 - \\lambda _ { l } ) , \\ldots , ( 1 - \\lambda _ { N } ) , 1 \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 724, + 828, + 775 + ], + "page_idx": 18 + } +] \ No newline at end of file diff --git a/parse/train/qiydAcw6Re/qiydAcw6Re_middle.json b/parse/train/qiydAcw6Re/qiydAcw6Re_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..95f0836166605461b4b4e4eb19420e7a0749dd5f --- /dev/null +++ b/parse/train/qiydAcw6Re/qiydAcw6Re_middle.json @@ -0,0 +1,69011 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 78, + 384, + 96 + ], + "lines": [ + { + "bbox": [ + 106, + 78, + 386, + 98 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 386, + 98 + ], + "score": 1.0, + "content": "GEOMETRY OF PROGRAM SYNTHESIS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 112, + 115, + 244, + 137 + ], + "lines": [ + { + "bbox": [ + 113, + 115, + 201, + 127 + ], + "spans": [ + { + "bbox": [ + 113, + 115, + 201, + 127 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 111, + 126, + 245, + 138 + ], + "spans": [ + { + "bbox": [ + 111, + 126, + 245, + 138 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 278, + 167, + 333, + 179 + ], + "lines": [ + { + "bbox": [ + 277, + 166, + 335, + 180 + ], + "spans": [ + { + "bbox": [ + 277, + 166, + 335, + 180 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 142, + 192, + 468, + 270 + ], + "lines": [ + { + "bbox": [ + 141, + 192, + 469, + 207 + ], + "spans": [ + { + "bbox": [ + 141, + 192, + 469, + 207 + ], + "score": 1.0, + "content": "We present a new perspective on program synthesis in which programs may be", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 203, + 469, + 217 + ], + "spans": [ + { + "bbox": [ + 141, + 203, + 469, + 217 + ], + "score": 1.0, + "content": "identified with singularities of analytic functions. As an example, Turing ma-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 214, + 469, + 228 + ], + "spans": [ + { + "bbox": [ + 141, + 214, + 469, + 228 + ], + "score": 1.0, + "content": "chines are synthesised from input-output examples by propagating uncertainty", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 226, + 469, + 238 + ], + "spans": [ + { + "bbox": [ + 141, + 226, + 469, + 238 + ], + "score": 1.0, + "content": "through a smooth relaxation of a universal Turing machine. The posterior distri-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 237, + 469, + 249 + ], + "spans": [ + { + "bbox": [ + 141, + 237, + 469, + 249 + ], + "score": 1.0, + "content": "bution over weights is approximated using Markov chain Monte Carlo and bounds", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 248, + 469, + 260 + ], + "spans": [ + { + "bbox": [ + 142, + 248, + 469, + 260 + ], + "score": 1.0, + "content": "on the generalisation error of these models is estimated using the real log canoni-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 258, + 406, + 272 + ], + "spans": [ + { + "bbox": [ + 142, + 258, + 406, + 272 + ], + "score": 1.0, + "content": "cal threshold, a geometric invariant from singular learning theory.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 108, + 294, + 205, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 208, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 208, + 309 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 320, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "The idea of program synthesis dates back to the birth of modern computation itself (Turing, 1948)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "and is recognised as one of the most important open problems in computer science (Gulwani et al.,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 341, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 355 + ], + "score": 1.0, + "content": "2017). However, there appear to be serious obstacles to synthesising programs by gradient descent", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 353, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 365 + ], + "score": 1.0, + "content": "at scale (Neelakantan et al., 2016; Kaiser & Sutskever, 2016; Bunel et al., 2016; Gaunt et al., 2016;", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 363, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 377 + ], + "score": 1.0, + "content": "Evans & Grefenstette, 2018; Chen et al., 2018) and these problems suggest that it would be appro-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "score": 1.0, + "content": "priate to make a fundamental study of the geometry of loss surfaces in program synthesis, since", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "this geometry determines the learning process. To that end, in this paper we explain a new point of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "view on program synthesis using the singular learning theory of Watanabe (2009) and the smooth", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 407, + 343, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 343, + 420 + ], + "score": 1.0, + "content": "relaxation of Turing machines from Clift & Murfet (2018).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 108, + 424, + 415, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 422, + 417, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 417, + 439 + ], + "score": 1.0, + "content": "In broad strokes this new geometric point of view on program synthesis says:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 132, + 446, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 132, + 444, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 132, + 444, + 436, + 459 + ], + "score": 1.0, + "content": "• Programs to be synthesised are singularities of analytic functions. If", + "type": "text" + }, + { + "bbox": [ + 437, + 445, + 472, + 457 + ], + "score": 0.91, + "content": "U \\subseteq \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 444, + 506, + 459 + ], + "score": 1.0, + "content": "is open", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 141, + 456, + 160, + 470 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 160, + 457, + 220, + 467 + ], + "score": 0.91, + "content": "K : U \\longrightarrow \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 456, + 289, + 470 + ], + "score": 1.0, + "content": "is analytic, then", + "type": "text" + }, + { + "bbox": [ + 289, + 457, + 318, + 467 + ], + "score": 0.91, + "content": "x \\in U", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 456, + 404, + 470 + ], + "score": 1.0, + "content": "is a critical point of", + "type": "text" + }, + { + "bbox": [ + 405, + 457, + 415, + 467 + ], + "score": 0.85, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 456, + 426, + 470 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 426, + 457, + 479, + 469 + ], + "score": 0.92, + "content": "\\nabla K ( x ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "and a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 142, + 468, + 419, + 481 + ], + "spans": [ + { + "bbox": [ + 142, + 469, + 249, + 481 + ], + "score": 1.0, + "content": "singularity of the function", + "type": "text" + }, + { + "bbox": [ + 249, + 469, + 260, + 478 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 469, + 373, + 481 + ], + "score": 1.0, + "content": "if it is a critical point where", + "type": "text" + }, + { + "bbox": [ + 374, + 468, + 415, + 480 + ], + "score": 0.93, + "content": "K ( x ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 469, + 419, + 481 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 138, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 138, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "The Kolmogorov complexity of a program is related to a geometric invariant of the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 141, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "associated singularity called the Real Log Canonical Threshold (RLCT). This invariant", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 141, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "controls both the generalisation error and the learning process, and is therefore an appro-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 517, + 464, + 531 + ], + "spans": [ + { + "bbox": [ + 141, + 517, + 464, + 531 + ], + "score": 1.0, + "content": "priate measure of “complexity” in continuous program synthesis. See Section 3.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 137, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 137, + 533, + 505, + 547 + ], + "score": 1.0, + "content": "The geometry has concrete practical implications. For example, a MCMC-based ap-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 141, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "proach to program synthesis will find, with high probability, a solution that is of low com-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 142, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 142, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "plexity (if it finds a solution at all). We sketch a novel point of view on the problem of “bad", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 142, + 567, + 428, + 578 + ], + "spans": [ + { + "bbox": [ + 142, + 567, + 428, + 578 + ], + "score": 1.0, + "content": "local minima” (Gaunt et al., 2016) based on these ideas. See Section 4.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 109, + 588, + 488, + 599 + ], + "lines": [ + { + "bbox": [ + 107, + 587, + 489, + 601 + ], + "spans": [ + { + "bbox": [ + 107, + 587, + 489, + 601 + ], + "score": 1.0, + "content": "We demonstrate all of these principles in experiments with toy examples of synthesis problems.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "Program synthesis as inference. We use Turing machines, but mutatis mutandis everything applies", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 258, + 628 + ], + "score": 1.0, + "content": "to other programming languages. 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As an example, Turing ma-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 214, + 469, + 228 + ], + "spans": [ + { + "bbox": [ + 141, + 214, + 469, + 228 + ], + "score": 1.0, + "content": "chines are synthesised from input-output examples by propagating uncertainty", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 226, + 469, + 238 + ], + "spans": [ + { + "bbox": [ + 141, + 226, + 469, + 238 + ], + "score": 1.0, + "content": "through a smooth relaxation of a universal Turing machine. The posterior distri-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 237, + 469, + 249 + ], + "spans": [ + { + "bbox": [ + 141, + 237, + 469, + 249 + ], + "score": 1.0, + "content": "bution over weights is approximated using Markov chain Monte Carlo and bounds", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 248, + 469, + 260 + ], + "spans": [ + { + "bbox": [ + 142, + 248, + 469, + 260 + ], + "score": 1.0, + "content": "on the generalisation error of these models is estimated using the real log canoni-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 258, + 406, + 272 + ], + "spans": [ + { + "bbox": [ + 142, + 258, + 406, + 272 + ], + "score": 1.0, + "content": "cal threshold, a geometric invariant from singular learning theory.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7, + "bbox_fs": [ + 141, + 192, + 469, + 272 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 294, + 205, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 208, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 208, + 309 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 320, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "The idea of program synthesis dates back to the birth of modern computation itself (Turing, 1948)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "and is recognised as one of the most important open problems in computer science (Gulwani et al.,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 341, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 355 + ], + "score": 1.0, + "content": "2017). However, there appear to be serious obstacles to synthesising programs by gradient descent", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 353, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 365 + ], + "score": 1.0, + "content": "at scale (Neelakantan et al., 2016; Kaiser & Sutskever, 2016; Bunel et al., 2016; Gaunt et al., 2016;", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 363, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 377 + ], + "score": 1.0, + "content": "Evans & Grefenstette, 2018; Chen et al., 2018) and these problems suggest that it would be appro-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "score": 1.0, + "content": "priate to make a fundamental study of the geometry of loss surfaces in program synthesis, since", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "this geometry determines the learning process. To that end, in this paper we explain a new point of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "view on program synthesis using the singular learning theory of Watanabe (2009) and the smooth", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 407, + 343, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 343, + 420 + ], + "score": 1.0, + "content": "relaxation of Turing machines from Clift & Murfet (2018).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 320, + 506, + 420 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 424, + 415, + 436 + ], + "lines": [ + { + "bbox": [ + 105, + 422, + 417, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 417, + 439 + ], + "score": 1.0, + "content": "In broad strokes this new geometric point of view on program synthesis says:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 422, + 417, + 439 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 446, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 132, + 444, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 132, + 444, + 436, + 459 + ], + "score": 1.0, + "content": "• Programs to be synthesised are singularities of analytic functions. If", + "type": "text" + }, + { + "bbox": [ + 437, + 445, + 472, + 457 + ], + "score": 0.91, + "content": "U \\subseteq \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 444, + 506, + 459 + ], + "score": 1.0, + "content": "is open", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 141, + 456, + 160, + 470 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 160, + 457, + 220, + 467 + ], + "score": 0.91, + "content": "K : U \\longrightarrow \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 456, + 289, + 470 + ], + "score": 1.0, + "content": "is analytic, then", + "type": "text" + }, + { + "bbox": [ + 289, + 457, + 318, + 467 + ], + "score": 0.91, + "content": "x \\in U", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 456, + 404, + 470 + ], + "score": 1.0, + "content": "is a critical point of", + "type": "text" + }, + { + "bbox": [ + 405, + 457, + 415, + 467 + ], + "score": 0.85, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 456, + 426, + 470 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 426, + 457, + 479, + 469 + ], + "score": 0.92, + "content": "\\nabla K ( x ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "and a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 142, + 468, + 419, + 481 + ], + "spans": [ + { + "bbox": [ + 142, + 469, + 249, + 481 + ], + "score": 1.0, + "content": "singularity of the function", + "type": "text" + }, + { + "bbox": [ + 249, + 469, + 260, + 478 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 469, + 373, + 481 + ], + "score": 1.0, + "content": "if it is a critical point where", + "type": "text" + }, + { + "bbox": [ + 374, + 468, + 415, + 480 + ], + "score": 0.93, + "content": "K ( x ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 469, + 419, + 481 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 138, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 138, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "The Kolmogorov complexity of a program is related to a geometric invariant of the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 141, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "associated singularity called the Real Log Canonical Threshold (RLCT). This invariant", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 141, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "controls both the generalisation error and the learning process, and is therefore an appro-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 517, + 464, + 531 + ], + "spans": [ + { + "bbox": [ + 141, + 517, + 464, + 531 + ], + "score": 1.0, + "content": "priate measure of “complexity” in continuous program synthesis. See Section 3.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 137, + 533, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 137, + 533, + 505, + 547 + ], + "score": 1.0, + "content": "The geometry has concrete practical implications. For example, a MCMC-based ap-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 141, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "proach to program synthesis will find, with high probability, a solution that is of low com-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 142, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 142, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "plexity (if it finds a solution at all). We sketch a novel point of view on the problem of “bad", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 142, + 567, + 428, + 578 + ], + "spans": [ + { + "bbox": [ + 142, + 567, + 428, + 578 + ], + "score": 1.0, + "content": "local minima” (Gaunt et al., 2016) based on these ideas. See Section 4.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27, + "bbox_fs": [ + 132, + 444, + 506, + 578 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 588, + 488, + 599 + ], + "lines": [ + { + "bbox": [ + 107, + 587, + 489, + 601 + ], + "spans": [ + { + "bbox": [ + 107, + 587, + 489, + 601 + ], + "score": 1.0, + "content": "We demonstrate all of these principles in experiments with toy examples of synthesis problems.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 107, + 587, + 489, + 601 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 604, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "Program synthesis as inference. We use Turing machines, but mutatis mutandis everything applies", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 258, + 628 + ], + "score": 1.0, + "content": "to other programming languages. 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One approach is to seek a smooth relaxation of the synthesis problem consisting", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 178, + 504, + 194 + ], + "spans": [ + { + "bbox": [ + 104, + 178, + 203, + 194 + ], + "score": 1.0, + "content": "of an analytic manifold", + "type": "text" + }, + { + "bbox": [ + 203, + 181, + 255, + 192 + ], + "score": 0.92, + "content": "W \\bar { \\supseteq } W ^ { c o d e }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 178, + 338, + 194 + ], + "score": 1.0, + "content": "and an extension of", + "type": "text" + }, + { + "bbox": [ + 339, + 182, + 349, + 191 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 178, + 443, + 194 + ], + "score": 1.0, + "content": "to an analytic function", + "type": "text" + }, + { + "bbox": [ + 444, + 181, + 504, + 191 + ], + "score": 0.91, + "content": "K : W \\longrightarrow \\mathbb { R }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 262, + 205 + ], + "score": 1.0, + "content": "so that we can search for the zeros of", + "type": "text" + }, + { + "bbox": [ + 262, + 193, + 273, + 202 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "using gradient descent. Perhaps the most natural way to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 202, + 504, + 216 + ], + "spans": [ + { + "bbox": [ + 104, + 202, + 283, + 216 + ], + "score": 1.0, + "content": "construct such a smooth relaxation is to take", + "type": "text" + }, + { + "bbox": [ + 283, + 204, + 295, + 213 + ], + "score": 0.76, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 202, + 476, + 216 + ], + "score": 1.0, + "content": "to be a space of probability distributions over", + "type": "text" + }, + { + "bbox": [ + 476, + 203, + 504, + 213 + ], + "score": 0.87, + "content": "W ^ { \\dot { c } o d e }", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 194, + 227 + ], + "score": 1.0, + "content": "and prescribe a model", + "type": "text" + }, + { + "bbox": [ + 195, + 214, + 234, + 226 + ], + "score": 0.93, + "content": "p ( \\boldsymbol { y } | \\boldsymbol { x } , \\boldsymbol { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "for propagating uncertainty about codes to uncertainty about outputs", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "(Gaunt et al., 2016; Evans & Grefenstette, 2018). The particular model we choose is based on the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "semantics of linear logic (Clift & Murfet, 2018). Supposing that such a smooth relaxation has been", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 246, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 226, + 260 + ], + "score": 1.0, + "content": "chosen together with a prior", + "type": "text" + }, + { + "bbox": [ + 226, + 247, + 249, + 259 + ], + "score": 0.92, + "content": "\\varphi ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 246, + 272, + 260 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 272, + 248, + 284, + 257 + ], + "score": 0.67, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 246, + 506, + 260 + ], + "score": 1.0, + "content": ", smooth program synthesis becomes the study of the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 258, + 296, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 259, + 271 + ], + "score": 1.0, + "content": "statistical learning theory of the triple", + "type": "text" + }, + { + "bbox": [ + 259, + 258, + 293, + 270 + ], + "score": 0.92, + "content": "( p , q , \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 258, + 296, + 271 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 275, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "There are perhaps two primary reasons to consider the smooth relaxation. Firstly, one might hope", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "that stochastic gradient descent or techniques like Markov chain Monte Carlo will be effective means", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "of solving the original combinatorial optimisation problem. This is not a new idea (Gulwani et al.,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "score": 1.0, + "content": "2017, §6) but so far its effectiveness for large programs has not been proven. Independently, one", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "might hope to find powerful new mathematical ideas that apply to the relaxed problem and shed light", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 330, + 411, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 411, + 343 + ], + "score": 1.0, + "content": "on the nature of program synthesis. This is the purpose of the present paper.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 345, + 420, + 359 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 421, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 273, + 361 + ], + "score": 1.0, + "content": "Singular learning theory. We denote by", + "type": "text" + }, + { + "bbox": [ + 273, + 346, + 390, + 359 + ], + "score": 0.92, + "content": "W _ { 0 } = \\{ w \\in W | K ( w ) = 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 345, + 421, + 361 + ], + "score": 1.0, + "content": "so that", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 253, + 364, + 358, + 378 + ], + "lines": [ + { + "bbox": [ + 253, + 364, + 358, + 378 + ], + "spans": [ + { + "bbox": [ + 253, + 364, + 358, + 378 + ], + "score": 0.91, + "content": "W _ { 0 } \\cap W ^ { c o d e } \\subseteq W _ { 0 } \\subseteq W", + "type": "interline_equation", + "image_path": "86ade0437cc02ec74cf772862fcd0f85b59d0d58456410aeb8aa4fceae3052a3.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 253, + 364, + 358, + 378 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 505, + 440 + ], + "lines": [ + { + "bbox": [ + 106, + 383, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 133, + 397 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 384, + 183, + 396 + ], + "score": 0.92, + "content": "W _ { 0 } \\cap W ^ { c o d e }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 383, + 506, + 397 + ], + "score": 1.0, + "content": "is the discrete set of solutions to the original synthesis problem. We refer to these", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 396, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 406, + 407 + ], + "score": 1.0, + "content": "as the classical solutions. As the vanishing locus of an analytic function,", + "type": "text" + }, + { + "bbox": [ + 407, + 396, + 422, + 407 + ], + "score": 0.9, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 396, + 505, + 407 + ], + "score": 1.0, + "content": "is an analytic space", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 126, + 420 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 126, + 407, + 135, + 416 + ], + "score": 0.78, + "content": "\\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 406, + 205, + 420 + ], + "score": 1.0, + "content": "(Hironaka, 1964,", + "type": "text" + }, + { + "bbox": [ + 205, + 407, + 226, + 418 + ], + "score": 0.52, + "content": "\\ S 0 . 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 406, + 506, + 420 + ], + "score": 1.0, + "content": ", (Griffith & Harris, 1978) and it is interesting to study the geometry of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 416, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 288, + 430 + ], + "score": 1.0, + "content": "this space near the classical solutions. Since", + "type": "text" + }, + { + "bbox": [ + 288, + 418, + 298, + 427 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 416, + 505, + 430 + ], + "score": 1.0, + "content": "is a Kullback-Leibler divergence it is non-negative", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 225, + 441 + ], + "score": 1.0, + "content": "and so it not only vanishes on", + "type": "text" + }, + { + "bbox": [ + 225, + 429, + 239, + 439 + ], + "score": 0.89, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 428, + 255, + 441 + ], + "score": 1.0, + "content": "but", + "type": "text" + }, + { + "bbox": [ + 255, + 429, + 274, + 439 + ], + "score": 0.85, + "content": "\\nabla K", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 428, + 414, + 441 + ], + "score": 1.0, + "content": "also vanishes, hence every point of", + "type": "text" + }, + { + "bbox": [ + 414, + 429, + 429, + 439 + ], + "score": 0.89, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "is a singular point.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 220, + 458 + ], + "score": 1.0, + "content": "Beyond this the geometry of", + "type": "text" + }, + { + "bbox": [ + 221, + 446, + 235, + 456 + ], + "score": 0.89, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 444, + 364, + 458 + ], + "score": 1.0, + "content": "depends on the particular model", + "type": "text" + }, + { + "bbox": [ + 364, + 445, + 403, + 457 + ], + "score": 0.93, + "content": "p ( \\boldsymbol { y } | \\boldsymbol { x } , \\boldsymbol { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 444, + 506, + 458 + ], + "score": 1.0, + "content": "that has been chosen, but", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 456, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 429, + 468 + ], + "score": 1.0, + "content": "some aspects are universal: the nature of program synthesis means that typically", + "type": "text" + }, + { + "bbox": [ + 429, + 456, + 444, + 467 + ], + "score": 0.89, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 456, + 506, + 468 + ], + "score": 1.0, + "content": "is an extended", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "object (i.e. it contains points other than the classical solutions) and the Hessian matrix of second", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 479, + 504, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 215, + 490 + ], + "score": 1.0, + "content": "order partial derivatives of", + "type": "text" + }, + { + "bbox": [ + 215, + 479, + 225, + 488 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 479, + 504, + 490 + ], + "score": 1.0, + "content": "at a classical solution is not invertible - that is, the classical solutions", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 236, + 501 + ], + "score": 1.0, + "content": "are degenerate critical points of", + "type": "text" + }, + { + "bbox": [ + 236, + 490, + 247, + 499 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 489, + 505, + 501 + ], + "score": 1.0, + "content": ". This means that singularity theory is the appropriate branch of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 275, + 513 + ], + "score": 1.0, + "content": "mathematics for studying the geometry of", + "type": "text" + }, + { + "bbox": [ + 275, + 500, + 290, + 511 + ], + "score": 0.9, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "near a classical solution. It also means that the Fisher", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 510, + 185, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 185, + 523 + ], + "score": 1.0, + "content": "information matrix", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 525, + 453, + 553 + ], + "lines": [ + { + "bbox": [ + 156, + 525, + 453, + 553 + ], + "spans": [ + { + "bbox": [ + 156, + 525, + 453, + 553 + ], + "score": 0.93, + "content": "I ( w ) _ { i j } = \\int \\int \\frac { \\partial } { \\partial w _ { i } } \\big [ \\log p ( y | x , w ) \\big ] \\frac { \\partial } { \\partial w _ { j } } \\big [ \\log p ( y | x , w ) \\big ] q ( y | x ) q ( x ) d x d y ,", + "type": "interline_equation", + "image_path": "b8dfea439a1747d3611410930a42886e436c6fc67f00c4da375cd2997dad9780.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 156, + 525, + 453, + 534.3333333333334 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 156, + 534.3333333333334, + 453, + 543.6666666666667 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 156, + 543.6666666666667, + 453, + 553.0000000000001 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 557, + 504, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 506, + 571 + ], + "score": 1.0, + "content": "is degenerate at a classical solution, so that the appropriate branch of statistical learning theory is", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "singular learning theory (Watanabe, 2007; 2009). For an introduction to singular learning theory in", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 580, + 321, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 321, + 592 + ], + "score": 1.0, + "content": "the context of deep learning see (Murfet et al., 2020).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 108, + 596, + 503, + 630 + ], + "lines": [ + { + "bbox": [ + 106, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "Broadly speaking the contribution of this paper is to realise program synthesis within the frame-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "score": 1.0, + "content": "work of singular learning theory, at both a theoretical and an experimental level. In more detail the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 619, + 210, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 210, + 631 + ], + "score": 1.0, + "content": "contents of the paper are:", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 132, + 639, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 133, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 133, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "• We define a staged pseudo-UTM (Appendix E) which is well-suited to experiments with the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 141, + 650, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 141, + 650, + 506, + 664 + ], + "score": 1.0, + "content": "ideas discussed above. Propagating uncertainty about the code through this UTM using the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 141, + 660, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 141, + 660, + 331, + 675 + ], + "score": 1.0, + "content": "ideas of (Clift & Murfet, 2018) defines a triple", + "type": "text" + }, + { + "bbox": [ + 332, + 662, + 365, + 673 + ], + "score": 0.92, + "content": "( p , q , \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 660, + 505, + 675 + ], + "score": 1.0, + "content": "associated to a synthesis problem.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 141, + 672, + 435, + 686 + ], + "spans": [ + { + "bbox": [ + 141, + 672, + 435, + 686 + ], + "score": 1.0, + "content": "This formally embeds program synthesis within singular learning theory.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 140, + 686, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 140, + 686, + 505, + 702 + ], + "score": 1.0, + "content": "We realise this embedding in code by providing an implementation in PyTorch of this prop-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 141, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 141, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "agation of uncertainty through a UTM. 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One approach is to seek a smooth relaxation of the synthesis problem consisting", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 178, + 504, + 194 + ], + "spans": [ + { + "bbox": [ + 104, + 178, + 203, + 194 + ], + "score": 1.0, + "content": "of an analytic manifold", + "type": "text" + }, + { + "bbox": [ + 203, + 181, + 255, + 192 + ], + "score": 0.92, + "content": "W \\bar { \\supseteq } W ^ { c o d e }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 178, + 338, + 194 + ], + "score": 1.0, + "content": "and an extension of", + "type": "text" + }, + { + "bbox": [ + 339, + 182, + 349, + 191 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 178, + 443, + 194 + ], + "score": 1.0, + "content": "to an analytic function", + "type": "text" + }, + { + "bbox": [ + 444, + 181, + 504, + 191 + ], + "score": 0.91, + "content": "K : W \\longrightarrow \\mathbb { R }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 262, + 205 + ], + "score": 1.0, + "content": "so that we can search for the zeros of", + "type": "text" + }, + { + "bbox": [ + 262, + 193, + 273, + 202 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "using gradient descent. 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The particular model we choose is based on the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "semantics of linear logic (Clift & Murfet, 2018). Supposing that such a smooth relaxation has been", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 246, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 226, + 260 + ], + "score": 1.0, + "content": "chosen together with a prior", + "type": "text" + }, + { + "bbox": [ + 226, + 247, + 249, + 259 + ], + "score": 0.92, + "content": "\\varphi ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 246, + 272, + 260 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 272, + 248, + 284, + 257 + ], + "score": 0.67, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 246, + 506, + 260 + ], + "score": 1.0, + "content": ", smooth program synthesis becomes the study of the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 258, + 296, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 259, + 271 + ], + "score": 1.0, + "content": "statistical learning theory of the triple", + "type": "text" + }, + { + "bbox": [ + 259, + 258, + 293, + 270 + ], + "score": 0.92, + "content": "( p , q , \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 258, + 296, + 271 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10, + "bbox_fs": [ + 104, + 168, + 506, + 271 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 275, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "There are perhaps two primary reasons to consider the smooth relaxation. Firstly, one might hope", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "that stochastic gradient descent or techniques like Markov chain Monte Carlo will be effective means", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "of solving the original combinatorial optimisation problem. This is not a new idea (Gulwani et al.,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "score": 1.0, + "content": "2017, §6) but so far its effectiveness for large programs has not been proven. Independently, one", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "might hope to find powerful new mathematical ideas that apply to the relaxed problem and shed light", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 330, + 411, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 411, + 343 + ], + "score": 1.0, + "content": "on the nature of program synthesis. This is the purpose of the present paper.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 275, + 506, + 343 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 345, + 420, + 359 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 421, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 273, + 361 + ], + "score": 1.0, + "content": "Singular learning theory. We denote by", + "type": "text" + }, + { + "bbox": [ + 273, + 346, + 390, + 359 + ], + "score": 0.92, + "content": "W _ { 0 } = \\{ w \\in W | K ( w ) = 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 345, + 421, + 361 + ], + "score": 1.0, + "content": "so that", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 345, + 421, + 361 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 253, + 364, + 358, + 378 + ], + "lines": [ + { + "bbox": [ + 253, + 364, + 358, + 378 + ], + "spans": [ + { + "bbox": [ + 253, + 364, + 358, + 378 + ], + "score": 0.91, + "content": "W _ { 0 } \\cap W ^ { c o d e } \\subseteq W _ { 0 } \\subseteq W", + "type": "interline_equation", + "image_path": "86ade0437cc02ec74cf772862fcd0f85b59d0d58456410aeb8aa4fceae3052a3.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 253, + 364, + 358, + 378 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 505, + 440 + ], + "lines": [ + { + "bbox": [ + 106, + 383, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 133, + 397 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 384, + 183, + 396 + ], + "score": 0.92, + "content": "W _ { 0 } \\cap W ^ { c o d e }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 383, + 506, + 397 + ], + "score": 1.0, + "content": "is the discrete set of solutions to the original synthesis problem. We refer to these", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 396, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 406, + 407 + ], + "score": 1.0, + "content": "as the classical solutions. 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Since", + "type": "text" + }, + { + "bbox": [ + 288, + 418, + 298, + 427 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 416, + 505, + 430 + ], + "score": 1.0, + "content": "is a Kullback-Leibler divergence it is non-negative", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 225, + 441 + ], + "score": 1.0, + "content": "and so it not only vanishes on", + "type": "text" + }, + { + "bbox": [ + 225, + 429, + 239, + 439 + ], + "score": 0.89, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 428, + 255, + 441 + ], + "score": 1.0, + "content": "but", + "type": "text" + }, + { + "bbox": [ + 255, + 429, + 274, + 439 + ], + "score": 0.85, + "content": "\\nabla K", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 428, + 414, + 441 + ], + "score": 1.0, + "content": "also vanishes, hence every point of", + "type": "text" + }, + { + "bbox": [ + 414, + 429, + 429, + 439 + ], + "score": 0.89, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "is a singular point.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 383, + 506, + 441 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 220, + 458 + ], + "score": 1.0, + "content": "Beyond this the geometry of", + "type": "text" + }, + { + "bbox": [ + 221, + 446, + 235, + 456 + ], + "score": 0.89, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 444, + 364, + 458 + ], + "score": 1.0, + "content": "depends on the particular model", + "type": "text" + }, + { + "bbox": [ + 364, + 445, + 403, + 457 + ], + "score": 0.93, + "content": "p ( \\boldsymbol { y } | \\boldsymbol { x } , \\boldsymbol { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 444, + 506, + 458 + ], + "score": 1.0, + "content": "that has been chosen, but", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 456, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 429, + 468 + ], + "score": 1.0, + "content": "some aspects are universal: the nature of program synthesis means that typically", + "type": "text" + }, + { + "bbox": [ + 429, + 456, + 444, + 467 + ], + "score": 0.89, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 456, + 506, + 468 + ], + "score": 1.0, + "content": "is an extended", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "object (i.e. it contains points other than the classical solutions) and the Hessian matrix of second", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 479, + 504, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 215, + 490 + ], + "score": 1.0, + "content": "order partial derivatives of", + "type": "text" + }, + { + "bbox": [ + 215, + 479, + 225, + 488 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 479, + 504, + 490 + ], + "score": 1.0, + "content": "at a classical solution is not invertible - that is, the classical solutions", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 236, + 501 + ], + "score": 1.0, + "content": "are degenerate critical points of", + "type": "text" + }, + { + "bbox": [ + 236, + 490, + 247, + 499 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 489, + 505, + 501 + ], + "score": 1.0, + "content": ". 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For an introduction to singular learning theory in", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 580, + 321, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 321, + 592 + ], + "score": 1.0, + "content": "the context of deep learning see (Murfet et al., 2020).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 557, + 506, + 592 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 596, + 503, + 630 + ], + "lines": [ + { + "bbox": [ + 106, + 596, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 610 + ], + "score": 1.0, + "content": "Broadly speaking the contribution of this paper is to realise program synthesis within the frame-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 620 + ], + "score": 1.0, + "content": "work of singular learning theory, at both a theoretical and an experimental level. In more detail the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 619, + 210, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 210, + 631 + ], + "score": 1.0, + "content": "contents of the paper are:", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42, + "bbox_fs": [ + 106, + 596, + 505, + 631 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 639, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 133, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 133, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "• We define a staged pseudo-UTM (Appendix E) which is well-suited to experiments with the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 141, + 650, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 141, + 650, + 506, + 664 + ], + "score": 1.0, + "content": "ideas discussed above. Propagating uncertainty about the code through this UTM using the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 141, + 660, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 141, + 660, + 331, + 675 + ], + "score": 1.0, + "content": "ideas of (Clift & Murfet, 2018) defines a triple", + "type": "text" + }, + { + "bbox": [ + 332, + 662, + 365, + 673 + ], + "score": 0.92, + "content": "( p , q , \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 660, + 505, + 675 + ], + "score": 1.0, + "content": "associated to a synthesis problem.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 141, + 672, + 435, + 686 + ], + "spans": [ + { + "bbox": [ + 141, + 672, + 435, + 686 + ], + "score": 1.0, + "content": "This formally embeds program synthesis within singular learning theory.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 140, + 686, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 140, + 686, + 505, + 702 + ], + "score": 1.0, + "content": "We realise this embedding in code by providing an implementation in PyTorch of this prop-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 141, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 141, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "agation of uncertainty through a UTM. 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We discuss how", + "type": "text" + }, + { + "bbox": [ + 400, + 158, + 415, + 169 + ], + "score": 0.89, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 158, + 505, + 171 + ], + "score": 1.0, + "content": "is an extended object", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 169, + 468, + 181 + ], + "spans": [ + { + "bbox": [ + 141, + 169, + 351, + 181 + ], + "score": 1.0, + "content": "and how the RLCT relates to the local dimension of", + "type": "text" + }, + { + "bbox": [ + 352, + 169, + 366, + 180 + ], + "score": 0.89, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 169, + 468, + 181 + ], + "score": 1.0, + "content": "near a classical solution.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 107, + 198, + 190, + 210 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 191, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 191, + 211 + ], + "score": 1.0, + "content": "RELATED WORK", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 223, + 505, + 312 + ], + "lines": [ + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "score": 1.0, + "content": "The idea of synthesising Turing machines can be traced back to the work of Solomonoff on inductive", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 235, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 505, + 247 + ], + "score": 1.0, + "content": "inference (Solomonoff, 1964). A more explicit form of the problem was given in Biermann (1972)", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "who proposed an algorithmic method. Machine learning based approaches appear in Schmidhuber", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 255, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 269 + ], + "score": 1.0, + "content": "(1997) and Hutter (2004), which pay particular attention to model complexity, and Gaunt et al.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "(2016) and Freer et al. (2014), the latter using the notion of “universal probabilistic Turing machine”", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "score": 1.0, + "content": "(De Leeuw et al., 1956). A different probabilistic extension of a universal Turing machine was", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "score": 1.0, + "content": "introduced in Clift & Murfet (2018) via linear logic. Studies of the singular geometry of learning", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 301, + 498, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 498, + 313 + ], + "score": 1.0, + "content": "models go back to Amari et al. (2003) and notably, the extensive work of Watanabe (2007; 2009).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 107, + 329, + 414, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 415, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 415, + 343 + ], + "score": 1.0, + "content": "2 TURING MACHINE SYNTHESIS AS SINGULAR LEARNING", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 106, + 355, + 504, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 504, + 367 + ], + "score": 1.0, + "content": "All known approaches to program synthesis can be formulated in terms of a singular learning prob-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 364, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 505, + 378 + ], + "score": 1.0, + "content": "lem. Singular learning theory is the extension of statistical learning theory to account for the fact", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 239, + 389 + ], + "score": 1.0, + "content": "that the set of learned parameters", + "type": "text" + }, + { + "bbox": [ + 240, + 376, + 254, + 388 + ], + "score": 0.89, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 375, + 505, + 389 + ], + "score": 1.0, + "content": "has the structure of an analytic space as opposed to an analytic", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 372, + 400 + ], + "score": 1.0, + "content": "manifold (Watanabe, 2007; 2009). It is organised around triples", + "type": "text" + }, + { + "bbox": [ + 372, + 388, + 406, + 399 + ], + "score": 0.92, + "content": "( p , q , \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "consisting of a class of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 398, + 415, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 138, + 411 + ], + "score": 1.0, + "content": "models", + "type": "text" + }, + { + "bbox": [ + 138, + 398, + 226, + 410 + ], + "score": 0.93, + "content": "\\{ p ( y | x , w ) : w \\in W \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 398, + 303, + 411 + ], + "score": 1.0, + "content": ", a true distribution", + "type": "text" + }, + { + "bbox": [ + 303, + 398, + 330, + 410 + ], + "score": 0.93, + "content": "q ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 398, + 377, + 411 + ], + "score": 1.0, + "content": "and a prior", + "type": "text" + }, + { + "bbox": [ + 377, + 400, + 385, + 410 + ], + "score": 0.83, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 398, + 398, + 411 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 399, + 399, + 410, + 408 + ], + "score": 0.79, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 398, + 415, + 411 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 415, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 104, + 414, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 104, + 414, + 393, + 429 + ], + "score": 1.0, + "content": "In our approach we fix a Universal Turing Machine (UTM), denoted", + "type": "text" + }, + { + "bbox": [ + 393, + 416, + 402, + 425 + ], + "score": 0.66, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 414, + 505, + 429 + ], + "score": 1.0, + "content": ", with a description tape", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 425, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 440 + ], + "score": 1.0, + "content": "(which specifies the code of the Turing machine to be executed), a work tape (simulating the tape", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 435, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 104, + 435, + 506, + 452 + ], + "score": 1.0, + "content": "of that Turing machine during its operation) and a state tape (simulating the state of that Turing", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 447, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 104, + 447, + 426, + 462 + ], + "score": 1.0, + "content": "machine). The general statistical learning problem that can be formulated using", + "type": "text" + }, + { + "bbox": [ + 427, + 448, + 436, + 459 + ], + "score": 0.69, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 447, + 505, + 462 + ], + "score": 1.0, + "content": "is the following:", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 104, + 459, + 209, + 472 + ], + "score": 1.0, + "content": "given some initial string", + "type": "text" + }, + { + "bbox": [ + 210, + 461, + 217, + 469 + ], + "score": 0.69, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "on the work tape, predict the state of the simulated machine and the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 460, + 483 + ], + "score": 1.0, + "content": "contents of the work tape after some specified number of steps (Clift & Murfet, 2018,", + "type": "text" + }, + { + "bbox": [ + 461, + 470, + 481, + 482 + ], + "score": 0.65, + "content": "\\ S 7 . 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 470, + 505, + 483 + ], + "score": 1.0, + "content": ". 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We discuss how", + "type": "text" + }, + { + "bbox": [ + 400, + 158, + 415, + 169 + ], + "score": 0.89, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 158, + 505, + 171 + ], + "score": 1.0, + "content": "is an extended object", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 169, + 468, + 181 + ], + "spans": [ + { + "bbox": [ + 141, + 169, + 351, + 181 + ], + "score": 1.0, + "content": "and how the RLCT relates to the local dimension of", + "type": "text" + }, + { + "bbox": [ + 352, + 169, + 366, + 180 + ], + "score": 0.89, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 169, + 468, + 181 + ], + "score": 1.0, + "content": "near a classical solution.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5, + "bbox_fs": [ + 132, + 82, + 507, + 181 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 198, + 190, + 210 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 191, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 191, + 211 + ], + "score": 1.0, + "content": "RELATED WORK", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 223, + 505, + 312 + ], + "lines": [ + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 236 + ], + "score": 1.0, + "content": "The idea of synthesising Turing machines can be traced back to the work of Solomonoff on inductive", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 235, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 505, + 247 + ], + "score": 1.0, + "content": "inference (Solomonoff, 1964). A more explicit form of the problem was given in Biermann (1972)", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "who proposed an algorithmic method. Machine learning based approaches appear in Schmidhuber", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 255, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 269 + ], + "score": 1.0, + "content": "(1997) and Hutter (2004), which pay particular attention to model complexity, and Gaunt et al.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "(2016) and Freer et al. (2014), the latter using the notion of “universal probabilistic Turing machine”", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "score": 1.0, + "content": "(De Leeuw et al., 1956). A different probabilistic extension of a universal Turing machine was", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "score": 1.0, + "content": "introduced in Clift & Murfet (2018) via linear logic. Studies of the singular geometry of learning", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 301, + 498, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 498, + 313 + ], + "score": 1.0, + "content": "models go back to Amari et al. (2003) and notably, the extensive work of Watanabe (2007; 2009).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 223, + 506, + 313 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 329, + 414, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 415, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 415, + 343 + ], + "score": 1.0, + "content": "2 TURING MACHINE SYNTHESIS AS SINGULAR LEARNING", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 106, + 355, + 504, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 504, + 367 + ], + "score": 1.0, + "content": "All known approaches to program synthesis can be formulated in terms of a singular learning prob-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 364, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 505, + 378 + ], + "score": 1.0, + "content": "lem. 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The triple", + "type": "text" + }, + { + "bbox": [ + 212, + 518, + 245, + 530 + ], + "score": 0.92, + "content": "( p , q , \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 517, + 436, + 532 + ], + "score": 1.0, + "content": "associated to a synthesis problem is the model", + "type": "text" + }, + { + "bbox": [ + 436, + 520, + 443, + 529 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 517, + 506, + 532 + ], + "score": 1.0, + "content": "of (5) together", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 528, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 209, + 542 + ], + "score": 1.0, + "content": "with the true distribution", + "type": "text" + }, + { + "bbox": [ + 209, + 531, + 216, + 540 + ], + "score": 0.81, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 528, + 292, + 542 + ], + "score": 1.0, + "content": "and uniform prior", + "type": "text" + }, + { + "bbox": [ + 292, + 531, + 300, + 540 + ], + "score": 0.82, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 528, + 397, + 542 + ], + "score": 1.0, + "content": "on the parameter space", + "type": "text" + }, + { + "bbox": [ + 398, + 529, + 410, + 539 + ], + "score": 0.67, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 528, + 506, + 542 + ], + "score": 1.0, + "content": ". The Kullback-Leibler", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 142, + 552 + ], + "score": 1.0, + "content": "function", + "type": "text" + }, + { + "bbox": [ + 142, + 540, + 168, + 551 + ], + "score": 0.94, + "content": "K ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "of the synthesis problem is defined by (1) and a solution to the synthesis problem is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 550, + 354, + 563 + ], + "spans": [ + { + "bbox": [ + 104, + 550, + 147, + 563 + ], + "score": 1.0, + "content": "a point of", + "type": "text" + }, + { + "bbox": [ + 147, + 552, + 162, + 562 + ], + "score": 0.89, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 550, + 298, + 563 + ], + "score": 1.0, + "content": ". 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We", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 627, + 457, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 156, + 640 + ], + "score": 1.0, + "content": "assume that", + "type": "text" + }, + { + "bbox": [ + 156, + 627, + 183, + 639 + ], + "score": 0.92, + "content": "{ \\bar { q } } ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 627, + 312, + 640 + ], + "score": 1.0, + "content": "is realisable that is, there exists", + "type": "text" + }, + { + "bbox": [ + 312, + 627, + 348, + 639 + ], + "score": 0.92, + "content": "w _ { 0 } \\in W", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 627, + 369, + 640 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 370, + 627, + 452, + 639 + ], + "score": 0.93, + "content": "q ( y | x ) = p ( y | x , w _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 627, + 457, + 640 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 644, + 505, + 710 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 504, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 140, + 657 + ], + "score": 1.0, + "content": "A triple", + "type": "text" + }, + { + "bbox": [ + 140, + 644, + 173, + 656 + ], + "score": 0.92, + "content": "( p , q , \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 642, + 392, + 657 + ], + "score": 1.0, + "content": "is regular if the model is identifiable, ie. for all inputs", + "type": "text" + }, + { + "bbox": [ + 392, + 644, + 423, + 654 + ], + "score": 0.91, + "content": "x \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 642, + 495, + 657 + ], + "score": 1.0, + "content": ", the map sending", + "type": "text" + }, + { + "bbox": [ + 495, + 646, + 504, + 654 + ], + "score": 0.64, + "content": "w", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 272, + 668 + ], + "score": 1.0, + "content": "to the conditional probability distribution", + "type": "text" + }, + { + "bbox": [ + 272, + 655, + 311, + 667 + ], + "score": 0.93, + "content": "p ( \\boldsymbol { y } | \\boldsymbol { x } , \\boldsymbol { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "is one-to-one, and the Fisher information matrix", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 472, + 679 + ], + "score": 1.0, + "content": "is non-degenerate. 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We", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 627, + 457, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 156, + 640 + ], + "score": 1.0, + "content": "assume that", + "type": "text" + }, + { + "bbox": [ + 156, + 627, + 183, + 639 + ], + "score": 0.92, + "content": "{ \\bar { q } } ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 627, + 312, + 640 + ], + "score": 1.0, + "content": "is realisable that is, there exists", + "type": "text" + }, + { + "bbox": [ + 312, + 627, + 348, + 639 + ], + "score": 0.92, + "content": "w _ { 0 } \\in W", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 627, + 369, + 640 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 370, + 627, + 452, + 639 + ], + "score": 0.93, + "content": "q ( y | x ) = p ( y | x , w _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 627, + 457, + 640 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 571, + 506, + 640 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 644, + 505, + 710 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 504, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 140, + 657 + ], + "score": 1.0, + "content": "A triple", + "type": "text" + }, + { + "bbox": [ + 140, + 644, + 173, + 656 + ], + "score": 0.92, + "content": "( p , q , \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 642, + 392, + 657 + ], + "score": 1.0, + "content": "is regular if the model is identifiable, ie. for all inputs", + "type": "text" + }, + { + "bbox": [ + 392, + 644, + 423, + 654 + ], + "score": 0.91, + "content": "x \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 642, + 495, + 657 + ], + "score": 1.0, + "content": ", the map sending", + "type": "text" + }, + { + "bbox": [ + 495, + 646, + 504, + 654 + ], + "score": 0.64, + "content": "w", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 272, + 668 + ], + "score": 1.0, + "content": "to the conditional probability distribution", + "type": "text" + }, + { + "bbox": [ + 272, + 655, + 311, + 667 + ], + "score": 0.93, + "content": "p ( \\boldsymbol { y } | \\boldsymbol { x } , \\boldsymbol { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "is one-to-one, and the Fisher information matrix", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 472, + 679 + ], + "score": 1.0, + "content": "is non-degenerate. Otherwise, the learning machine is strictly singular (Watanabe, 2009,", + "type": "text" + }, + { + "bbox": [ + 473, + 666, + 500, + 678 + ], + "score": 0.74, + "content": "\\ S 1 . 2 . 1 ", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 666, + 505, + 679 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "Triples arising from synthesis problems are typically singular: in Example 2.5 below we show an", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 285, + 701 + ], + "score": 1.0, + "content": "explicit example where multiple parameters", + "type": "text" + }, + { + "bbox": [ + 286, + 690, + 294, + 698 + ], + "score": 0.73, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "determine the same model, and in Example C.2 we", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 698, + 258, + 712 + ], + "score": 1.0, + "content": "give an example where the Hessian of", + "type": "text" + }, + { + "bbox": [ + 259, + 700, + 268, + 708 + ], + "score": 0.84, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 698, + 384, + 712 + ], + "score": 1.0, + "content": "is degenerate everywhere on", + "type": "text" + }, + { + "bbox": [ + 385, + 699, + 399, + 710 + ], + "score": 0.89, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "(Watanabe, 2009, §1.1.3).", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 642, + 506, + 712 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "Remark 2.3. Non-deterministic synthesis problems arise naturally in various contexts, for example", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "in the fitting of algorithms to the behaviour of deep reinforcement learning agents. Suppose an agent", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 344, + 117 + ], + "score": 1.0, + "content": "is acting in an environment with starting states encoded by", + "type": "text" + }, + { + "bbox": [ + 345, + 105, + 376, + 115 + ], + "score": 0.91, + "content": "x \\in \\Sigma ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "and possible episode end states", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 119, + 128 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 116, + 147, + 127 + ], + "score": 0.91, + "content": "y \\in Q", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 115, + 445, + 128 + ], + "score": 1.0, + "content": ". Even if the optimal policy is known to determine a computable function", + "type": "text" + }, + { + "bbox": [ + 445, + 115, + 489, + 126 + ], + "score": 0.9, + "content": "\\Sigma ^ { * } \\longrightarrow Q", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 504, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 453, + 139 + ], + "score": 1.0, + "content": "statistics of the observed behaviour after finite training time will only provide a function", + "type": "text" + }, + { + "bbox": [ + 453, + 127, + 504, + 138 + ], + "score": 0.9, + "content": "\\Sigma ^ { * } \\longrightarrow \\Delta Q", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 495, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 495, + 149 + ], + "score": 1.0, + "content": "and if we wish to fit algorithms to behaviour it makes sense to deal with this uncertainty directly.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 152, + 505, + 186 + ], + "lines": [ + { + "bbox": [ + 105, + 151, + 504, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 185, + 165 + ], + "score": 1.0, + "content": "Definition 2.4. Let", + "type": "text" + }, + { + "bbox": [ + 185, + 152, + 219, + 164 + ], + "score": 0.92, + "content": "( p , q , \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 151, + 504, + 165 + ], + "score": 1.0, + "content": "be the triple associated to a synthesis problem. 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In", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 217, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 230 + ], + "score": 1.0, + "content": "Section 3 we relate the RLCT to Kolmogorov complexity and in Section 5 we estimate the RLCT of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 228, + 471, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 471, + 241 + ], + "score": 1.0, + "content": "the synthesis problem detectA given below, using the method explained in Appendix A.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 243, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 433, + 255 + ], + "score": 1.0, + "content": "Example 2.5 (detectA). The deterministic synthesis problem detectA has", + "type": "text" + }, + { + "bbox": [ + 433, + 243, + 501, + 255 + ], + "score": 0.92, + "content": "\\Sigma = \\{ \\boxed \\} , A , B \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 242, + 506, + 255 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 107, + 254, + 196, + 266 + ], + "score": 0.9, + "content": "Q = \\{ { \\mathrm { r e j e c t } } , { \\mathrm { a c c e p t } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 254, + 214, + 267 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 214, + 254, + 242, + 266 + ], + "score": 0.94, + "content": "q ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 254, + 434, + 267 + ], + "score": 1.0, + "content": "is determined by the function taking in a string", + "type": "text" + }, + { + "bbox": [ + 434, + 257, + 442, + 264 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 254, + 454, + 267 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 454, + 255, + 463, + 264 + ], + "score": 0.79, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 254, + 488, + 267 + ], + "score": 1.0, + "content": "’s and", + "type": "text" + }, + { + "bbox": [ + 488, + 255, + 497, + 264 + ], + "score": 0.8, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "’s", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 322, + 277 + ], + "score": 1.0, + "content": "and returning the state accept if the string contains an", + "type": "text" + }, + { + "bbox": [ + 322, + 266, + 331, + 275 + ], + "score": 0.69, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "and state reject otherwise. The conditional", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 476, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 173, + 289 + ], + "score": 1.0, + "content": "true distribution", + "type": "text" + }, + { + "bbox": [ + 173, + 276, + 200, + 288 + ], + "score": 0.93, + "content": "q ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 276, + 476, + 289 + ], + "score": 1.0, + "content": "is realisable because this function is computed by a Turing machine.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 293, + 505, + 360 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 377, + 306 + ], + "score": 1.0, + "content": "Two solutions are shown in Figure 2. On the left is a parameter", + "type": "text" + }, + { + "bbox": [ + 377, + 292, + 456, + 305 + ], + "score": 0.91, + "content": "w _ { l } \\in \\ b { W } _ { 0 } \\setminus \\ b { W } ^ { c o d e }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 291, + 506, + 306 + ], + "score": 1.0, + "content": "and on the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 300, + 507, + 317 + ], + "spans": [ + { + "bbox": [ + 104, + 300, + 138, + 317 + ], + "score": 1.0, + "content": "right is", + "type": "text" + }, + { + "bbox": [ + 138, + 304, + 216, + 315 + ], + "score": 0.91, + "content": "w _ { r } \\in W _ { 0 } \\cap W ^ { c o d e }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 300, + 335, + 317 + ], + "score": 1.0, + "content": ". Varying the distributions in", + "type": "text" + }, + { + "bbox": [ + 335, + 305, + 347, + 315 + ], + "score": 0.82, + "content": "w _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 300, + 507, + 317 + ], + "score": 1.0, + "content": "that have nonzero entropy we obtain a", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 159, + 327 + ], + "score": 1.0, + "content": "submanifold", + "type": "text" + }, + { + "bbox": [ + 159, + 316, + 195, + 326 + ], + "score": 0.9, + "content": "V \\subseteq W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 315, + 241, + 327 + ], + "score": 1.0, + "content": "containing", + "type": "text" + }, + { + "bbox": [ + 241, + 317, + 253, + 326 + ], + "score": 0.78, + "content": "w _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "of dimension 14. This leads by (Watanabe, 2009, Remark 7.3)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 325, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 219, + 338 + ], + "score": 1.0, + "content": "to a bound on the RLCT of", + "type": "text" + }, + { + "bbox": [ + 219, + 325, + 306, + 339 + ], + "score": 0.92, + "content": "\\lambda \\le \\frac { 1 } { 2 } ( 3 0 - 1 4 ) = 8", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 326, + 505, + 338 + ], + "score": 1.0, + "content": "which is consistent with the experimental results", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 336, + 504, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 488, + 350 + ], + "score": 1.0, + "content": "in Table 1. This highlights that solutions need not lie at vertices of the probability simplex, and", + "type": "text" + }, + { + "bbox": [ + 489, + 337, + 504, + 348 + ], + "score": 0.88, + "content": "W _ { 0 }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 348, + 424, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 424, + 361 + ], + "score": 1.0, + "content": "may contain a high-dimensional submanifold around a given classical solution.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "image", + "bbox": [ + 117, + 371, + 492, + 463 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 371, + 492, + 463 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 371, + 492, + 463 + ], + "spans": [ + { + "bbox": [ + 117, + 371, + 492, + 463 + ], + "score": 0.968, + "type": "image", + "image_path": "94e2cb3be00f1414e5f172ce0b09cdea4b00fe6fbc48f36b360a03838c63cdbb.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 117, + 371, + 492, + 401.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 117, + 401.6666666666667, + 492, + 432.33333333333337 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 117, + 432.33333333333337, + 492, + 463.00000000000006 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 147, + 473, + 462, + 486 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 147, + 471, + 464, + 488 + ], + "spans": [ + { + "bbox": [ + 147, + 471, + 464, + 488 + ], + "score": 1.0, + "content": "Figure 2: Visualisation of two solutions for the synthesis problem detectA .", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + } + ], + "index": 25.0 + }, + { + "type": "title", + "bbox": [ + 107, + 505, + 243, + 517 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 244, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 244, + 519 + ], + "score": 1.0, + "content": "2.1 THE SYNTHESIS PROCESS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 526, + 505, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 526, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 506, + 539 + ], + "score": 1.0, + "content": "Synthesis is a problem because we do not assume that the true distribution is known: for example, if", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 107, + 537, + 134, + 550 + ], + "score": 0.91, + "content": "q { \\dot { ( } } y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 537, + 320, + 550 + ], + "score": 1.0, + "content": "is deterministic and the associated function is", + "type": "text" + }, + { + "bbox": [ + 320, + 538, + 378, + 549 + ], + "score": 0.91, + "content": "f : \\Sigma ^ { * } \\longrightarrow Q", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 537, + 505, + 550 + ], + "score": 1.0, + "content": ", we assume that some example", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 547, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 129, + 561 + ], + "score": 1.0, + "content": "pairs", + "type": "text" + }, + { + "bbox": [ + 129, + 549, + 167, + 560 + ], + "score": 0.92, + "content": "( x , f ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 547, + 373, + 561 + ], + "score": 1.0, + "content": "are known but no general algorithm for computing", + "type": "text" + }, + { + "bbox": [ + 374, + 549, + 381, + 560 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 547, + 505, + 561 + ], + "score": 1.0, + "content": "is known (if it were, synthesis", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 555, + 504, + 576 + ], + "spans": [ + { + "bbox": [ + 104, + 555, + 423, + 576 + ], + "score": 1.0, + "content": "would have already been performed). In practice synthesis starts with a sample", + "type": "text" + }, + { + "bbox": [ + 423, + 559, + 504, + 572 + ], + "score": 0.93, + "content": "D _ { n } = \\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { n }", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 568, + 370, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 128, + 584 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 129, + 570, + 157, + 582 + ], + "score": 0.93, + "content": "q ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 568, + 370, + 584 + ], + "score": 1.0, + "content": "with associated empirical Kullback-Leibler distance", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 587, + 374, + 620 + ], + "lines": [ + { + "bbox": [ + 237, + 587, + 374, + 620 + ], + "spans": [ + { + "bbox": [ + 237, + 587, + 374, + 620 + ], + "score": 0.95, + "content": "K _ { n } ( w ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\log \\frac { q ( y _ { i } | x _ { i } ) } { p ( y _ { i } | x _ { i } , w ) } .", + "type": "interline_equation", + "image_path": "7d14e8fbfe73c032c6de880f843e24e7d2034b028aec0b312f8aaa04795f064c.jpg" + } + ] + } + ], + "index": 33.5, + "virtual_lines": [ + { + "bbox": [ + 237, + 587, + 374, + 603.5 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 237, + 603.5, + 374, + 620.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 626, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 104, + 624, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 104, + 624, + 288, + 640 + ], + "score": 1.0, + "content": "If the synthesis problem is deterministic and", + "type": "text" + }, + { + "bbox": [ + 288, + 626, + 334, + 637 + ], + "score": 0.91, + "content": "u \\in W ^ { c o d e }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 624, + 356, + 640 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 356, + 627, + 404, + 639 + ], + "score": 0.93, + "content": "K _ { n } ( u ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 624, + 461, + 640 + ], + "score": 1.0, + "content": "if and only if", + "type": "text" + }, + { + "bbox": [ + 461, + 629, + 468, + 637 + ], + "score": 0.74, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 624, + 505, + 640 + ], + "score": 1.0, + "content": "explains", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 211, + 650 + ], + "score": 1.0, + "content": "the data in the sense that", + "type": "text" + }, + { + "bbox": [ + 212, + 638, + 284, + 650 + ], + "score": 0.91, + "content": "\\operatorname { s t e p } ^ { t } ( x _ { i } , u ) = y _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 637, + 301, + 650 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 302, + 639, + 349, + 649 + ], + "score": 0.9, + "content": "1 \\leq i \\leq n", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 637, + 506, + 650 + ], + "score": 1.0, + "content": ". We now review two natural ways of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 647, + 338, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 338, + 663 + ], + "score": 1.0, + "content": "finding such solutions in the context of machine learning.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "Synthesis by stochastic gradient descent (SGD). The first approach is to view the process of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 368, + 690 + ], + "score": 1.0, + "content": "program synthesis as stochastic gradient descent for the function", + "type": "text" + }, + { + "bbox": [ + 369, + 677, + 428, + 687 + ], + "score": 0.9, + "content": "K : W \\longrightarrow \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 677, + 470, + 690 + ], + "score": 1.0, + "content": ". We view", + "type": "text" + }, + { + "bbox": [ + 471, + 677, + 486, + 688 + ], + "score": 0.89, + "content": "D _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "as a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 286, + 700 + ], + "score": 1.0, + "content": "large training set and further sample subsets", + "type": "text" + }, + { + "bbox": [ + 286, + 688, + 303, + 699 + ], + "score": 0.89, + "content": "D _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 687, + 325, + 700 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 325, + 689, + 357, + 698 + ], + "score": 0.87, + "content": "m \\ll n", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 687, + 413, + 700 + ], + "score": 1.0, + "content": "and compute", + "type": "text" + }, + { + "bbox": [ + 413, + 688, + 438, + 699 + ], + "score": 0.91, + "content": "\\nabla K _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "to take gradient", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 162, + 712 + ], + "score": 1.0, + "content": "descent steps", + "type": "text" + }, + { + "bbox": [ + 162, + 699, + 270, + 711 + ], + "score": 0.9, + "content": "w _ { i + 1 } = w _ { i } - \\eta \\nabla K _ { m } ( w _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 698, + 364, + 712 + ], + "score": 1.0, + "content": "for some learning rate", + "type": "text" + }, + { + "bbox": [ + 365, + 701, + 371, + 710 + ], + "score": 0.8, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 698, + 506, + 712 + ], + "score": 1.0, + "content": ". Stochastic gradient descent has", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 413, + 722 + ], + "score": 1.0, + "content": "the advantage (in principle) of scaling to high-dimensional parameter spaces", + "type": "text" + }, + { + "bbox": [ + 413, + 710, + 425, + 720 + ], + "score": 0.74, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 709, + 506, + 722 + ], + "score": 1.0, + "content": ", but in practice it is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 720, + 414, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 315, + 733 + ], + "score": 1.0, + "content": "challenging to use gradient descent to find points of", + "type": "text" + }, + { + "bbox": [ + 315, + 721, + 330, + 732 + ], + "score": 0.88, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 720, + 414, + 733 + ], + "score": 1.0, + "content": "(Gaunt et al., 2016).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "Remark 2.3. Non-deterministic synthesis problems arise naturally in various contexts, for example", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "in the fitting of algorithms to the behaviour of deep reinforcement learning agents. Suppose an agent", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 344, + 117 + ], + "score": 1.0, + "content": "is acting in an environment with starting states encoded by", + "type": "text" + }, + { + "bbox": [ + 345, + 105, + 376, + 115 + ], + "score": 0.91, + "content": "x \\in \\Sigma ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "and possible episode end states", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 119, + 128 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 116, + 147, + 127 + ], + "score": 0.91, + "content": "y \\in Q", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 115, + 445, + 128 + ], + "score": 1.0, + "content": ". Even if the optimal policy is known to determine a computable function", + "type": "text" + }, + { + "bbox": [ + 445, + 115, + 489, + 126 + ], + "score": 0.9, + "content": "\\Sigma ^ { * } \\longrightarrow Q", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 504, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 453, + 139 + ], + "score": 1.0, + "content": "statistics of the observed behaviour after finite training time will only provide a function", + "type": "text" + }, + { + "bbox": [ + 453, + 127, + 504, + 138 + ], + "score": 0.9, + "content": "\\Sigma ^ { * } \\longrightarrow \\Delta Q", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 495, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 495, + 149 + ], + "score": 1.0, + "content": "and if we wish to fit algorithms to behaviour it makes sense to deal with this uncertainty directly.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 81, + 506, + 149 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 152, + 505, + 186 + ], + "lines": [ + { + "bbox": [ + 105, + 151, + 504, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 185, + 165 + ], + "score": 1.0, + "content": "Definition 2.4. Let", + "type": "text" + }, + { + "bbox": [ + 185, + 152, + 219, + 164 + ], + "score": 0.92, + "content": "( p , q , \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 151, + 504, + 165 + ], + "score": 1.0, + "content": "be the triple associated to a synthesis problem. 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In", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 217, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 230 + ], + "score": 1.0, + "content": "Section 3 we relate the RLCT to Kolmogorov complexity and in Section 5 we estimate the RLCT of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 228, + 471, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 471, + 241 + ], + "score": 1.0, + "content": "the synthesis problem detectA given below, using the method explained in Appendix A.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 195, + 506, + 241 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 243, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 242, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 433, + 255 + ], + "score": 1.0, + "content": "Example 2.5 (detectA). The deterministic synthesis problem detectA has", + "type": "text" + }, + { + "bbox": [ + 433, + 243, + 501, + 255 + ], + "score": 0.92, + "content": "\\Sigma = \\{ \\boxed \\} , A , B \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 242, + 506, + 255 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 107, + 254, + 196, + 266 + ], + "score": 0.9, + "content": "Q = \\{ { \\mathrm { r e j e c t } } , { \\mathrm { a c c e p t } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 254, + 214, + 267 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 214, + 254, + 242, + 266 + ], + "score": 0.94, + "content": "q ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 254, + 434, + 267 + ], + "score": 1.0, + "content": "is determined by the function taking in a string", + "type": "text" + }, + { + "bbox": [ + 434, + 257, + 442, + 264 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 254, + 454, + 267 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 454, + 255, + 463, + 264 + ], + "score": 0.79, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 254, + 488, + 267 + ], + "score": 1.0, + "content": "’s and", + "type": "text" + }, + { + "bbox": [ + 488, + 255, + 497, + 264 + ], + "score": 0.8, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "’s", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 322, + 277 + ], + "score": 1.0, + "content": "and returning the state accept if the string contains an", + "type": "text" + }, + { + "bbox": [ + 322, + 266, + 331, + 275 + ], + "score": 0.69, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "and state reject otherwise. 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On the left is a parameter", + "type": "text" + }, + { + "bbox": [ + 377, + 292, + 456, + 305 + ], + "score": 0.91, + "content": "w _ { l } \\in \\ b { W } _ { 0 } \\setminus \\ b { W } ^ { c o d e }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 291, + 506, + 306 + ], + "score": 1.0, + "content": "and on the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 300, + 507, + 317 + ], + "spans": [ + { + "bbox": [ + 104, + 300, + 138, + 317 + ], + "score": 1.0, + "content": "right is", + "type": "text" + }, + { + "bbox": [ + 138, + 304, + 216, + 315 + ], + "score": 0.91, + "content": "w _ { r } \\in W _ { 0 } \\cap W ^ { c o d e }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 300, + 335, + 317 + ], + "score": 1.0, + "content": ". Varying the distributions in", + "type": "text" + }, + { + "bbox": [ + 335, + 305, + 347, + 315 + ], + "score": 0.82, + "content": "w _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 300, + 507, + 317 + ], + "score": 1.0, + "content": "that have nonzero entropy we obtain a", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 159, + 327 + ], + "score": 1.0, + "content": "submanifold", + "type": "text" + }, + { + "bbox": [ + 159, + 316, + 195, + 326 + ], + "score": 0.9, + "content": "V \\subseteq W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 315, + 241, + 327 + ], + "score": 1.0, + "content": "containing", + "type": "text" + }, + { + "bbox": [ + 241, + 317, + 253, + 326 + ], + "score": 0.78, + "content": "w _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "of dimension 14. This leads by (Watanabe, 2009, Remark 7.3)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 325, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 219, + 338 + ], + "score": 1.0, + "content": "to a bound on the RLCT of", + "type": "text" + }, + { + "bbox": [ + 219, + 325, + 306, + 339 + ], + "score": 0.92, + "content": "\\lambda \\le \\frac { 1 } { 2 } ( 3 0 - 1 4 ) = 8", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 326, + 505, + 338 + ], + "score": 1.0, + "content": "which is consistent with the experimental results", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 336, + 504, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 488, + 350 + ], + "score": 1.0, + "content": "in Table 1. This highlights that solutions need not lie at vertices of the probability simplex, and", + "type": "text" + }, + { + "bbox": [ + 489, + 337, + 504, + 348 + ], + "score": 0.88, + "content": "W _ { 0 }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 348, + 424, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 424, + 361 + ], + "score": 1.0, + "content": "may contain a high-dimensional submanifold around a given classical solution.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5, + "bbox_fs": [ + 104, + 291, + 507, + 361 + ] + }, + { + "type": "image", + "bbox": [ + 117, + 371, + 492, + 463 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 117, + 371, + 492, + 463 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 371, + 492, + 463 + ], + "spans": [ + { + "bbox": [ + 117, + 371, + 492, + 463 + ], + "score": 0.968, + "type": "image", + "image_path": "94e2cb3be00f1414e5f172ce0b09cdea4b00fe6fbc48f36b360a03838c63cdbb.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 117, + 371, + 492, + 401.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 117, + 401.6666666666667, + 492, + 432.33333333333337 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 117, + 432.33333333333337, + 492, + 463.00000000000006 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 147, + 473, + 462, + 486 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 147, + 471, + 464, + 488 + ], + "spans": [ + { + "bbox": [ + 147, + 471, + 464, + 488 + ], + "score": 1.0, + "content": "Figure 2: Visualisation of two solutions for the synthesis problem detectA .", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + } + ], + "index": 25.0 + }, + { + "type": "title", + "bbox": [ + 107, + 505, + 243, + 517 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 244, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 244, + 519 + ], + "score": 1.0, + "content": "2.1 THE SYNTHESIS PROCESS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 526, + 505, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 526, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 506, + 539 + ], + "score": 1.0, + "content": "Synthesis is a problem because we do not assume that the true distribution is known: for example, if", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 107, + 537, + 134, + 550 + ], + "score": 0.91, + "content": "q { \\dot { ( } } y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 537, + 320, + 550 + ], + "score": 1.0, + "content": "is deterministic and the associated function is", + "type": "text" + }, + { + "bbox": [ + 320, + 538, + 378, + 549 + ], + "score": 0.91, + "content": "f : \\Sigma ^ { * } \\longrightarrow Q", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 537, + 505, + 550 + ], + "score": 1.0, + "content": ", we assume that some example", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 547, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 129, + 561 + ], + "score": 1.0, + "content": "pairs", + "type": "text" + }, + { + "bbox": [ + 129, + 549, + 167, + 560 + ], + "score": 0.92, + "content": "( x , f ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 547, + 373, + 561 + ], + "score": 1.0, + "content": "are known but no general algorithm for computing", + "type": "text" + }, + { + "bbox": [ + 374, + 549, + 381, + 560 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 547, + 505, + 561 + ], + "score": 1.0, + "content": "is known (if it were, synthesis", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 555, + 504, + 576 + ], + "spans": [ + { + "bbox": [ + 104, + 555, + 423, + 576 + ], + "score": 1.0, + "content": "would have already been performed). 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We now review two natural ways of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 647, + 338, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 338, + 663 + ], + "score": 1.0, + "content": "finding such solutions in the context of machine learning.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 624, + 506, + 663 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "Synthesis by stochastic gradient descent (SGD). The first approach is to view the process of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 368, + 690 + ], + "score": 1.0, + "content": "program synthesis as stochastic gradient descent for the function", + "type": "text" + }, + { + "bbox": [ + 369, + 677, + 428, + 687 + ], + "score": 0.9, + "content": "K : W \\longrightarrow \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 677, + 470, + 690 + ], + "score": 1.0, + "content": ". We view", + "type": "text" + }, + { + "bbox": [ + 471, + 677, + 486, + 688 + ], + "score": 0.89, + "content": "D _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "as a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 286, + 700 + ], + "score": 1.0, + "content": "large training set and further sample subsets", + "type": "text" + }, + { + "bbox": [ + 286, + 688, + 303, + 699 + ], + "score": 0.89, + "content": "D _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 687, + 325, + 700 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 325, + 689, + 357, + 698 + ], + "score": 0.87, + "content": "m \\ll n", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 687, + 413, + 700 + ], + "score": 1.0, + "content": "and compute", + "type": "text" + }, + { + "bbox": [ + 413, + 688, + 438, + 699 + ], + "score": 0.91, + "content": "\\nabla K _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "to take gradient", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 162, + 712 + ], + "score": 1.0, + "content": "descent steps", + "type": "text" + }, + { + "bbox": [ + 162, + 699, + 270, + 711 + ], + "score": 0.9, + "content": "w _ { i + 1 } = w _ { i } - \\eta \\nabla K _ { m } ( w _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 698, + 364, + 712 + ], + "score": 1.0, + "content": "for some learning rate", + "type": "text" + }, + { + "bbox": [ + 365, + 701, + 371, + 710 + ], + "score": 0.8, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 698, + 506, + 712 + ], + "score": 1.0, + "content": ". Stochastic gradient descent has", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 413, + 722 + ], + "score": 1.0, + "content": "the advantage (in principle) of scaling to high-dimensional parameter spaces", + "type": "text" + }, + { + "bbox": [ + 413, + 710, + 425, + 720 + ], + "score": 0.74, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 709, + 506, + 722 + ], + "score": 1.0, + "content": ", but in practice it is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 720, + 414, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 315, + 733 + ], + "score": 1.0, + "content": "challenging to use gradient descent to find points of", + "type": "text" + }, + { + "bbox": [ + 315, + 721, + 330, + 732 + ], + "score": 0.88, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 720, + 414, + 733 + ], + "score": 1.0, + "content": "(Gaunt et al., 2016).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5, + "bbox_fs": [ + 104, + 666, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Synthesis by sampling. The second approach is to consider the Bayesian posterior associated to the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 491, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 419, + 108 + ], + "score": 1.0, + "content": "synthesis problem, which can be viewed as an update on the prior distribution", + "type": "text" + }, + { + "bbox": [ + 419, + 96, + 427, + 105 + ], + "score": 0.82, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 93, + 476, + 108 + ], + "score": 1.0, + "content": "after seeing", + "type": "text" + }, + { + "bbox": [ + 477, + 94, + 491, + 105 + ], + "score": 0.87, + "content": "D _ { n }", + "type": "inline_equation" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 110, + 490, + 144 + ], + "lines": [ + { + "bbox": [ + 120, + 110, + 490, + 144 + ], + "spans": [ + { + "bbox": [ + 120, + 110, + 490, + 144 + ], + "score": 0.94, + "content": "p ( w | D _ { n } ) = { \\frac { p ( D _ { n } | w ) p ( w ) } { p ( D _ { n } ) } } = { \\frac { 1 } { Z _ { n } } } \\varphi ( w ) \\prod _ { i = 1 } ^ { n } p ( y _ { i } | x _ { i } , w ) = { \\frac { 1 } { Z _ { n } ^ { 0 } } } \\exp \\{ - n K _ { n } ( w ) + \\log \\varphi ( w ) \\}", + "type": "interline_equation", + "image_path": "85dfd95a7a7c20074844aa594f0db4f5329f3fcaec6759bced1b5a0c8373d2d2.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 120, + 110, + 490, + 121.33333333333333 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 120, + 121.33333333333333, + 490, + 132.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 120, + 132.66666666666666, + 490, + 144.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 149, + 505, + 228 + ], + "lines": [ + { + "bbox": [ + 105, + 148, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 133, + 164 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 149, + 270, + 162 + ], + "score": 0.92, + "content": "\\begin{array} { r } { Z _ { n } ^ { 0 } = \\int \\varphi ( w ) \\exp ( - n K _ { n } ( w ) ) d w } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 148, + 283, + 164 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 283, + 152, + 290, + 160 + ], + "score": 0.78, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 148, + 506, + 164 + ], + "score": 1.0, + "content": "is large the posterior distribution concentrates around", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 145, + 174 + ], + "score": 1.0, + "content": "solutions", + "type": "text" + }, + { + "bbox": [ + 145, + 163, + 179, + 172 + ], + "score": 0.9, + "content": "w \\in W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 161, + 505, + 174 + ], + "score": 1.0, + "content": "and so sampling from the posterior will tend to produce machines that are (nearly)", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 505, + 185 + ], + "score": 1.0, + "content": "solutions. The gold standard sampling is Markov Chain Monte Carlo (MCMC). Scaling MCMC to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 133, + 196 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 183, + 145, + 193 + ], + "score": 0.6, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 182, + 505, + 196 + ], + "score": 1.0, + "content": "is high-dimensional is a challenging task with many attempts to bridge the gap with SGD", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "score": 1.0, + "content": "(Welling & Teh, 2011; Chen et al., 2014; Ding et al., 2014; Zhang et al., 2020). Nonetheless in", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 218 + ], + "score": 1.0, + "content": "simple cases we demonstrate experimentally in Section 5 that machines may be synthesised by", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 216, + 282, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 282, + 229 + ], + "score": 1.0, + "content": "using MCMC to sample from the posterior.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 108, + 244, + 273, + 257 + ], + "lines": [ + { + "bbox": [ + 104, + 242, + 275, + 259 + ], + "spans": [ + { + "bbox": [ + 104, + 242, + 275, + 259 + ], + "score": 1.0, + "content": "3 COMPLEXITY OF PROGRAMS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 269, + 505, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 281 + ], + "score": 1.0, + "content": "Every Turing machine is the solution of a deterministic synthesis problem, so Section 2 associates to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 281, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 336, + 293 + ], + "score": 1.0, + "content": "any Turing machine a singularity of a semi-analytic space", + "type": "text" + }, + { + "bbox": [ + 336, + 281, + 351, + 292 + ], + "score": 0.88, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 281, + 506, + 293 + ], + "score": 1.0, + "content": ". To indicate that this connection is not", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 292, + 504, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 504, + 304 + ], + "score": 1.0, + "content": "vacuous, we sketch how the complexity of a program is related to the real log canonical threshold", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 303, + 374, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 374, + 315 + ], + "score": 1.0, + "content": "of a singularity. 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Let", + "type": "text" + }, + { + "bbox": [ + 372, + 331, + 387, + 342 + ], + "score": 0.89, + "content": "D _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 330, + 457, + 343 + ], + "score": 1.0, + "content": "be sampled from", + "type": "text" + }, + { + "bbox": [ + 458, + 330, + 487, + 342 + ], + "score": 0.92, + "content": "q ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 340, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 104, + 340, + 119, + 357 + ], + "score": 1.0, + "content": "let", + "type": "text" + }, + { + "bbox": [ + 119, + 342, + 201, + 354 + ], + "score": 0.91, + "content": "u , v \\in W ^ { c o d e } \\cap W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 340, + 412, + 357 + ], + "score": 1.0, + "content": "be two explanations for the sample in the sense that", + "type": "text" + }, + { + "bbox": [ + 413, + 343, + 502, + 355 + ], + "score": 0.92, + "content": "K _ { n } ( u ) = K _ { n } ( v ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 340, + 506, + 357 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "score": 1.0, + "content": "Which explanation for the data should we prefer? The classical answer based on Occam’s razor", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "(Solomonoff, 1964) is that we should prefer the shorter program, that is, the one using the fewest", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 376, + 187, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 187, + 388 + ], + "score": 1.0, + "content": "states and symbols.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 122, + 405 + ], + "score": 1.0, + "content": "Set", + "type": "text" + }, + { + "bbox": [ + 122, + 393, + 160, + 405 + ], + "score": 0.92, + "content": "N = | \\Sigma |", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 392, + 179, + 405 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 179, + 393, + 219, + 405 + ], + "score": 0.92, + "content": "M = | Q |", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 392, + 311, + 405 + ], + "score": 1.0, + "content": ". Any Turing machine", + "type": "text" + }, + { + "bbox": [ + 311, + 393, + 320, + 403 + ], + "score": 0.79, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 392, + 345, + 405 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 346, + 393, + 383, + 404 + ], + "score": 0.92, + "content": "N ^ { \\prime } \\leq N", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 392, + 438, + 405 + ], + "score": 1.0, + "content": "symbols and", + "type": "text" + }, + { + "bbox": [ + 438, + 393, + 479, + 403 + ], + "score": 0.92, + "content": "M ^ { \\prime } \\leq M", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 392, + 506, + 405 + ], + "score": 1.0, + "content": "states", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 104, + 402, + 165, + 416 + ], + "score": 1.0, + "content": "has a code for", + "type": "text" + }, + { + "bbox": [ + 165, + 405, + 174, + 414 + ], + "score": 0.79, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 402, + 214, + 416 + ], + "score": 1.0, + "content": "of length", + "type": "text" + }, + { + "bbox": [ + 214, + 404, + 245, + 414 + ], + "score": 0.89, + "content": "c M ^ { \\prime } N ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 402, + 274, + 416 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 274, + 406, + 280, + 414 + ], + "score": 0.77, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 402, + 403, + 416 + ], + "score": 1.0, + "content": "is a constant. We assume that", + "type": "text" + }, + { + "bbox": [ + 403, + 404, + 432, + 416 + ], + "score": 0.91, + "content": "\\Sigma _ { i n p u t }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "is included in the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 414, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 172, + 428 + ], + "score": 1.0, + "content": "tape alphabet of", + "type": "text" + }, + { + "bbox": [ + 173, + 415, + 181, + 424 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 414, + 210, + 428 + ], + "score": 1.0, + "content": "so that", + "type": "text" + }, + { + "bbox": [ + 211, + 414, + 270, + 426 + ], + "score": 0.92, + "content": "N ^ { \\prime } \\geq | \\Sigma _ { i n p u t } |", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 414, + 437, + 428 + ], + "score": 1.0, + "content": "and define the Kolmogorov complexity of", + "type": "text" + }, + { + "bbox": [ + 437, + 416, + 443, + 426 + ], + "score": 0.81, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 414, + 506, + 428 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 424, + 482, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 115, + 436 + ], + "score": 0.75, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 424, + 189, + 440 + ], + "score": 1.0, + "content": "to be the infimum", + "type": "text" + }, + { + "bbox": [ + 190, + 425, + 207, + 438 + ], + "score": 0.87, + "content": "{ \\mathfrak { c } } ( q )", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 424, + 219, + 440 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 219, + 426, + 245, + 436 + ], + "score": 0.87, + "content": "M ^ { \\prime } N ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 424, + 336, + 440 + ], + "score": 1.0, + "content": "over Turing machines", + "type": "text" + }, + { + "bbox": [ + 336, + 426, + 345, + 435 + ], + "score": 0.79, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 424, + 471, + 440 + ], + "score": 1.0, + "content": "that give classical solutions for", + "type": "text" + }, + { + "bbox": [ + 471, + 428, + 477, + 437 + ], + "score": 0.77, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 424, + 482, + 440 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 104, + 442, + 478, + 454 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 479, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 122, + 456 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 443, + 129, + 452 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 442, + 234, + 456 + ], + "score": 1.0, + "content": "be the RLCT of the triple", + "type": "text" + }, + { + "bbox": [ + 235, + 442, + 268, + 455 + ], + "score": 0.92, + "content": "( p , q , \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 442, + 479, + 456 + ], + "score": 1.0, + "content": "associated to the synthesis problem (Definition 2.4).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 108, + 456, + 253, + 470 + ], + "lines": [ + { + "bbox": [ + 105, + 454, + 254, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 167, + 473 + ], + "score": 1.0, + "content": "Theorem 3.1.", + "type": "text" + }, + { + "bbox": [ + 168, + 456, + 250, + 471 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\lambda \\le \\frac { 1 } { 2 } ( M + N ) \\mathfrak { c } ( q ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 454, + 254, + 473 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 484, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 151, + 499 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 152, + 485, + 221, + 497 + ], + "score": 0.92, + "content": "u \\in W ^ { c o d e } \\cap W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 483, + 506, + 499 + ], + "score": 1.0, + "content": "be the code of a Turing machine realising the infimum in the definition", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 441, + 509 + ], + "score": 1.0, + "content": "of the Kolmogorov complexity and suppose that this machine only uses symbols in", + "type": "text" + }, + { + "bbox": [ + 441, + 497, + 452, + 507 + ], + "score": 0.87, + "content": "\\Sigma ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "and states in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 506, + 504, + 521 + ], + "spans": [ + { + "bbox": [ + 107, + 508, + 118, + 519 + ], + "score": 0.88, + "content": "Q ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 506, + 140, + 521 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 141, + 507, + 183, + 519 + ], + "score": 0.92, + "content": "N ^ { \\prime } = | \\Sigma ^ { \\prime } |", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 506, + 202, + 521 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 202, + 507, + 247, + 519 + ], + "score": 0.93, + "content": "\\bar { M } ^ { \\prime } = | Q ^ { \\prime } |", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 506, + 441, + 521 + ], + "score": 1.0, + "content": ". The time evolution of the staged pseudo-UTM", + "type": "text" + }, + { + "bbox": [ + 442, + 508, + 451, + 518 + ], + "score": 0.67, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 506, + 497, + 521 + ], + "score": 1.0, + "content": "simulating", + "type": "text" + }, + { + "bbox": [ + 497, + 510, + 504, + 518 + ], + "score": 0.7, + "content": "u", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 103, + 516, + 507, + 535 + ], + "spans": [ + { + "bbox": [ + 103, + 516, + 119, + 535 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 119, + 519, + 165, + 532 + ], + "score": 0.91, + "content": "x \\in \\Sigma _ { i n p u t } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 516, + 507, + 535 + ], + "score": 1.0, + "content": "is independent of the entries on the description tape that belong to tuples of the form", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 529, + 504, + 545 + ], + "spans": [ + { + "bbox": [ + 107, + 531, + 157, + 543 + ], + "score": 0.87, + "content": "( \\sigma , q , ? , ? , ? )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 529, + 180, + 545 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 180, + 531, + 250, + 542 + ], + "score": 0.89, + "content": "( \\sigma , q ) \\notin \\Sigma ^ { \\prime } \\times Q ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 529, + 271, + 545 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 271, + 531, + 306, + 542 + ], + "score": 0.9, + "content": "V \\subseteq W", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 529, + 497, + 545 + ], + "score": 1.0, + "content": "be the submanifold of points which agree with", + "type": "text" + }, + { + "bbox": [ + 497, + 533, + 504, + 541 + ], + "score": 0.74, + "content": "u", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 540, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 104, + 540, + 180, + 557 + ], + "score": 1.0, + "content": "on all tuples with", + "type": "text" + }, + { + "bbox": [ + 180, + 543, + 250, + 554 + ], + "score": 0.9, + "content": "( \\sigma , q ) \\in \\Sigma ^ { \\prime } \\times Q ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 540, + 372, + 557 + ], + "score": 1.0, + "content": "and are otherwise free. Then", + "type": "text" + }, + { + "bbox": [ + 372, + 542, + 430, + 553 + ], + "score": 0.89, + "content": "u \\in V \\subseteq W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 540, + 449, + 557 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 450, + 542, + 505, + 554 + ], + "score": 0.8, + "content": "\\operatorname { c o d i m } ( V ) =", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 551, + 504, + 567 + ], + "spans": [ + { + "bbox": [ + 107, + 553, + 173, + 565 + ], + "score": 0.93, + "content": "M ^ { \\prime } N ^ { \\prime } ( \\bar { M } + N )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 551, + 367, + 567 + ], + "score": 1.0, + "content": "and by (Watanabe, 2009, Theorem 7.3) we have", + "type": "text" + }, + { + "bbox": [ + 368, + 553, + 438, + 566 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\lambda \\le \\frac 1 2 \\bmod { \\mathrm { i m } } ( V ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 551, + 442, + 567 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 496, + 554, + 504, + 561 + ], + "score": 0.997, + "content": "□", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 573, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "Remark 3.2. The Kolmogorov complexity depends only on the number of symbols and states used.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "The RLCT is a more refined invariant since it also depends on how each symbol and state is used", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 595, + 501, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 408, + 608 + ], + "score": 1.0, + "content": "(Clift & Murfet, 2018, Remark 7.8) as this affects the polynomials defining", + "type": "text" + }, + { + "bbox": [ + 409, + 595, + 424, + 606 + ], + "score": 0.88, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 595, + 501, + 608 + ], + "score": 1.0, + "content": "(see Appendix D).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "title", + "bbox": [ + 108, + 623, + 262, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 264, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 264, + 639 + ], + "score": 1.0, + "content": "4 PRACTICAL IMPLICATIONS", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 648, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "Using singular learning theory we have explained how programs to be synthesised are singularities", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "of analytic functions, and how the Kolmogorov complexity of a program bounds the RLCT of the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 671, + 496, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 496, + 684 + ], + "score": 1.0, + "content": "associated singularity. We now sketch some practical insights that follow from this point of view.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Synthesis minimises the free energy: the sampling-based approach to synthesis (Section 2.1) aims", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 426, + 712 + ], + "score": 1.0, + "content": "to approximate, via MCMC, sampling from the Bayesian posterior for the triple", + "type": "text" + }, + { + "bbox": [ + 427, + 699, + 460, + 711 + ], + "score": 0.92, + "content": "( p , q , \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "associated", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "to a synthesis problem. To understand the behaviour of these Markov chains we follow the asymp-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 356, + 732 + ], + "score": 1.0, + "content": "totic analysis of (Watanabe, 2009, Section 7.6). If we cover", + "type": "text" + }, + { + "bbox": [ + 357, + 721, + 369, + 731 + ], + "score": 0.67, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 720, + 460, + 732 + ], + "score": 1.0, + "content": "by small closed balls", + "type": "text" + }, + { + "bbox": [ + 460, + 721, + 473, + 732 + ], + "score": 0.89, + "content": "V _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 720, + 506, + 732 + ], + "score": 1.0, + "content": "around", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "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": "text", + "bbox": [ + 105, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Synthesis by sampling. The second approach is to consider the Bayesian posterior associated to the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 491, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 419, + 108 + ], + "score": 1.0, + "content": "synthesis problem, which can be viewed as an update on the prior distribution", + "type": "text" + }, + { + "bbox": [ + 419, + 96, + 427, + 105 + ], + "score": 0.82, + "content": "\\varphi", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 93, + 476, + 108 + ], + "score": 1.0, + "content": "after seeing", + "type": "text" + }, + { + "bbox": [ + 477, + 94, + 491, + 105 + ], + "score": 0.87, + "content": "D _ { n }", + "type": "inline_equation" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 505, + 108 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 110, + 490, + 144 + ], + "lines": [ + { + "bbox": [ + 120, + 110, + 490, + 144 + ], + "spans": [ + { + "bbox": [ + 120, + 110, + 490, + 144 + ], + "score": 0.94, + "content": "p ( w | D _ { n } ) = { \\frac { p ( D _ { n } | w ) p ( w ) } { p ( D _ { n } ) } } = { \\frac { 1 } { Z _ { n } } } \\varphi ( w ) \\prod _ { i = 1 } ^ { n } p ( y _ { i } | x _ { i } , w ) = { \\frac { 1 } { Z _ { n } ^ { 0 } } } \\exp \\{ - n K _ { n } ( w ) + \\log \\varphi ( w ) \\}", + "type": "interline_equation", + "image_path": "85dfd95a7a7c20074844aa594f0db4f5329f3fcaec6759bced1b5a0c8373d2d2.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 120, + 110, + 490, + 121.33333333333333 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 120, + 121.33333333333333, + 490, + 132.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 120, + 132.66666666666666, + 490, + 144.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 149, + 505, + 228 + ], + "lines": [ + { + "bbox": [ + 105, + 148, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 133, + 164 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 149, + 270, + 162 + ], + "score": 0.92, + "content": "\\begin{array} { r } { Z _ { n } ^ { 0 } = \\int \\varphi ( w ) \\exp ( - n K _ { n } ( w ) ) d w } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 148, + 283, + 164 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 283, + 152, + 290, + 160 + ], + "score": 0.78, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 148, + 506, + 164 + ], + "score": 1.0, + "content": "is large the posterior distribution concentrates around", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 145, + 174 + ], + "score": 1.0, + "content": "solutions", + "type": "text" + }, + { + "bbox": [ + 145, + 163, + 179, + 172 + ], + "score": 0.9, + "content": "w \\in W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 161, + 505, + 174 + ], + "score": 1.0, + "content": "and so sampling from the posterior will tend to produce machines that are (nearly)", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 505, + 185 + ], + "score": 1.0, + "content": "solutions. The gold standard sampling is Markov Chain Monte Carlo (MCMC). Scaling MCMC to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 133, + 196 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 183, + 145, + 193 + ], + "score": 0.6, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 182, + 505, + 196 + ], + "score": 1.0, + "content": "is high-dimensional is a challenging task with many attempts to bridge the gap with SGD", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "score": 1.0, + "content": "(Welling & Teh, 2011; Chen et al., 2014; Ding et al., 2014; Zhang et al., 2020). Nonetheless in", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 505, + 218 + ], + "score": 1.0, + "content": "simple cases we demonstrate experimentally in Section 5 that machines may be synthesised by", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 216, + 282, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 282, + 229 + ], + "score": 1.0, + "content": "using MCMC to sample from the posterior.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 148, + 506, + 229 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 244, + 273, + 257 + ], + "lines": [ + { + "bbox": [ + 104, + 242, + 275, + 259 + ], + "spans": [ + { + "bbox": [ + 104, + 242, + 275, + 259 + ], + "score": 1.0, + "content": "3 COMPLEXITY OF PROGRAMS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 269, + 505, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 281 + ], + "score": 1.0, + "content": "Every Turing machine is the solution of a deterministic synthesis problem, so Section 2 associates to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 281, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 336, + 293 + ], + "score": 1.0, + "content": "any Turing machine a singularity of a semi-analytic space", + "type": "text" + }, + { + "bbox": [ + 336, + 281, + 351, + 292 + ], + "score": 0.88, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 281, + 506, + 293 + ], + "score": 1.0, + "content": ". To indicate that this connection is not", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 292, + 504, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 504, + 304 + ], + "score": 1.0, + "content": "vacuous, we sketch how the complexity of a program is related to the real log canonical threshold", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 303, + 374, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 374, + 315 + ], + "score": 1.0, + "content": "of a singularity. 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We assume that", + "type": "text" + }, + { + "bbox": [ + 403, + 404, + 432, + 416 + ], + "score": 0.91, + "content": "\\Sigma _ { i n p u t }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "is included in the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 414, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 172, + 428 + ], + "score": 1.0, + "content": "tape alphabet of", + "type": "text" + }, + { + "bbox": [ + 173, + 415, + 181, + 424 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 414, + 210, + 428 + ], + "score": 1.0, + "content": "so that", + "type": "text" + }, + { + "bbox": [ + 211, + 414, + 270, + 426 + ], + "score": 0.92, + "content": "N ^ { \\prime } \\geq | \\Sigma _ { i n p u t } |", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 414, + 437, + 428 + ], + "score": 1.0, + "content": "and define the Kolmogorov complexity of", + "type": "text" + }, + { + "bbox": [ + 437, + 416, + 443, + 426 + ], + "score": 0.81, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 414, + 506, + 428 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 424, + 482, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 115, + 436 + ], + "score": 0.75, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 424, + 189, + 440 + ], + "score": 1.0, + "content": "to be the infimum", + "type": "text" + }, + { + "bbox": [ + 190, + 425, + 207, + 438 + ], + "score": 0.87, + "content": "{ \\mathfrak { c } } ( q )", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 424, + 219, + 440 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 219, + 426, + 245, + 436 + ], + "score": 0.87, + "content": "M ^ { \\prime } N ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 424, + 336, + 440 + ], + "score": 1.0, + "content": "over Turing machines", + "type": "text" + }, + { + "bbox": [ + 336, + 426, + 345, + 435 + ], + "score": 0.79, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 424, + 471, + 440 + ], + "score": 1.0, + "content": "that give classical solutions for", + "type": "text" + }, + { + "bbox": [ + 471, + 428, + 477, + 437 + ], + "score": 0.77, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 424, + 482, + 440 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 392, + 506, + 440 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 442, + 478, + 454 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 479, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 122, + 456 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 443, + 129, + 452 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 442, + 234, + 456 + ], + "score": 1.0, + "content": "be the RLCT of the triple", + "type": "text" + }, + { + "bbox": [ + 235, + 442, + 268, + 455 + ], + "score": 0.92, + "content": "( p , q , \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 442, + 479, + 456 + ], + "score": 1.0, + "content": "associated to the synthesis problem (Definition 2.4).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 442, + 479, + 456 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 456, + 253, + 470 + ], + "lines": [ + { + "bbox": [ + 105, + 454, + 254, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 167, + 473 + ], + "score": 1.0, + "content": "Theorem 3.1.", + "type": "text" + }, + { + "bbox": [ + 168, + 456, + 250, + 471 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\lambda \\le \\frac { 1 } { 2 } ( M + N ) \\mathfrak { c } ( q ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 454, + 254, + 473 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 454, + 254, + 473 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 484, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 151, + 499 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 152, + 485, + 221, + 497 + ], + "score": 0.92, + "content": "u \\in W ^ { c o d e } \\cap W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 483, + 506, + 499 + ], + "score": 1.0, + "content": "be the code of a Turing machine realising the infimum in the definition", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 441, + 509 + ], + "score": 1.0, + "content": "of the Kolmogorov complexity and suppose that this machine only uses symbols in", + "type": "text" + }, + { + "bbox": [ + 441, + 497, + 452, + 507 + ], + "score": 0.87, + "content": "\\Sigma ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "and states in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 506, + 504, + 521 + ], + "spans": [ + { + "bbox": [ + 107, + 508, + 118, + 519 + ], + "score": 0.88, + "content": "Q ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 506, + 140, + 521 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 141, + 507, + 183, + 519 + ], + "score": 0.92, + "content": "N ^ { \\prime } = | \\Sigma ^ { \\prime } |", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 506, + 202, + 521 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 202, + 507, + 247, + 519 + ], + "score": 0.93, + "content": "\\bar { M } ^ { \\prime } = | Q ^ { \\prime } |", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 506, + 441, + 521 + ], + "score": 1.0, + "content": ". The time evolution of the staged pseudo-UTM", + "type": "text" + }, + { + "bbox": [ + 442, + 508, + 451, + 518 + ], + "score": 0.67, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 506, + 497, + 521 + ], + "score": 1.0, + "content": "simulating", + "type": "text" + }, + { + "bbox": [ + 497, + 510, + 504, + 518 + ], + "score": 0.7, + "content": "u", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 103, + 516, + 507, + 535 + ], + "spans": [ + { + "bbox": [ + 103, + 516, + 119, + 535 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 119, + 519, + 165, + 532 + ], + "score": 0.91, + "content": "x \\in \\Sigma _ { i n p u t } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 516, + 507, + 535 + ], + "score": 1.0, + "content": "is independent of the entries on the description tape that belong to tuples of the form", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 529, + 504, + 545 + ], + "spans": [ + { + "bbox": [ + 107, + 531, + 157, + 543 + ], + "score": 0.87, + "content": "( \\sigma , q , ? , ? , ? )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 529, + 180, + 545 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 180, + 531, + 250, + 542 + ], + "score": 0.89, + "content": "( \\sigma , q ) \\notin \\Sigma ^ { \\prime } \\times Q ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 529, + 271, + 545 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 271, + 531, + 306, + 542 + ], + "score": 0.9, + "content": "V \\subseteq W", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 529, + 497, + 545 + ], + "score": 1.0, + "content": "be the submanifold of points which agree with", + "type": "text" + }, + { + "bbox": [ + 497, + 533, + 504, + 541 + ], + "score": 0.74, + "content": "u", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 540, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 104, + 540, + 180, + 557 + ], + "score": 1.0, + "content": "on all tuples with", + "type": "text" + }, + { + "bbox": [ + 180, + 543, + 250, + 554 + ], + "score": 0.9, + "content": "( \\sigma , q ) \\in \\Sigma ^ { \\prime } \\times Q ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 540, + 372, + 557 + ], + "score": 1.0, + "content": "and are otherwise free. Then", + "type": "text" + }, + { + "bbox": [ + 372, + 542, + 430, + 553 + ], + "score": 0.89, + "content": "u \\in V \\subseteq W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 540, + 449, + 557 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 450, + 542, + 505, + 554 + ], + "score": 0.8, + "content": "\\operatorname { c o d i m } ( V ) =", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 551, + 504, + 567 + ], + "spans": [ + { + "bbox": [ + 107, + 553, + 173, + 565 + ], + "score": 0.93, + "content": "M ^ { \\prime } N ^ { \\prime } ( \\bar { M } + N )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 551, + 367, + 567 + ], + "score": 1.0, + "content": "and by (Watanabe, 2009, Theorem 7.3) we have", + "type": "text" + }, + { + "bbox": [ + 368, + 553, + 438, + 566 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\lambda \\le \\frac 1 2 \\bmod { \\mathrm { i m } } ( V ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 551, + 442, + 567 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 496, + 554, + 504, + 561 + ], + "score": 0.997, + "content": "□", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32, + "bbox_fs": [ + 103, + 483, + 507, + 567 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 573, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "Remark 3.2. The Kolmogorov complexity depends only on the number of symbols and states used.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "The RLCT is a more refined invariant since it also depends on how each symbol and state is used", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 595, + 501, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 408, + 608 + ], + "score": 1.0, + "content": "(Clift & Murfet, 2018, Remark 7.8) as this affects the polynomials defining", + "type": "text" + }, + { + "bbox": [ + 409, + 595, + 424, + 606 + ], + "score": 0.88, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 595, + 501, + 608 + ], + "score": 1.0, + "content": "(see Appendix D).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 573, + 505, + 608 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 623, + 262, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 264, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 264, + 639 + ], + "score": 1.0, + "content": "4 PRACTICAL IMPLICATIONS", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 648, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "Using singular learning theory we have explained how programs to be synthesised are singularities", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "of analytic functions, and how the Kolmogorov complexity of a program bounds the RLCT of the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 671, + 496, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 496, + 684 + ], + "score": 1.0, + "content": "associated singularity. We now sketch some practical insights that follow from this point of view.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 648, + 506, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Synthesis minimises the free energy: the sampling-based approach to synthesis (Section 2.1) aims", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 426, + 712 + ], + "score": 1.0, + "content": "to approximate, via MCMC, sampling from the Bayesian posterior for the triple", + "type": "text" + }, + { + "bbox": [ + 427, + 699, + 460, + 711 + ], + "score": 0.92, + "content": "( p , q , \\varphi )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "associated", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "to a synthesis problem. To understand the behaviour of these Markov chains we follow the asymp-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 356, + 732 + ], + "score": 1.0, + "content": "totic analysis of (Watanabe, 2009, Section 7.6). 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Here", + "type": "text" + }, + { + "bbox": [ + 227, + 175, + 243, + 185 + ], + "score": 0.89, + "content": "K _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 172, + 467, + 188 + ], + "score": 1.0, + "content": "is the smallest value of the Kullback-Leibler divergence", + "type": "text" + }, + { + "bbox": [ + 468, + 175, + 478, + 184 + ], + "score": 0.84, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 172, + 491, + 188 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 491, + 174, + 504, + 185 + ], + "score": 0.88, + "content": "V _ { \\alpha }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 185, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 123, + 198 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 185, + 136, + 196 + ], + "score": 0.89, + "content": "\\lambda _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 185, + 225, + 198 + ], + "score": 1.0, + "content": "is the RLCT of the set", + "type": "text" + }, + { + "bbox": [ + 225, + 186, + 268, + 197 + ], + "score": 0.93, + "content": "W _ { K _ { \\alpha } } \\cap V _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 185, + 295, + 198 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 295, + 185, + 411, + 197 + ], + "score": 0.92, + "content": "W _ { c } = \\{ w \\in W | K ( w ) = c \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 185, + 472, + 198 + ], + "score": 1.0, + "content": "is a level set of", + "type": "text" + }, + { + "bbox": [ + 473, + 186, + 483, + 195 + ], + "score": 0.84, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 185, + 506, + 198 + ], + "score": 1.0, + "content": ". 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By Section 3 one may think of these", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "points as the “lowest complexity” solutions. However it is possible that there are other local minima", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 257, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 269 + ], + "score": 1.0, + "content": "of the free energy. 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While synthesis by SGD", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 333, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 346 + ], + "score": 1.0, + "content": "and sampling are different, it is a reasonable hypothesis that these siren minima are a significant", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 344, + 240, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 240, + 358 + ], + "score": 1.0, + "content": "contributing factor in both cases.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 361, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 107, + 362, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 107, + 362, + 263, + 372 + ], + "score": 1.0, + "content": "Can we avoid siren minima? 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At any given value of", + "type": "text" + }, + { + "bbox": [ + 383, + 398, + 390, + 406 + ], + "score": 0.79, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 396, + 505, + 409 + ], + "score": 1.0, + "content": "there is a “siren free” region", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 405, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 156, + 424 + ], + "score": 1.0, + "content": "in the range", + "type": "text" + }, + { + "bbox": [ + 157, + 407, + 205, + 423 + ], + "score": 0.94, + "content": "c \\geq { \\frac { \\lambda \\log ( n ) } { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 405, + 506, + 424 + ], + "score": 1.0, + "content": "since the RLCT is non-negative (Figure 3). Thus the learning process will", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 421, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 223, + 438 + ], + "score": 1.0, + "content": "be more reliable the smaller", + "type": "text" + }, + { + "bbox": [ + 223, + 421, + 254, + 437 + ], + "score": 0.93, + "content": "\\frac { \\lambda \\log ( n ) } { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 421, + 428, + 438 + ], + "score": 1.0, + "content": "is. This can arranged either by increasing", + "type": "text" + }, + { + "bbox": [ + 428, + 426, + 435, + 434 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 421, + 506, + 438 + ], + "score": 1.0, + "content": "(providing more", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 435, + 217, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 205, + 448 + ], + "score": 1.0, + "content": "examples) or decreasing", + "type": "text" + }, + { + "bbox": [ + 206, + 435, + 213, + 445 + ], + "score": 0.74, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 435, + 217, + 448 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 451, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "score": 1.0, + "content": "While the RLCT is determined by the synthesis problem, it is possible to change its value by chang-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 463, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 222, + 474 + ], + "score": 1.0, + "content": "ing the structure of the UTM", + "type": "text" + }, + { + "bbox": [ + 223, + 463, + 232, + 473 + ], + "score": 0.56, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 464, + 323, + 474 + ], + "score": 1.0, + "content": ". 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The", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "score": 1.0, + "content": "Markov chains used to generate approximate samples from the posterior are attempting to minimise", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 207, + 484, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 353, + 220 + ], + "score": 1.0, + "content": "the free energy, which involves a tradeoff between the energy", + "type": "text" + }, + { + "bbox": [ + 353, + 208, + 375, + 218 + ], + "score": 0.91, + "content": "K _ { \\alpha } n", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 207, + 440, + 220 + ], + "score": 1.0, + "content": "and the entropy", + "type": "text" + }, + { + "bbox": [ + 441, + 207, + 481, + 219 + ], + "score": 0.93, + "content": "\\lambda _ { \\alpha } \\log ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 207, + 484, + 220 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5, + "bbox_fs": [ + 104, + 172, + 506, + 220 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 223, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 106, + 224, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 505, + 236 + ], + "score": 1.0, + "content": "Why synthesis gets stuck: the kind of local minimum of the free energy that we want the synthesis", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 235, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 104, + 235, + 222, + 247 + ], + "score": 1.0, + "content": "process to find are solutions", + "type": "text" + }, + { + "bbox": [ + 222, + 235, + 263, + 246 + ], + "score": 0.92, + "content": "w _ { \\alpha } \\in W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 235, + 291, + 247 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 292, + 235, + 305, + 246 + ], + "score": 0.89, + "content": "\\lambda _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 235, + 506, + 247 + ], + "score": 1.0, + "content": "is minimal. By Section 3 one may think of these", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "points as the “lowest complexity” solutions. However it is possible that there are other local minima", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 257, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 269 + ], + "score": 1.0, + "content": "of the free energy. Indeed, there may be local minima where the free energy is lower than the free", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 252, + 281 + ], + "score": 1.0, + "content": "energy at any solution since at finite", + "type": "text" + }, + { + "bbox": [ + 253, + 270, + 260, + 278 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 268, + 414, + 281 + ], + "score": 1.0, + "content": "it is possible to tradeoff an increase in", + "type": "text" + }, + { + "bbox": [ + 414, + 268, + 429, + 279 + ], + "score": 0.89, + "content": "K _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "against a decrease", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 278, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 159, + 292 + ], + "score": 1.0, + "content": "in the RLCT", + "type": "text" + }, + { + "bbox": [ + 160, + 279, + 172, + 290 + ], + "score": 0.88, + "content": "\\lambda _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 278, + 506, + 292 + ], + "score": 1.0, + "content": ". In practice, the existence of such “siren minima” of the free energy may manifest", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "itself as regions where the synthesis process gets stuck and fails to converge to a solution. 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In practice", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 311, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 325 + ], + "score": 1.0, + "content": "it has been observed that program synthesis by gradient descent often fails for complex problems", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 322, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 506, + 336 + ], + "score": 1.0, + "content": "in the sense that it fails to converge to a solution (Gaunt et al., 2016). While synthesis by SGD", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 333, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 346 + ], + "score": 1.0, + "content": "and sampling are different, it is a reasonable hypothesis that these siren minima are a significant", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 344, + 240, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 240, + 358 + ], + "score": 1.0, + "content": "contributing factor in both cases.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 224, + 506, + 358 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 361, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 107, + 362, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 107, + 362, + 263, + 372 + ], + "score": 1.0, + "content": "Can we avoid siren minima? 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At any given value of", + "type": "text" + }, + { + "bbox": [ + 383, + 398, + 390, + 406 + ], + "score": 0.79, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 396, + 505, + 409 + ], + "score": 1.0, + "content": "there is a “siren free” region", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 405, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 156, + 424 + ], + "score": 1.0, + "content": "in the range", + "type": "text" + }, + { + "bbox": [ + 157, + 407, + 205, + 423 + ], + "score": 0.94, + "content": "c \\geq { \\frac { \\lambda \\log ( n ) } { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 405, + 506, + 424 + ], + "score": 1.0, + "content": "since the RLCT is non-negative (Figure 3). Thus the learning process will", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 421, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 223, + 438 + ], + "score": 1.0, + "content": "be more reliable the smaller", + "type": "text" + }, + { + "bbox": [ + 223, + 421, + 254, + 437 + ], + "score": 0.93, + "content": "\\frac { \\lambda \\log ( n ) } { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 421, + 428, + 438 + ], + "score": 1.0, + "content": "is. This can arranged either by increasing", + "type": "text" + }, + { + "bbox": [ + 428, + 426, + 435, + 434 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 421, + 506, + 438 + ], + "score": 1.0, + "content": "(providing more", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 435, + 217, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 205, + 448 + ], + "score": 1.0, + "content": "examples) or decreasing", + "type": "text" + }, + { + "bbox": [ + 206, + 435, + 213, + 445 + ], + "score": 0.74, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 435, + 217, + 448 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 362, + 506, + 448 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 451, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 465 + ], + "score": 1.0, + "content": "While the RLCT is determined by the synthesis problem, it is possible to change its value by chang-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 463, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 222, + 474 + ], + "score": 1.0, + "content": "ing the structure of the UTM", + "type": "text" + }, + { + "bbox": [ + 223, + 463, + 232, + 473 + ], + "score": 0.56, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 464, + 323, + 474 + ], + "score": 1.0, + "content": ". 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Our RLCT estimates are contained", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 151, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 151, + 253 + ], + "score": 1.0, + "content": "in Table 2.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5 + }, + { + "type": "table", + "bbox": [ + 173, + 261, + 437, + 315 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 173, + 261, + 437, + 315 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 173, + 261, + 437, + 315 + ], + "spans": [ + { + "bbox": [ + 173, + 261, + 437, + 315 + ], + "score": 0.976, + "html": "
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", + "type": "table", + "image_path": "509b50040e6f76ba80e26f4d6f57fd36e0f2da05f22a5bcf5029ea5b32f517d6.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 173, + 261, + 437, + 279.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 173, + 279.0, + 437, + 297.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 173, + 297.0, + 437, + 315.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 212, + 327, + 399, + 339 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 210, + 326, + 399, + 341 + ], + "spans": [ + { + "bbox": [ + 210, + 326, + 399, + 341 + ], + "score": 1.0, + "content": "Table 2: RLCT estimates for parityCheck.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + } + ], + "index": 17.0 + }, + { + "type": "title", + "bbox": [ + 107, + 364, + 190, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 192, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 192, + 379 + ], + "score": 1.0, + "content": "6 DISCUSSION", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 505, + 444 + ], + "lines": [ + { + "bbox": [ + 106, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "We have developed a theoretical framework in which all programs can in principle be learnt from", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "input-output examples via an existing optimisation procedure. This is done by associating to each", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 410, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 104, + 410, + 506, + 424 + ], + "score": 1.0, + "content": "program a smooth relaxation which, based on Clift & Murfet (2018), can be argued to be more", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 422, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 435 + ], + "score": 1.0, + "content": "canonical than existing approaches. This realization has important implications for the building of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 432, + 186, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 186, + 447 + ], + "score": 1.0, + "content": "intelligent systems.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 450, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "In approaches to program synthesis based on gradient descent there is a tendency to think of solutions", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 385, + 473 + ], + "score": 1.0, + "content": "to the synthesis problem as isolated critical points of the loss function", + "type": "text" + }, + { + "bbox": [ + 386, + 461, + 396, + 471 + ], + "score": 0.79, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 460, + 505, + 473 + ], + "score": 1.0, + "content": ", but this is a false intuition", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 473, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 484 + ], + "score": 1.0, + "content": "based on regular models. 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This is done by associating to each", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 410, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 104, + 410, + 506, + 424 + ], + "score": 1.0, + "content": "program a smooth relaxation which, based on Clift & Murfet (2018), can be argued to be more", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 422, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 435 + ], + "score": 1.0, + "content": "canonical than existing approaches. 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Input: range of β's, set of training sets T each of size n, approximate samples {w1,..,WR} from pβ(w|Dn) for each training set Dn and each β
for training set Dn ∈ T do
for β in range of β's do
ples from pβ(w|Dn)
end for
Perform generalised least squares to fit X in Equation (7),call result λ(Dn)
end for
Output: ∑Dn∈T λ(Dn)
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Input: range of β's, set of training sets T each of size n, approximate samples {w1,..,WR} from pβ(w|Dn) for each training set Dn and each β
for training set Dn ∈ T do
for β in range of β's do
ples from pβ(w|Dn)
end for
Perform generalised least squares to fit X in Equation (7),call result λ(Dn)
end for
Output: ∑Dn∈T λ(Dn)
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HyperparameterdetectAparityCheck
Dataset size (n)200100
Minimum sequence length (a)41
Maximum sequence length (b)7/8/9/105/6/7
Number of samples (R)20.0002.000
Number of burn-in steps1,000500
Number of datasets (|T|)43
Target accept probability0.80.8
Concentration (α)1.01.0
Chain temperature (T)log(500)/log(1000)log(300)
Number of timesteps (t)1042
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HyperparameterdetectAparityCheck
Dataset size (n)200100
Minimum sequence length (a)41
Maximum sequence length (b)7/8/9/105/6/7
Number of samples (R)20.0002.000
Number of burn-in steps1,000500
Number of datasets (|T|)43
Target accept probability0.80.8
Concentration (α)1.01.0
Chain temperature (T)log(500)/log(1000)log(300)
Number of timesteps (t)1042
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Suppose", + "type": "text" + }, + { + "bbox": [ + 202, + 119, + 285, + 132 + ], + "score": 0.93, + "content": "q ( y | x ) = p ( y | x , w _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 118, + 313, + 133 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 313, + 119, + 361, + 131 + ], + "score": 0.93, + "content": "w _ { 0 } = ( 1 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 118, + 448, + 133 + ], + "score": 1.0, + "content": ". It is easy to see that", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 167, + 135, + 444, + 165 + ], + "lines": [ + { + "bbox": [ + 167, + 135, + 444, + 165 + ], + "spans": [ + { + "bbox": [ + 167, + 135, + 444, + 165 + ], + "score": 0.92, + "content": "K ( w ) = - { \\frac { 1 } { 4 } } \\sum _ { a _ { 2 } , a _ { 3 } } \\log p \\big ( y = a _ { 3 } | x = ( a _ { 2 } , a _ { 3 } ) , w \\big ) = - { \\frac { 1 } { 2 } } \\log [ g ( h , k ) ]", + "type": "interline_equation", + "image_path": "b29a76eb679442a67367c43a891a13bfa9eff320e573767248836bc96aa07018.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 167, + 135, + 444, + 145.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 167, + 145.0, + 444, + 155.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 167, + 155.0, + 444, + 165.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 169, + 462, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 169, + 462, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 133, + 185 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 170, + 344, + 185 + ], + "score": 0.9, + "content": "g ( h , k ) = \\left( ( 1 - h ) ^ { 2 } k + h ^ { 2 } \\right) \\left( ( 1 - h ) ^ { 2 } ( 1 - k ) + h ^ { 2 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 169, + 421, + 185 + ], + "score": 1.0, + "content": "is a polynomial in", + "type": "text" + }, + { + "bbox": [ + 421, + 174, + 429, + 182 + ], + "score": 0.77, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 169, + 462, + 185 + ], + "score": 1.0, + "content": ". Hence", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 189, + 419, + 204 + ], + "lines": [ + { + "bbox": [ + 191, + 189, + 419, + 204 + ], + "spans": [ + { + "bbox": [ + 191, + 189, + 419, + 204 + ], + "score": 0.86, + "content": "W _ { 0 } = \\{ ( h , k ) \\in W : g ( h , k ) = 1 \\} = \\mathbb { V } ( g - 1 ) \\cap [ 0 , 1 ] ^ { 2 }", + "type": "interline_equation", + "image_path": "908fcb4fd665de455f62bb90a7952c7ac7c4614cac4247c979470c2625d932ef.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 191, + 189, + 419, + 204 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 208, + 503, + 231 + ], + "lines": [ + { + "bbox": [ + 104, + 206, + 504, + 222 + ], + "spans": [ + { + "bbox": [ + 104, + 206, + 482, + 222 + ], + "score": 1.0, + "content": "is a semi-algebraic variety, that is, it is defined by polynomial equations and inequalities. Here", + "type": "text" + }, + { + "bbox": [ + 482, + 208, + 504, + 220 + ], + "score": 0.89, + "content": "\\mathbb { V } ( h )", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 219, + 282, + 231 + ], + "spans": [ + { + "bbox": [ + 107, + 219, + 272, + 231 + ], + "score": 1.0, + "content": "denotes the vanishing locus of a function", + "type": "text" + }, + { + "bbox": [ + 272, + 220, + 279, + 229 + ], + "score": 0.82, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 219, + 282, + 231 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 232, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 203, + 247 + ], + "score": 1.0, + "content": "Example C.2. Suppose", + "type": "text" + }, + { + "bbox": [ + 204, + 234, + 254, + 246 + ], + "score": 0.92, + "content": "q ( A B ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 232, + 273, + 247 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 273, + 233, + 389, + 246 + ], + "score": 0.92, + "content": "\\begin{array} { r } { q ( y | x = A B ) = \\frac { 1 } { 2 } A + \\frac { 1 } { 2 } B } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 232, + 506, + 247 + ], + "score": 1.0, + "content": ". 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We assume", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 309, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 195, + 117 + ], + "score": 1.0, + "content": "that some distribution", + "type": "text" + }, + { + "bbox": [ + 196, + 105, + 215, + 117 + ], + "score": 0.91, + "content": "q ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 104, + 236, + 117 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 237, + 104, + 272, + 117 + ], + "score": 0.93, + "content": "\\{ A , B \\} ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 104, + 309, + 117 + ], + "score": 1.0, + "content": "is given.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 82, + 506, + 117 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 118, + 447, + 132 + ], + "lines": [ + { + "bbox": [ + 105, + 118, + 448, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 202, + 133 + ], + "score": 1.0, + "content": "Example C.1. Suppose", + "type": "text" + }, + { + "bbox": [ + 202, + 119, + 285, + 132 + ], + "score": 0.93, + "content": "q ( y | x ) = p ( y | x , w _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 118, + 313, + 133 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 313, + 119, + 361, + 131 + ], + "score": 0.93, + "content": "w _ { 0 } = ( 1 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 118, + 448, + 133 + ], + "score": 1.0, + "content": ". It is easy to see that", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 118, + 448, + 133 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 167, + 135, + 444, + 165 + ], + "lines": [ + { + "bbox": [ + 167, + 135, + 444, + 165 + ], + "spans": [ + { + "bbox": [ + 167, + 135, + 444, + 165 + ], + "score": 0.92, + "content": "K ( w ) = - { \\frac { 1 } { 4 } } \\sum _ { a _ { 2 } , a _ { 3 } } \\log p \\big ( y = a _ { 3 } | x = ( a _ { 2 } , a _ { 3 } ) , w \\big ) = - { \\frac { 1 } { 2 } } \\log [ g ( h , k ) ]", + "type": "interline_equation", + "image_path": "b29a76eb679442a67367c43a891a13bfa9eff320e573767248836bc96aa07018.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 167, + 135, + 444, + 145.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 167, + 145.0, + 444, + 155.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 167, + 155.0, + 444, + 165.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 169, + 462, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 169, + 462, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 133, + 185 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 170, + 344, + 185 + ], + "score": 0.9, + "content": "g ( h , k ) = \\left( ( 1 - h ) ^ { 2 } k + h ^ { 2 } \\right) \\left( ( 1 - h ) ^ { 2 } ( 1 - k ) + h ^ { 2 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 169, + 421, + 185 + ], + "score": 1.0, + "content": "is a polynomial in", + "type": "text" + }, + { + "bbox": [ + 421, + 174, + 429, + 182 + ], + "score": 0.77, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 169, + 462, + 185 + ], + "score": 1.0, + "content": ". Hence", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 169, + 462, + 185 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 189, + 419, + 204 + ], + "lines": [ + { + "bbox": [ + 191, + 189, + 419, + 204 + ], + "spans": [ + { + "bbox": [ + 191, + 189, + 419, + 204 + ], + "score": 0.86, + "content": "W _ { 0 } = \\{ ( h , k ) \\in W : g ( h , k ) = 1 \\} = \\mathbb { V } ( g - 1 ) \\cap [ 0 , 1 ] ^ { 2 }", + "type": "interline_equation", + "image_path": "908fcb4fd665de455f62bb90a7952c7ac7c4614cac4247c979470c2625d932ef.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 191, + 189, + 419, + 204 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 208, + 503, + 231 + ], + "lines": [ + { + "bbox": [ + 104, + 206, + 504, + 222 + ], + "spans": [ + { + "bbox": [ + 104, + 206, + 482, + 222 + ], + "score": 1.0, + "content": "is a semi-algebraic variety, that is, it is defined by polynomial equations and inequalities. Here", + "type": "text" + }, + { + "bbox": [ + 482, + 208, + 504, + 220 + ], + "score": 0.89, + "content": "\\mathbb { V } ( h )", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 219, + 282, + 231 + ], + "spans": [ + { + "bbox": [ + 107, + 219, + 272, + 231 + ], + "score": 1.0, + "content": "denotes the vanishing locus of a function", + "type": "text" + }, + { + "bbox": [ + 272, + 220, + 279, + 229 + ], + "score": 0.82, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 219, + 282, + 231 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 104, + 206, + 504, + 231 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 232, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 203, + 247 + ], + "score": 1.0, + "content": "Example C.2. Suppose", + "type": "text" + }, + { + "bbox": [ + 204, + 234, + 254, + 246 + ], + "score": 0.92, + "content": "q ( A B ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 232, + 273, + 247 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 273, + 233, + 389, + 246 + ], + "score": 0.92, + "content": "\\begin{array} { r } { q ( y | x = A B ) = \\frac { 1 } { 2 } A + \\frac { 1 } { 2 } B } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 232, + 506, + 247 + ], + "score": 1.0, + "content": ". Then the Kullback-Leibler", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 245, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 163, + 260 + ], + "score": 1.0, + "content": "divergence is", + "type": "text" + }, + { + "bbox": [ + 163, + 246, + 290, + 260 + ], + "score": 0.9, + "content": "\\begin{array} { r } { K ( h , k ) = - \\frac { 1 } { 2 } \\log ( 4 f ( 1 - f ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 245, + 320, + 260 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 320, + 246, + 439, + 259 + ], + "score": 0.9, + "content": "f = ( 1 - h ) ^ { 2 } k + 2 h ( 1 - h )", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 245, + 473, + 260 + ], + "score": 1.0, + "content": ". Hence", + "type": "text" + }, + { + "bbox": [ + 473, + 246, + 505, + 258 + ], + "score": 0.86, + "content": "\\nabla K =", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 258, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 107, + 258, + 182, + 274 + ], + "score": 0.92, + "content": "\\begin{array} { r } { ( f - \\frac { 1 } { 2 } ) \\frac { 1 } { f ( 1 - f ) } \\nabla f . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 258, + 224, + 273 + ], + "score": 1.0, + "content": ". 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Choose", + "type": "text" + }, + { + "bbox": [ + 169, + 244, + 199, + 254 + ], + "score": 0.91, + "content": "x \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 242, + 222, + 256 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 222, + 244, + 262, + 256 + ], + "score": 0.93, + "content": "q ( x ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 242, + 294, + 256 + ], + "score": 1.0, + "content": "and let", + "type": "text" + }, + { + "bbox": [ + 295, + 246, + 302, + 255 + ], + "score": 0.8, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 242, + 355, + 256 + ], + "score": 1.0, + "content": "be such that", + "type": "text" + }, + { + "bbox": [ + 355, + 244, + 403, + 256 + ], + "score": 0.93, + "content": "q ( y | x ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 242, + 425, + 256 + ], + "score": 1.0, + "content": ". 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The set", + "type": "text" + }, + { + "bbox": [ + 195, + 284, + 209, + 295 + ], + "score": 0.89, + "content": "W _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 283, + 299, + 297 + ], + "score": 1.0, + "content": "is semi-algebraic and", + "type": "text" + }, + { + "bbox": [ + 300, + 284, + 344, + 295 + ], + "score": 0.89, + "content": "W _ { 0 } \\subseteq \\partial W", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 283, + 348, + 297 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 307, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 306, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 163, + 321 + ], + "score": 1.0, + "content": "Proof. Given", + "type": "text" + }, + { + "bbox": [ + 163, + 308, + 195, + 318 + ], + "score": 0.91, + "content": "x \\in \\Sigma ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 306, + 218, + 321 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 218, + 307, + 257, + 319 + ], + "score": 0.92, + "content": "q ( x ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 306, + 297, + 321 + ], + "score": 1.0, + "content": "we write", + "type": "text" + }, + { + "bbox": [ + 297, + 308, + 336, + 320 + ], + "score": 0.93, + "content": "y = y ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 306, + 439, + 321 + ], + "score": 1.0, + "content": "for the unique state with", + "type": "text" + }, + { + "bbox": [ + 440, + 307, + 489, + 320 + ], + "score": 0.92, + "content": "q ( x , y ) \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 306, + 505, + 321 + ], + "score": 1.0, + "content": ". In", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 317, + 300, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 300, + 331 + ], + "score": 1.0, + "content": "this notation the Kullback-Leibler divergence is", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 335, + 462, + 362 + ], + "lines": [ + { + "bbox": [ + 149, + 335, + 462, + 362 + ], + "spans": [ + { + "bbox": [ + 149, + 335, + 462, + 362 + ], + "score": 0.91, + "content": "K ( w ) = \\sum _ { x \\in \\mathcal { X } } c D _ { K L } ( y | | F ^ { x } ( w ) ) = - c \\sum _ { x \\in \\mathcal { X } } \\log F _ { y } ^ { x } ( w ) = - c \\log \\prod _ { x \\in \\mathcal { X } } F _ { y } ^ { x } ( w ) .", + "type": "interline_equation", + "image_path": "0c762935a362819fdbd7538cdfac2d59d015b87dc485620b7b2c813c4c593257.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 149, + 335, + 462, + 362 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 366, + 134, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 366, + 135, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 135, + 379 + ], + "score": 1.0, + "content": "Hence", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 380, + 371, + 407 + ], + "lines": [ + { + "bbox": [ + 240, + 380, + 371, + 407 + ], + "spans": [ + { + "bbox": [ + 240, + 380, + 371, + 407 + ], + "score": 0.93, + "content": "W _ { 0 } = W \\cap \\bigcap _ { x \\in \\mathcal { X } } \\mathbb { V } ( 1 - F _ { y } ^ { x } ( w ) )", + "type": "interline_equation", + "image_path": "2c5fd9c205eb37e6c4fab6c9fb80af46c80b3cb65c7707540728e36317f80812.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 240, + 380, + 371, + 407 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 412, + 178, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 179, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 179, + 424 + ], + "score": 1.0, + "content": "is semi-algebraic.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 506, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 207, + 442 + ], + "score": 1.0, + "content": "Recall that the function", + "type": "text" + }, + { + "bbox": [ + 207, + 429, + 239, + 440 + ], + "score": 0.89, + "content": "\\Delta \\mathrm { s t e p } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 427, + 506, + 442 + ], + "score": 1.0, + "content": "is associated to an encoding of the UTM in linear logic by the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "Sweedler semantics (Clift & Murfet, 2018) and the particular polynomials involved have a form", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 450, + 504, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 504, + 462 + ], + "score": 1.0, + "content": "that is determined by the details of that encoding (Clift & Murfet, 2018, Proposition 4.3). 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Choose", + "type": "text" + }, + { + "bbox": [ + 169, + 244, + 199, + 254 + ], + "score": 0.91, + "content": "x \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 242, + 222, + 256 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 222, + 244, + 262, + 256 + ], + "score": 0.93, + "content": "q ( x ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 242, + 294, + 256 + ], + "score": 1.0, + "content": "and let", + "type": "text" + }, + { + "bbox": [ + 295, + 246, + 302, + 255 + ], + "score": 0.8, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 242, + 355, + 256 + ], + "score": 1.0, + "content": "be such that", + "type": "text" + }, + { + "bbox": [ + 355, + 244, + 403, + 256 + ], + "score": 0.93, + "content": "q ( y | x ) = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 242, + 425, + 256 + ], + "score": 1.0, + "content": ". 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In", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 317, + 300, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 300, + 331 + ], + "score": 1.0, + "content": "this notation the Kullback-Leibler divergence is", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 306, + 505, + 331 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 335, + 462, + 362 + ], + "lines": [ + { + "bbox": [ + 149, + 335, + 462, + 362 + ], + "spans": [ + { + "bbox": [ + 149, + 335, + 462, + 362 + ], + "score": 0.91, + "content": "K ( w ) = \\sum _ { x \\in \\mathcal { X } } c D _ { K L } ( y | | F ^ { x } ( w ) ) = - c \\sum _ { x \\in \\mathcal { X } } \\log F _ { y } ^ { x } ( w ) = - c \\log \\prod _ { x \\in \\mathcal { X } } F _ { y } ^ { x } ( w ) .", + "type": "interline_equation", + "image_path": "0c762935a362819fdbd7538cdfac2d59d015b87dc485620b7b2c813c4c593257.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 149, + 335, + 462, + 362 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 366, + 134, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 366, + 135, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 135, + 379 + ], + "score": 1.0, + "content": "Hence", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 366, + 135, + 379 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 380, + 371, + 407 + ], + "lines": [ + { + "bbox": [ + 240, + 380, + 371, + 407 + ], + "spans": [ + { + "bbox": [ + 240, + 380, + 371, + 407 + ], + "score": 0.93, + "content": "W _ { 0 } = W \\cap \\bigcap _ { x \\in \\mathcal { X } } \\mathbb { V } ( 1 - F _ { y } ^ { x } ( w ) )", + "type": "interline_equation", + "image_path": "2c5fd9c205eb37e6c4fab6c9fb80af46c80b3cb65c7707540728e36317f80812.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 240, + 380, + 371, + 407 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 412, + 178, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 179, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 179, + 424 + ], + "score": 1.0, + "content": "is semi-algebraic.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 411, + 179, + 424 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 428, + 506, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 207, + 442 + ], + "score": 1.0, + "content": "Recall that the function", + "type": "text" + }, + { + "bbox": [ + 207, + 429, + 239, + 440 + ], + "score": 0.89, + "content": "\\Delta \\mathrm { s t e p } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 427, + 506, + 442 + ], + "score": 1.0, + "content": "is associated to an encoding of the UTM in linear logic by the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "Sweedler semantics (Clift & Murfet, 2018) and the particular polynomials involved have a form", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 450, + 504, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 504, + 462 + ], + "score": 1.0, + "content": "that is determined by the details of that encoding (Clift & Murfet, 2018, Proposition 4.3). 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With this in hand we may compute", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 588, + 368, + 605 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 607, + 372, + 663 + ], + "lines": [ + { + "bbox": [ + 239, + 607, + 372, + 663 + ], + "spans": [ + { + "bbox": [ + 239, + 607, + 372, + 663 + ], + "score": 0.94, + "content": "\\begin{array} { r } { W _ { 0 } = W \\cap \\displaystyle \\bigcap _ { x \\in \\mathcal { X } } \\mathbb { V } ( 1 - F _ { y } ^ { x } ( w ) ) } \\\\ { = W \\cap \\displaystyle \\bigcap _ { x \\in \\mathcal { X } } \\bigcap _ { s \\neq y } \\mathbb { V } ( F _ { s } ^ { x } ( w ) ) . } \\end{array}", + "type": "interline_equation", + "image_path": "ac771f11c67d3891be5cece10a275aa657d6e1044f6dc0a9a7feef0fd986cedf.jpg" + } + ] + } + ], + "index": 32.5, + "virtual_lines": [ + { + "bbox": [ + 239, + 607, + 372, + 635.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 239, + 635.0, + 372, + 663.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 668, + 506, + 705 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 123, + 680 + ], + "score": 1.0, + "content": "But", + "type": "text" + }, + { + "bbox": [ + 124, + 669, + 137, + 680 + ], + "score": 0.89, + "content": "F _ { s } ^ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 667, + 448, + 680 + ], + "score": 1.0, + "content": "is a polynomial with non-negative integer coefficients, which takes values in", + "type": "text" + }, + { + "bbox": [ + 449, + 668, + 469, + 680 + ], + "score": 0.56, + "content": "[ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 667, + 485, + 680 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 486, + 669, + 505, + 679 + ], + "score": 0.84, + "content": "w \\in", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 107, + 680, + 118, + 689 + ], + "score": 0.62, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 678, + 210, + 691 + ], + "score": 1.0, + "content": ". Hence it vanishes on", + "type": "text" + }, + { + "bbox": [ + 210, + 682, + 218, + 689 + ], + "score": 0.76, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 678, + 334, + 691 + ], + "score": 1.0, + "content": "if and only if for each triple", + "type": "text" + }, + { + "bbox": [ + 335, + 680, + 361, + 691 + ], + "score": 0.92, + "content": "\\mu , \\zeta , \\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 678, + 382, + 691 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 383, + 679, + 441, + 691 + ], + "score": 0.94, + "content": "s = \\pi ( \\mu , \\zeta , \\xi )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "one or more of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 103, + 689, + 376, + 708 + ], + "spans": [ + { + "bbox": [ + 103, + 689, + 317, + 708 + ], + "score": 1.0, + "content": "the coordinate functions xσ,qµ(σ,q,i), yσ,qζ(σ,q,j), zσ,qξ(σ,q,k)", + "type": "text" + }, + { + "bbox": [ + 314, + 691, + 364, + 702 + ], + "score": 1.0, + "content": "vanishes on", + "type": "text" + }, + { + "bbox": [ + 364, + 693, + 372, + 700 + ], + "score": 0.76, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 691, + 376, + 702 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 103, + 667, + 506, + 708 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 707, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 299, + 723 + ], + "score": 1.0, + "content": "The desired conclusion follows unless for every", + "type": "text" + }, + { + "bbox": [ + 299, + 710, + 326, + 720 + ], + "score": 0.89, + "content": "x \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 707, + 345, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 345, + 709, + 398, + 722 + ], + "score": 0.92, + "content": "( \\mu , \\zeta , \\xi ) \\in \\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 707, + 434, + 723 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 434, + 709, + 493, + 722 + ], + "score": 0.91, + "content": "\\pi ( \\mu , \\zeta , \\xi ) = y", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 707, + 506, + 723 + ], + "score": 1.0, + "content": "so", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 719, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 124, + 733 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 721, + 156, + 732 + ], + "score": 0.92, + "content": "F _ { s } ^ { x } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 719, + 184, + 733 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 184, + 721, + 208, + 732 + ], + "score": 0.9, + "content": "s \\neq y", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 719, + 296, + 733 + ], + "score": 1.0, + "content": ". But in this case case", + "type": "text" + }, + { + "bbox": [ + 297, + 721, + 335, + 732 + ], + "score": 0.9, + "content": "W _ { 0 } = W", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 719, + 464, + 733 + ], + "score": 1.0, + "content": "which contradicts Lemma D.1.", + "type": "text" + }, + { + "bbox": [ + 494, + 721, + 506, + 731 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 707, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 80, + 247, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 248, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 248, + 96 + ], + "score": 1.0, + "content": "E STAGED PSEUDO-UTM", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 223, + 118 + ], + "score": 1.0, + "content": "Simulating a Turing machine", + "type": "text" + }, + { + "bbox": [ + 223, + 106, + 235, + 116 + ], + "score": 0.8, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 105, + 309, + 118 + ], + "score": 1.0, + "content": "with tape alphabet", + "type": "text" + }, + { + "bbox": [ + 310, + 106, + 318, + 115 + ], + "score": 0.83, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 105, + 382, + 118 + ], + "score": 1.0, + "content": "and set of states", + "type": "text" + }, + { + "bbox": [ + 383, + 106, + 392, + 117 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "on a standard UTM requires", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 245, + 128 + ], + "score": 1.0, + "content": "the specification of an encoding of", + "type": "text" + }, + { + "bbox": [ + 245, + 117, + 253, + 126 + ], + "score": 0.83, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 117, + 271, + 128 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 271, + 117, + 280, + 128 + ], + "score": 0.87, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 117, + 505, + 128 + ], + "score": 1.0, + "content": "in the tape alphabet of the UTM. From the point of view", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "of exploring the geometry of program synthesis, this additional complexity is uninteresting and so", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 139, + 341, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 341, + 150 + ], + "score": 1.0, + "content": "here we consider a staged pseudo-UTM whose alphabet is", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 152, + 384, + 165 + ], + "lines": [ + { + "bbox": [ + 226, + 152, + 384, + 165 + ], + "spans": [ + { + "bbox": [ + 226, + 152, + 384, + 165 + ], + "score": 0.92, + "content": "\\Sigma _ { \\mathrm { U T M } } = \\Sigma \\cup Q \\cup \\{ L , R , S \\} \\cup \\{ X , \\sqsubseteq \\}", + "type": "interline_equation", + "image_path": "47f784d767f57b579cb9a41c8e56f298fa8eea94a0fa855856aa2e9b613efbd7.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 226, + 152, + 384, + 165 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 167, + 505, + 211 + ], + "lines": [ + { + "bbox": [ + 106, + 167, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 240, + 178 + ], + "score": 1.0, + "content": "where the union is disjoint where", + "type": "text" + }, + { + "bbox": [ + 241, + 167, + 250, + 177 + ], + "score": 0.75, + "content": "\\boxed { \\begin{array} { r l } \\end{array} }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 167, + 506, + 178 + ], + "score": 1.0, + "content": "is the blank symbol (which is distinct from the blank symbol of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 177, + 504, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 119, + 188 + ], + "score": 0.64, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 177, + 419, + 190 + ], + "score": 1.0, + "content": "). Such a machine is capable of simulating any machine with tape alphabet", + "type": "text" + }, + { + "bbox": [ + 419, + 178, + 427, + 188 + ], + "score": 0.82, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 177, + 495, + 190 + ], + "score": 1.0, + "content": "and set of states", + "type": "text" + }, + { + "bbox": [ + 495, + 178, + 504, + 189 + ], + "score": 0.82, + "content": "Q", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "but cannot simulate arbitrary machines and is not a UTM in the standard sense. The adjective staged", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 200, + 409, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 409, + 212 + ], + "score": 1.0, + "content": "refers to the design of the UTM, which we now explain. The set of states is", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 212, + 425, + 256 + ], + "lines": [ + { + "bbox": [ + 162, + 212, + 425, + 256 + ], + "spans": [ + { + "bbox": [ + 162, + 212, + 425, + 256 + ], + "score": 0.35, + "content": "\\begin{array} { r } { Q _ { \\mathrm { U T M } } = \\{ \\mathrm { c o m p S y m b o l , c o m p S t a t e , c o p y S y m b o l , c o p y S t a t e , c o p y } \\mathrm { ~ } \\forall \\mathrm { t o p y ~ } \\mathrm { ~ c o p y ~ } \\mathrm { ~ c o p y ~ } \\mathrm { ~ c o p y ~ } \\mathrm { ~ c o p y ~ } \\mathrm { ~ c o p y ~ } \\mathrm { ~ c o p y ~ } \\mathrm { ~ c o m p ~ } } \\\\ \\mathrm { ~ \\ \" c o m p S t a t e , ~ } \\mathrm { \\ \" { c o p y S y m b o l , } \\mathrm { ~ \\ \" { c o p y S t a t e , } \\mathrm { ~ - c o p y S t a t e , } \\mathrm { ~ - c o p y D i r , } \\mathrm { ~ \\ ~ } } } \\\\ \\mathrm { ~ \\ \" { u p d a t e S y m b o l , u p d a t e S t a t e , u p d a t e D i r , r e s e t D e s c r ~ } \\} . } \\end{array}", + "type": "interline_equation", + "image_path": "b3a2fa940172cc778c9b8e4ee80141b2b78103b5513ade8d54750f3baf6c4d72.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 162, + 212, + 425, + 226.66666666666666 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 162, + 226.66666666666666, + 425, + 241.33333333333331 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 162, + 241.33333333333331, + 425, + 255.99999999999997 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 256, + 503, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 254, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 506, + 270 + ], + "score": 1.0, + "content": "The UTM has four tapes numbered from 0 to 3, which we refer to as the description tape, the staging", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 266, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 281 + ], + "score": 1.0, + "content": "tape, the state tape and the working tape respectively. Initially the description tape contains a string", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 277, + 154, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 154, + 290 + ], + "score": 1.0, + "content": "of the form", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 287, + 399, + 301 + ], + "lines": [ + { + "bbox": [ + 210, + 287, + 399, + 301 + ], + "spans": [ + { + "bbox": [ + 210, + 287, + 399, + 301 + ], + "score": 0.88, + "content": "X s _ { 0 } q _ { 0 } s _ { 0 } ^ { \\prime } q _ { 0 } ^ { \\prime } d _ { 0 } s _ { 1 } q _ { 1 } s _ { 1 } ^ { \\prime } q _ { 1 } ^ { \\prime } d _ { 1 } \\dots s _ { N } q _ { N } s _ { N } ^ { \\prime } q _ { N } ^ { \\prime } d _ { N } X ,", + "type": "interline_equation", + "image_path": "e20513688d9121bee8af67f0f13a1b91614ecd3c7df2358faec9690fbf851037.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 210, + 287, + 399, + 301 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 272, + 314 + ], + "score": 1.0, + "content": "corresponding to the tuples which define", + "type": "text" + }, + { + "bbox": [ + 272, + 302, + 284, + 312 + ], + "score": 0.82, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 302, + 411, + 314 + ], + "score": 1.0, + "content": ", with the tape head initially on", + "type": "text" + }, + { + "bbox": [ + 411, + 304, + 421, + 313 + ], + "score": 0.84, + "content": "s _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 302, + 506, + 314 + ], + "score": 1.0, + "content": ". The staging tape is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 174, + 326 + ], + "score": 1.0, + "content": "initially a string", + "type": "text" + }, + { + "bbox": [ + 174, + 313, + 203, + 323 + ], + "score": 0.83, + "content": "X X X", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 312, + 349, + 326 + ], + "score": 1.0, + "content": "with the tape head over the second", + "type": "text" + }, + { + "bbox": [ + 349, + 313, + 359, + 322 + ], + "score": 0.8, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 312, + 506, + 326 + ], + "score": 1.0, + "content": ". The state tape has a single square", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 233, + 336 + ], + "score": 1.0, + "content": "containing some distribution in", + "type": "text" + }, + { + "bbox": [ + 234, + 324, + 251, + 335 + ], + "score": 0.88, + "content": "\\Delta Q", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 324, + 489, + 336 + ], + "score": 1.0, + "content": ", corresponding to the initial state of the simulated machine", + "type": "text" + }, + { + "bbox": [ + 489, + 324, + 501, + 334 + ], + "score": 0.78, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 324, + 505, + 336 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 334, + 504, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 487, + 347 + ], + "score": 1.0, + "content": "with the tape head over that square. Each square on the the working tape is some distribution in", + "type": "text" + }, + { + "bbox": [ + 487, + 335, + 504, + 345 + ], + "score": 0.8, + "content": "\\Delta \\Sigma", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 310, + 358 + ], + "score": 1.0, + "content": "with only finitely many distributions different from", + "type": "text" + }, + { + "bbox": [ + 310, + 345, + 320, + 355 + ], + "score": 0.37, + "content": "\\boxed { \\begin{array} { r l } \\end{array} }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 345, + 505, + 358 + ], + "score": 1.0, + "content": ". The UTM is initialized in state compSymbol.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 361, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "The operation of the UTM is outlined in Figure 6. It consists of two phases; the scan phase (middle", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 374, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 506, + 385 + ], + "score": 1.0, + "content": "and right path), and the update phase (left path). During the scan phase, the description tape is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 506, + 396 + ], + "score": 1.0, + "content": "scanned from left to right, and the first two squares of each tuple are compared to the contents of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "the working tape and state tape respectively. If both agree, then the last three symbols of the tuple", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 406, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 419 + ], + "score": 1.0, + "content": "are written to the staging tape (middle path), otherwise the tuple is ignored (right path). Once the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 416, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 107, + 418, + 117, + 427 + ], + "score": 0.81, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 416, + 505, + 430 + ], + "score": 1.0, + "content": "at the end of the description tape is reached, the UTM begins the update phase, wherein the three", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "symbols on the staging tape are then used to print the new symbol on the working tape, to update", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "the simulated state on the state tape, and to move the working tape head in the appropriate direction.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 450, + 372, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 358, + 462 + ], + "score": 1.0, + "content": "The tape head on the description tape is then reset to the initial", + "type": "text" + }, + { + "bbox": [ + 359, + 451, + 369, + 460 + ], + "score": 0.82, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 450, + 372, + 462 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 463, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "Remark E.1. One could imagine a variant of the UTM which did not include a staging tape, instead", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "performing the actions on the work and state tape directly upon reading the appropriate tuple on the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 484, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 499 + ], + "score": 1.0, + "content": "description tape. However, this is problematic when the contents of the state or working tape are", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "distributions, as the exact time-step of the simulated machine can become unsynchronised, increas-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 435, + 520 + ], + "score": 1.0, + "content": "ing entropy. As a simple example, suppose that the contents of the state tape were", + "type": "text" + }, + { + "bbox": [ + 436, + 507, + 484, + 519 + ], + "score": 0.91, + "content": "0 . 5 q + 0 . 5 p", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 507, + 505, + 520 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 519, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 288, + 531 + ], + "score": 1.0, + "content": "the symbol under the working tape head was", + "type": "text" + }, + { + "bbox": [ + 289, + 521, + 294, + 528 + ], + "score": 0.66, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 519, + 416, + 531 + ], + "score": 1.0, + "content": ". Upon encountering the tuple", + "type": "text" + }, + { + "bbox": [ + 416, + 519, + 450, + 530 + ], + "score": 0.85, + "content": "s q s { ' } q { ' } R", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 519, + 505, + 531 + ], + "score": 1.0, + "content": ", the machine", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "would enter a superposition of states corresponding to the tape head having both moved right and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 540, + 293, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 293, + 552 + ], + "score": 1.0, + "content": "not moved, complicating the future behaviour.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 559, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 106, + 559, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 571 + ], + "score": 1.0, + "content": "We define the period of the UTM to be the smallest nonzero time interval taken for the tape head", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 288, + 582 + ], + "score": 1.0, + "content": "on the description tape to return to the initial", + "type": "text" + }, + { + "bbox": [ + 288, + 571, + 298, + 581 + ], + "score": 0.82, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 570, + 505, + 582 + ], + "score": 1.0, + "content": ", and the machine to reenter the state compSymbol.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 304, + 594 + ], + "score": 1.0, + "content": "If the number of tuples on the description tape is", + "type": "text" + }, + { + "bbox": [ + 305, + 582, + 315, + 591 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 582, + 442, + 594 + ], + "score": 1.0, + "content": ", then the period of the UTM is", + "type": "text" + }, + { + "bbox": [ + 442, + 582, + 501, + 592 + ], + "score": 0.91, + "content": "T = 1 0 N + 5", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 582, + 505, + 594 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 592, + 447, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 398, + 605 + ], + "score": 1.0, + "content": "Moreover, other than the working tape, the position of the tape heads are", + "type": "text" + }, + { + "bbox": [ + 398, + 593, + 407, + 603 + ], + "score": 0.79, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 592, + 447, + 605 + ], + "score": 1.0, + "content": "-periodic.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5 + }, + { + "type": "title", + "bbox": [ + 108, + 619, + 275, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 617, + 276, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 276, + 634 + ], + "score": 1.0, + "content": "F SMOOTH TURING MACHINES", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 122, + 657 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 644, + 132, + 654 + ], + "score": 0.72, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 643, + 402, + 657 + ], + "score": 1.0, + "content": "be the staged pseudo-UTM of Appendix E. In defining the model", + "type": "text" + }, + { + "bbox": [ + 402, + 644, + 441, + 656 + ], + "score": 0.94, + "content": "p ( \\boldsymbol { y } | \\boldsymbol { x } , \\boldsymbol { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "associated to a", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 341, + 667 + ], + "score": 1.0, + "content": "synthesis problem in Section 2 we use a smooth relaxation", + "type": "text" + }, + { + "bbox": [ + 342, + 654, + 374, + 666 + ], + "score": 0.84, + "content": "\\Delta \\mathrm { { s t e p } } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 654, + 464, + 667 + ], + "score": 1.0, + "content": "of the step function of", + "type": "text" + }, + { + "bbox": [ + 465, + 655, + 473, + 664 + ], + "score": 0.71, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 654, + 506, + 667 + ], + "score": 1.0, + "content": ". In this", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 666, + 502, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 502, + 679 + ], + "score": 1.0, + "content": "appendix we define the smooth relaxation of any Turing machine following Clift & Murfet (2018).", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 682, + 505, + 716 + ], + "lines": [ + { + "bbox": [ + 105, + 681, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 122, + 695 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 682, + 185, + 694 + ], + "score": 0.92, + "content": "M = ( \\Sigma , Q , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 681, + 384, + 695 + ], + "score": 1.0, + "content": "be a Turing machine with a finite set of symbols", + "type": "text" + }, + { + "bbox": [ + 385, + 683, + 393, + 693 + ], + "score": 0.81, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 681, + 477, + 695 + ], + "score": 1.0, + "content": ", a finite set of states", + "type": "text" + }, + { + "bbox": [ + 478, + 683, + 487, + 694 + ], + "score": 0.86, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 681, + 506, + 695 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 693, + 506, + 707 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 182, + 707 + ], + "score": 1.0, + "content": "transition function", + "type": "text" + }, + { + "bbox": [ + 182, + 693, + 317, + 705 + ], + "score": 0.92, + "content": "\\delta : \\Sigma \\times Q \\bar { \\to } \\Sigma \\times Q \\times \\{ - 1 , 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 693, + 360, + 707 + ], + "score": 1.0, + "content": ". We write", + "type": "text" + }, + { + "bbox": [ + 360, + 694, + 415, + 705 + ], + "score": 0.93, + "content": "\\delta _ { i } = \\mathsf { p r o j } _ { i } \\circ \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 693, + 444, + 707 + ], + "score": 1.0, + "content": "for the", + "type": "text" + }, + { + "bbox": [ + 445, + 695, + 450, + 703 + ], + "score": 0.37, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 693, + 506, + 707 + ], + "score": 1.0, + "content": "th component", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 704, + 255, + 717 + ], + "spans": [ + { + "bbox": [ + 106, + 704, + 117, + 717 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 705, + 123, + 714 + ], + "score": 0.8, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 704, + 139, + 717 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 139, + 704, + 189, + 717 + ], + "score": 0.93, + "content": "i \\in \\{ 1 , 2 , 3 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 704, + 209, + 717 + ], + "score": 1.0, + "content": ". For", + "type": "text" + }, + { + "bbox": [ + 210, + 705, + 238, + 715 + ], + "score": 0.88, + "content": "\\sqsubseteq \\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 704, + 255, + 717 + ], + "score": 1.0, + "content": ", let", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48 + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 719, + 424, + 734 + ], + "lines": [ + { + "bbox": [ + 187, + 719, + 424, + 734 + ], + "spans": [ + { + "bbox": [ + 187, + 719, + 424, + 734 + ], + "score": 0.3, + "content": "\\Sigma ^ { \\mathbb { Z } , \\sqcap } = \\{ f : \\mathbb { Z } \\to \\Sigma | f ( i ) = \\bigsqcup \\mathrm { e x c e p t ~ f o r ~ f i n i t e l y ~ m a n y ~ } i \\} .", + "type": "interline_equation", + "image_path": "f3487b835791ef405aca7831e8d5f0b285bc6d2fd3410dde01e3985b09ed852f.jpg" + } + ] + } + ], + "index": 50, + "virtual_lines": [ + { + "bbox": [ + 187, + 719, + 424, + 734 + ], + "spans": [], + "index": 50 + } + ] + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "15", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 106, + 26, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 80, + 247, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 248, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 248, + 96 + ], + "score": 1.0, + "content": "E STAGED PSEUDO-UTM", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 223, + 118 + ], + "score": 1.0, + "content": "Simulating a Turing machine", + "type": "text" + }, + { + "bbox": [ + 223, + 106, + 235, + 116 + ], + "score": 0.8, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 105, + 309, + 118 + ], + "score": 1.0, + "content": "with tape alphabet", + "type": "text" + }, + { + "bbox": [ + 310, + 106, + 318, + 115 + ], + "score": 0.83, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 105, + 382, + 118 + ], + "score": 1.0, + "content": "and set of states", + "type": "text" + }, + { + "bbox": [ + 383, + 106, + 392, + 117 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "on a standard UTM requires", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 245, + 128 + ], + "score": 1.0, + "content": "the specification of an encoding of", + "type": "text" + }, + { + "bbox": [ + 245, + 117, + 253, + 126 + ], + "score": 0.83, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 117, + 271, + 128 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 271, + 117, + 280, + 128 + ], + "score": 0.87, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 117, + 505, + 128 + ], + "score": 1.0, + "content": "in the tape alphabet of the UTM. From the point of view", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "of exploring the geometry of program synthesis, this additional complexity is uninteresting and so", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 139, + 341, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 341, + 150 + ], + "score": 1.0, + "content": "here we consider a staged pseudo-UTM whose alphabet is", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 105, + 505, + 150 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 152, + 384, + 165 + ], + "lines": [ + { + "bbox": [ + 226, + 152, + 384, + 165 + ], + "spans": [ + { + "bbox": [ + 226, + 152, + 384, + 165 + ], + "score": 0.92, + "content": "\\Sigma _ { \\mathrm { U T M } } = \\Sigma \\cup Q \\cup \\{ L , R , S \\} \\cup \\{ X , \\sqsubseteq \\}", + "type": "interline_equation", + "image_path": "47f784d767f57b579cb9a41c8e56f298fa8eea94a0fa855856aa2e9b613efbd7.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 226, + 152, + 384, + 165 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 167, + 505, + 211 + ], + "lines": [ + { + "bbox": [ + 106, + 167, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 240, + 178 + ], + "score": 1.0, + "content": "where the union is disjoint where", + "type": "text" + }, + { + "bbox": [ + 241, + 167, + 250, + 177 + ], + "score": 0.75, + "content": "\\boxed { \\begin{array} { r l } \\end{array} }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 167, + 506, + 178 + ], + "score": 1.0, + "content": "is the blank symbol (which is distinct from the blank symbol of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 177, + 504, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 119, + 188 + ], + "score": 0.64, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 177, + 419, + 190 + ], + "score": 1.0, + "content": "). Such a machine is capable of simulating any machine with tape alphabet", + "type": "text" + }, + { + "bbox": [ + 419, + 178, + 427, + 188 + ], + "score": 0.82, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 177, + 495, + 190 + ], + "score": 1.0, + "content": "and set of states", + "type": "text" + }, + { + "bbox": [ + 495, + 178, + 504, + 189 + ], + "score": 0.82, + "content": "Q", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "but cannot simulate arbitrary machines and is not a UTM in the standard sense. The adjective staged", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 200, + 409, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 409, + 212 + ], + "score": 1.0, + "content": "refers to the design of the UTM, which we now explain. The set of states is", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 106, + 167, + 506, + 212 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 212, + 425, + 256 + ], + "lines": [ + { + "bbox": [ + 162, + 212, + 425, + 256 + ], + "spans": [ + { + "bbox": [ + 162, + 212, + 425, + 256 + ], + "score": 0.35, + "content": "\\begin{array} { r } { Q _ { \\mathrm { U T M } } = \\{ \\mathrm { c o m p S y m b o l , c o m p S t a t e , c o p y S y m b o l , c o p y S t a t e , c o p y } \\mathrm { ~ } \\forall \\mathrm { t o p y ~ } \\mathrm { ~ c o p y ~ } \\mathrm { ~ c o p y ~ } \\mathrm { ~ c o p y ~ } \\mathrm { ~ c o p y ~ } \\mathrm { ~ c o p y ~ } \\mathrm { ~ c o p y ~ } \\mathrm { ~ c o m p ~ } } \\\\ \\mathrm { ~ \\ \" c o m p S t a t e , ~ } \\mathrm { \\ \" { c o p y S y m b o l , } \\mathrm { ~ \\ \" { c o p y S t a t e , } \\mathrm { ~ - c o p y S t a t e , } \\mathrm { ~ - c o p y D i r , } \\mathrm { ~ \\ ~ } } } \\\\ \\mathrm { ~ \\ \" { u p d a t e S y m b o l , u p d a t e S t a t e , u p d a t e D i r , r e s e t D e s c r ~ } \\} . } \\end{array}", + "type": "interline_equation", + "image_path": "b3a2fa940172cc778c9b8e4ee80141b2b78103b5513ade8d54750f3baf6c4d72.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 162, + 212, + 425, + 226.66666666666666 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 162, + 226.66666666666666, + 425, + 241.33333333333331 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 162, + 241.33333333333331, + 425, + 255.99999999999997 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 256, + 503, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 254, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 506, + 270 + ], + "score": 1.0, + "content": "The UTM has four tapes numbered from 0 to 3, which we refer to as the description tape, the staging", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 266, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 281 + ], + "score": 1.0, + "content": "tape, the state tape and the working tape respectively. Initially the description tape contains a string", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 277, + 154, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 154, + 290 + ], + "score": 1.0, + "content": "of the form", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 254, + 506, + 290 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 210, + 287, + 399, + 301 + ], + "lines": [ + { + "bbox": [ + 210, + 287, + 399, + 301 + ], + "spans": [ + { + "bbox": [ + 210, + 287, + 399, + 301 + ], + "score": 0.88, + "content": "X s _ { 0 } q _ { 0 } s _ { 0 } ^ { \\prime } q _ { 0 } ^ { \\prime } d _ { 0 } s _ { 1 } q _ { 1 } s _ { 1 } ^ { \\prime } q _ { 1 } ^ { \\prime } d _ { 1 } \\dots s _ { N } q _ { N } s _ { N } ^ { \\prime } q _ { N } ^ { \\prime } d _ { N } X ,", + "type": "interline_equation", + "image_path": "e20513688d9121bee8af67f0f13a1b91614ecd3c7df2358faec9690fbf851037.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 210, + 287, + 399, + 301 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 272, + 314 + ], + "score": 1.0, + "content": "corresponding to the tuples which define", + "type": "text" + }, + { + "bbox": [ + 272, + 302, + 284, + 312 + ], + "score": 0.82, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 302, + 411, + 314 + ], + "score": 1.0, + "content": ", with the tape head initially on", + "type": "text" + }, + { + "bbox": [ + 411, + 304, + 421, + 313 + ], + "score": 0.84, + "content": "s _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 302, + 506, + 314 + ], + "score": 1.0, + "content": ". The staging tape is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 174, + 326 + ], + "score": 1.0, + "content": "initially a string", + "type": "text" + }, + { + "bbox": [ + 174, + 313, + 203, + 323 + ], + "score": 0.83, + "content": "X X X", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 312, + 349, + 326 + ], + "score": 1.0, + "content": "with the tape head over the second", + "type": "text" + }, + { + "bbox": [ + 349, + 313, + 359, + 322 + ], + "score": 0.8, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 312, + 506, + 326 + ], + "score": 1.0, + "content": ". The state tape has a single square", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 233, + 336 + ], + "score": 1.0, + "content": "containing some distribution in", + "type": "text" + }, + { + "bbox": [ + 234, + 324, + 251, + 335 + ], + "score": 0.88, + "content": "\\Delta Q", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 324, + 489, + 336 + ], + "score": 1.0, + "content": ", corresponding to the initial state of the simulated machine", + "type": "text" + }, + { + "bbox": [ + 489, + 324, + 501, + 334 + ], + "score": 0.78, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 324, + 505, + 336 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 334, + 504, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 487, + 347 + ], + "score": 1.0, + "content": "with the tape head over that square. Each square on the the working tape is some distribution in", + "type": "text" + }, + { + "bbox": [ + 487, + 335, + 504, + 345 + ], + "score": 0.8, + "content": "\\Delta \\Sigma", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 310, + 358 + ], + "score": 1.0, + "content": "with only finitely many distributions different from", + "type": "text" + }, + { + "bbox": [ + 310, + 345, + 320, + 355 + ], + "score": 0.37, + "content": "\\boxed { \\begin{array} { r l } \\end{array} }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 345, + 505, + 358 + ], + "score": 1.0, + "content": ". The UTM is initialized in state compSymbol.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 302, + 506, + 358 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 361, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "The operation of the UTM is outlined in Figure 6. It consists of two phases; the scan phase (middle", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 374, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 506, + 385 + ], + "score": 1.0, + "content": "and right path), and the update phase (left path). During the scan phase, the description tape is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 506, + 396 + ], + "score": 1.0, + "content": "scanned from left to right, and the first two squares of each tuple are compared to the contents of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "the working tape and state tape respectively. If both agree, then the last three symbols of the tuple", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 406, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 419 + ], + "score": 1.0, + "content": "are written to the staging tape (middle path), otherwise the tuple is ignored (right path). Once the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 416, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 107, + 418, + 117, + 427 + ], + "score": 0.81, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 416, + 505, + 430 + ], + "score": 1.0, + "content": "at the end of the description tape is reached, the UTM begins the update phase, wherein the three", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "symbols on the staging tape are then used to print the new symbol on the working tape, to update", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "the simulated state on the state tape, and to move the working tape head in the appropriate direction.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 450, + 372, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 358, + 462 + ], + "score": 1.0, + "content": "The tape head on the description tape is then reset to the initial", + "type": "text" + }, + { + "bbox": [ + 359, + 451, + 369, + 460 + ], + "score": 0.82, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 450, + 372, + 462 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 362, + 506, + 462 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 463, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "Remark E.1. One could imagine a variant of the UTM which did not include a staging tape, instead", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "performing the actions on the work and state tape directly upon reading the appropriate tuple on the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 484, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 499 + ], + "score": 1.0, + "content": "description tape. However, this is problematic when the contents of the state or working tape are", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "distributions, as the exact time-step of the simulated machine can become unsynchronised, increas-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 435, + 520 + ], + "score": 1.0, + "content": "ing entropy. As a simple example, suppose that the contents of the state tape were", + "type": "text" + }, + { + "bbox": [ + 436, + 507, + 484, + 519 + ], + "score": 0.91, + "content": "0 . 5 q + 0 . 5 p", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 507, + 505, + 520 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 519, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 288, + 531 + ], + "score": 1.0, + "content": "the symbol under the working tape head was", + "type": "text" + }, + { + "bbox": [ + 289, + 521, + 294, + 528 + ], + "score": 0.66, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 519, + 416, + 531 + ], + "score": 1.0, + "content": ". Upon encountering the tuple", + "type": "text" + }, + { + "bbox": [ + 416, + 519, + 450, + 530 + ], + "score": 0.85, + "content": "s q s { ' } q { ' } R", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 519, + 505, + 531 + ], + "score": 1.0, + "content": ", the machine", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "would enter a superposition of states corresponding to the tape head having both moved right and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 540, + 293, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 293, + 552 + ], + "score": 1.0, + "content": "not moved, complicating the future behaviour.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 463, + 505, + 552 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 559, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 106, + 559, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 571 + ], + "score": 1.0, + "content": "We define the period of the UTM to be the smallest nonzero time interval taken for the tape head", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 288, + 582 + ], + "score": 1.0, + "content": "on the description tape to return to the initial", + "type": "text" + }, + { + "bbox": [ + 288, + 571, + 298, + 581 + ], + "score": 0.82, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 570, + 505, + 582 + ], + "score": 1.0, + "content": ", and the machine to reenter the state compSymbol.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 304, + 594 + ], + "score": 1.0, + "content": "If the number of tuples on the description tape is", + "type": "text" + }, + { + "bbox": [ + 305, + 582, + 315, + 591 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 582, + 442, + 594 + ], + "score": 1.0, + "content": ", then the period of the UTM is", + "type": "text" + }, + { + "bbox": [ + 442, + 582, + 501, + 592 + ], + "score": 0.91, + "content": "T = 1 0 N + 5", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 582, + 505, + 594 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 592, + 447, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 398, + 605 + ], + "score": 1.0, + "content": "Moreover, other than the working tape, the position of the tape heads are", + "type": "text" + }, + { + "bbox": [ + 398, + 593, + 407, + 603 + ], + "score": 0.79, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 592, + 447, + 605 + ], + "score": 1.0, + "content": "-periodic.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 559, + 505, + 605 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 619, + 275, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 617, + 276, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 276, + 634 + ], + "score": 1.0, + "content": "F SMOOTH TURING MACHINES", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 122, + 657 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 644, + 132, + 654 + ], + "score": 0.72, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 643, + 402, + 657 + ], + "score": 1.0, + "content": "be the staged pseudo-UTM of Appendix E. In defining the model", + "type": "text" + }, + { + "bbox": [ + 402, + 644, + 441, + 656 + ], + "score": 0.94, + "content": "p ( \\boldsymbol { y } | \\boldsymbol { x } , \\boldsymbol { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "associated to a", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 341, + 667 + ], + "score": 1.0, + "content": "synthesis problem in Section 2 we use a smooth relaxation", + "type": "text" + }, + { + "bbox": [ + 342, + 654, + 374, + 666 + ], + "score": 0.84, + "content": "\\Delta \\mathrm { { s t e p } } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 654, + 464, + 667 + ], + "score": 1.0, + "content": "of the step function of", + "type": "text" + }, + { + "bbox": [ + 465, + 655, + 473, + 664 + ], + "score": 0.71, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 654, + 506, + 667 + ], + "score": 1.0, + "content": ". 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0.814, + "content": "·", + "type": "text" + }, + { + "bbox": [ + 142, + 127, + 214, + 139 + ], + "score": 0.92, + "content": "M v _ { t } \\in \\{ L , S , R \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 126, + 415, + 139 + ], + "score": 1.0, + "content": ": the direction to move, in the transition from time", + "type": "text" + }, + { + "bbox": [ + 416, + 128, + 421, + 137 + ], + "score": 0.78, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 126, + 432, + 139 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 432, + 128, + 454, + 137 + ], + "score": 0.87, + "content": "t + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 126, + 458, + 139 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 147, + 504, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 147, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 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Let", + "type": "text" + }, + { + "bbox": [ + 184, + 173, + 245, + 186 + ], + "score": 0.91, + "content": "M = ( \\Sigma , Q , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 172, + 436, + 187 + ], + "score": 1.0, + "content": "be a Turing machine. The smooth relaxation of", + "type": "text" + }, + { + "bbox": [ + 437, + 174, + 448, + 184 + ], + "score": 0.85, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 172, + 505, + 187 + ], + "score": 1.0, + "content": "is the smooth", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 312, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 182, + 199 + ], + "score": 1.0, + "content": "dynamical system", + "type": "text" + }, + { + "bbox": [ + 182, + 186, + 283, + 198 + ], + "score": 0.87, + "content": "( ( \\Delta \\Sigma ) ^ { \\mathbb { Z } , \\sqsupset } \\times \\Delta Q , \\Delta \\mathrm { s t e p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 183, + 312, + 199 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 202, + 396, + 218 + ], + "lines": [ + { + "bbox": [ + 214, + 202, + 396, + 218 + ], + "spans": [ + { + "bbox": [ + 214, + 202, + 396, + 218 + ], + "score": 0.92, + "content": "\\Delta \\mathrm { s t e p } : ( \\Delta \\Sigma ) ^ { \\mathbb { Z } , \\square } \\times \\Delta Q ( \\Delta \\Sigma ) ^ { \\mathbb { Z } , \\square } \\times \\Delta Q", + "type": "interline_equation", + "image_path": "72b1aa44f2de884700842f66e48f63028a81392f430e1671c249d7cc87b59b8d.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 214, + 202, + 396, + 218 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 222, + 502, + 246 + ], + "lines": [ + { + "bbox": [ + 104, + 221, + 503, + 237 + ], + "spans": [ + { + "bbox": [ + 104, + 221, + 281, + 237 + ], + "score": 1.0, + "content": "is a smooth transformation sending a state", + "type": "text" + }, + { + "bbox": [ + 282, + 223, + 375, + 236 + ], + "score": 0.91, + "content": "( \\{ P ( Y _ { u , t } ) \\} _ { u \\in \\mathbb { Z } } , P ( S _ { t } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 221, + 389, + 237 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 389, + 223, + 503, + 236 + ], + "score": 0.88, + "content": "( \\{ P ( Y _ { u , t + 1 } ) \\} _ { u \\in \\mathbb { Z } } , P ( S _ { t + 1 } ) )", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 234, + 222, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 222, + 247 + ], + "score": 1.0, + "content": "determined by the equations", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 130, + 254, + 394, + 270 + ], + "lines": [ + { + "bbox": [ + 130, + 254, + 394, + 270 + ], + "spans": [ + { + "bbox": [ + 130, + 254, + 394, + 270 + ], + "score": 0.84, + "content": "\\begin{array} { r } { P ( M v _ { t } = d | C ) = \\sum _ { \\sigma , q } \\delta _ { \\delta _ { 3 } ( \\sigma , q ) = d } P ( Y _ { 0 , t } = \\sigma | C ) P ( S _ { t } = q | C ) , } \\end{array}", + "type": "interline_equation", + "image_path": "1b4007c46825393aef4aa4521ba4b324339f93f3df64105c7d175487d591569c.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 130, + 254, + 394, + 270 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 284, + 405, + 299 + ], + "lines": [ + { + "bbox": [ + 132, + 284, + 405, + 299 + ], + "spans": [ + { + "bbox": [ + 132, + 284, + 405, + 299 + ], + "score": 0.72, + "content": "\\begin{array} { r } { P ( W r _ { t } = \\sigma | C ) = \\sum _ { \\sigma ^ { \\prime } , q } \\delta _ { \\delta _ { 1 } ( \\sigma ^ { \\prime } , q ) = \\sigma } P ( Y _ { 0 , t } = \\sigma ^ { \\prime } | C ) P ( S _ { t } = q | C ) , } \\end{array}", + "type": "interline_equation", + "image_path": "919c45ad9588dbcda6b701e326612cb29b59ce549ab3de2174aa479d62c58d1d.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 132, + 284, + 405, + 299 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 313, + 408, + 329 + ], + "lines": [], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 131, + 313, + 408, + 329 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 341, + 504, + 402 + ], + "lines": [ + { + "bbox": [ + 132, + 341, + 504, + 402 + ], + "spans": [ + { + "bbox": [ + 132, + 341, + 504, + 402 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { P ( Y _ { u , t + 1 } = \\sigma | C ) = P ( M v _ { t } = L | C ) \\Big ( \\delta _ { u \\neq 1 } P ( Y _ { u - 1 , t } = \\sigma | C ) + \\delta _ { u = 1 } P ( W r _ { t } = \\sigma | C ) \\Big ) } \\\\ & { \\qquad + P ( M v _ { t } = S | C ) \\Big ( \\delta _ { u \\neq 0 } P ( Y _ { u , t } = \\sigma | C ) + \\delta _ { u = 0 } P ( W r _ { t } = \\sigma | C ) \\Big ) } \\\\ & { \\qquad + P ( M v _ { t } = R | C ) \\Big ( \\delta _ { u \\neq - 1 } P ( Y _ { u + 1 , t } = \\sigma | C ) + \\delta _ { u = - 1 } P ( W r _ { t } = \\sigma | C ) \\Big ) , } \\end{array}", + "type": "interline_equation", + "image_path": "ac6ffd9db28543066c5647c7c715ab7e342982d3a5b07f74916f12368ec08bbc.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 132, + 341, + 504, + 361.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 132, + 361.3333333333333, + 504, + 381.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 132, + 381.66666666666663, + 504, + 401.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 409, + 290, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 407, + 293, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 133, + 425 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 409, + 220, + 423 + ], + "score": 0.92, + "content": "C \\in ( \\Delta \\Sigma ) ^ { \\mathbb { Z } , \\square } \\times \\Delta Q", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 407, + 293, + 425 + ], + "score": 1.0, + "content": "is an initial state.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 430, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 446 + ], + "score": 1.0, + "content": "We will call the smooth relaxation of a Turing machine a smooth Turing machine. A smooth Turing", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 441, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 455 + ], + "score": 1.0, + "content": "machine encodes uncertainty in the initial configuration of a Turing machine together with an update", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "score": 1.0, + "content": "rule for how to propagate this uncertainty over time. We interpret the smooth step function as", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "updating the state of belief of a “naive” Bayesian observer. This nomenclature comes from the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 476, + 490, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 490, + 488 + ], + "score": 1.0, + "content": "assumption of conditional independence between random variables in our probability functions.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 489, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "score": 1.0, + "content": "Remark F.2. Propagating uncertainty using standard probability leads to a smooth dynamical sys-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 501, + 504, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 504, + 513 + ], + "score": 1.0, + "content": "tem which encodes the state evolution of an “ordinary” Bayesian observer of the Turing machine.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "score": 1.0, + "content": "This requires the calculation of various joint distributions which makes such an extension computa-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "tionally difficult to work with. Computation aside, the naive probabilistic extension is justified from", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "the point of view of derivatives of algorithms according to the denotational semantics of differential", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 545, + 339, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 339, + 558 + ], + "score": 1.0, + "content": "linear logic. See Clift & Murfet (2018) for further details.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 106, + 565, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 504, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 504, + 578 + ], + "score": 1.0, + "content": "We call the smooth extension of a universal Turing machine a smooth universal Turing machine.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 249, + 589 + ], + "score": 1.0, + "content": "Recall that the staged pseudo-UTM", + "type": "text" + }, + { + "bbox": [ + 249, + 577, + 258, + 587 + ], + "score": 0.6, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "has four tapes: the description tape, the staging tape, the state", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 587, + 435, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 303, + 601 + ], + "score": 1.0, + "content": "tape and working tape. 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Let", + "type": "text" + }, + { + "bbox": [ + 184, + 173, + 245, + 186 + ], + "score": 0.91, + "content": "M = ( \\Sigma , Q , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 172, + 436, + 187 + ], + "score": 1.0, + "content": "be a Turing machine. 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1 , t } = \\sigma | C ) + \\delta _ { u = 1 } P ( W r _ { t } = \\sigma | C ) \\Big ) } \\\\ & { \\qquad + P ( M v _ { t } = S | C ) \\Big ( \\delta _ { u \\neq 0 } P ( Y _ { u , t } = \\sigma | C ) + \\delta _ { u = 0 } P ( W r _ { t } = \\sigma | C ) \\Big ) } \\\\ & { \\qquad + P ( M v _ { t } = R | C ) \\Big ( \\delta _ { u \\neq - 1 } P ( Y _ { u + 1 , t } = \\sigma | C ) + \\delta _ { u = - 1 } P ( W r _ { t } = \\sigma | C ) \\Big ) , } \\end{array}", + "type": "interline_equation", + "image_path": "ac6ffd9db28543066c5647c7c715ab7e342982d3a5b07f74916f12368ec08bbc.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 132, + 341, + 504, + 361.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 132, + 361.3333333333333, + 504, + 381.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 132, + 381.66666666666663, + 504, + 401.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 409, + 290, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 407, + 293, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 133, + 425 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 409, + 220, + 423 + ], + "score": 0.92, + "content": "C \\in ( \\Delta \\Sigma ) ^ { \\mathbb { Z } , \\square } \\times \\Delta Q", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 407, + 293, + 425 + ], + "score": 1.0, + "content": "is an initial state.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 407, + 293, + 425 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 430, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 446 + ], + "score": 1.0, + "content": "We will call the smooth relaxation of a Turing machine a smooth Turing machine. A smooth Turing", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 441, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 455 + ], + "score": 1.0, + "content": "machine encodes uncertainty in the initial configuration of a Turing machine together with an update", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 467 + ], + "score": 1.0, + "content": "rule for how to propagate this uncertainty over time. We interpret the smooth step function as", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "updating the state of belief of a “naive” Bayesian observer. This nomenclature comes from the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 476, + 490, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 490, + 488 + ], + "score": 1.0, + "content": "assumption of conditional independence between random variables in our probability functions.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 429, + 506, + 488 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 489, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "score": 1.0, + "content": "Remark F.2. Propagating uncertainty using standard probability leads to a smooth dynamical sys-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 501, + 504, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 504, + 513 + ], + "score": 1.0, + "content": "tem which encodes the state evolution of an “ordinary” Bayesian observer of the Turing machine.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "score": 1.0, + "content": "This requires the calculation of various joint distributions which makes such an extension computa-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "tionally difficult to work with. Computation aside, the naive probabilistic extension is justified from", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "the point of view of derivatives of algorithms according to the denotational semantics of differential", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 545, + 339, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 339, + 558 + ], + "score": 1.0, + "content": "linear logic. See Clift & Murfet (2018) for further details.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 489, + 505, + 558 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 565, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 504, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 504, + 578 + ], + "score": 1.0, + "content": "We call the smooth extension of a universal Turing machine a smooth universal Turing machine.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 249, + 589 + ], + "score": 1.0, + "content": "Recall that the staged pseudo-UTM", + "type": "text" + }, + { + "bbox": [ + 249, + 577, + 258, + 587 + ], + "score": 0.6, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "has four tapes: the description tape, the staging tape, the state", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 587, + 435, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 303, + 601 + ], + "score": 1.0, + "content": "tape and working tape. The smooth relaxation of", + "type": "text" + }, + { + "bbox": [ + 303, + 588, + 312, + 597 + ], + "score": 0.82, + "content": "\\mathcal { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 587, + 435, + 601 + ], + "score": 1.0, + "content": "is a smooth dynamical system", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 106, + 565, + 505, + 601 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 167, + 604, + 443, + 620 + ], + "lines": [ + { + "bbox": [ + 167, + 604, + 443, + 620 + ], + "spans": [ + { + "bbox": [ + 167, + 604, + 443, + 620 + ], + "score": 0.88, + "content": "\\Delta \\mathrm { s t e p } _ { \\mathcal { U } } : [ ( \\Delta \\Sigma _ { \\mathrm { U T M } } ) ^ { \\mathbb { Z } , \\bigtriangledown } ] ^ { 4 } \\times \\Delta Q _ { \\mathrm { U T M } } \\to [ ( \\Delta \\Sigma _ { \\mathrm { U T M } } ) ^ { \\mathbb { Z } , \\bigtriangledown } ] ^ { 4 } \\times \\Delta Q _ { \\mathrm { U T M } } .", + "type": 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Given", + "type": "text" + }, + { + "bbox": [ + 195, + 196, + 218, + 207 + ], + "score": 0.91, + "content": "t \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 192, + 259, + 211 + ], + "score": 1.0, + "content": "we define", + "type": "text" + }, + { + "bbox": [ + 260, + 195, + 373, + 207 + ], + "score": 0.86, + "content": "\\Delta \\operatorname { s t e p } ^ { t } : \\Sigma ^ { * } \\times W \\longrightarrow \\Delta Q", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 192, + 387, + 211 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 210, + 376, + 225 + ], + "lines": [ + { + "bbox": [ + 235, + 210, + 376, + 225 + ], + "spans": [ + { + "bbox": [ + 235, + 210, + 376, + 225 + ], + "score": 0.9, + "content": "\\Delta \\operatorname { s t e p } ^ { t } ( x , w ) = \\Pi _ { Q } F ^ { t } ( x , w , \\operatorname { i n i t } )", + "type": "interline_equation", + "image_path": "74696f823bfc89d626700cc4f9a1a8bb8d8f406ee6306dab64b37d909b2ce88c.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 235, + 210, + 376, + 225 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 228, + 257, + 241 + ], + "lines": [ + { + "bbox": [ + 105, + 226, + 258, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 133, + 244 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 229, + 149, + 241 + ], + "score": 0.9, + "content": "\\Pi _ { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 226, + 237, + 244 + ], + "score": 1.0, + "content": "is the projection onto", + "type": "text" + }, + { + "bbox": [ + 237, + 229, + 254, + 240 + ], + "score": 0.89, + "content": "\\Delta Q", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 226, + 258, + 244 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 107, + 255, + 238, + 268 + ], + "lines": [ + { + "bbox": [ + 105, + 254, + 239, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 239, + 271 + ], + "score": 1.0, + "content": "G DIRECT SIMULATION", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 280, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "score": 1.0, + "content": "For computational efficiency in our PyTorch implementation of the staged pseudo-UTM we imple-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 292, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 129, + 304 + ], + "score": 1.0, + "content": "ment", + "type": "text" + }, + { + "bbox": [ + 129, + 292, + 138, + 302 + ], + "score": 0.83, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 292, + 212, + 304 + ], + "score": 1.0, + "content": "of (9) rather than", + "type": "text" + }, + { + "bbox": [ + 212, + 292, + 247, + 304 + ], + "score": 0.92, + "content": "\\Delta \\mathrm { s t e p } _ { \\mathcal { U } }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 292, + 506, + 304 + ], + "score": 1.0, + "content": ". 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The task then is to define functions", + "type": "text" + }, + { + "bbox": [ + 450, + 357, + 466, + 369 + ], + "score": 0.91, + "content": "f , g", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 329, + 506, + 369 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 372, + 347, + 388 + ], + "lines": [ + { + "bbox": [ + 263, + 372, + 347, + 388 + ], + "spans": [ + { + "bbox": [ + 263, + 372, + 347, + 388 + ], + "score": 0.86, + "content": "\\widetilde { S } ( t + 1 ) = f ( \\widetilde { S } ( t ) )", + "type": "interline_equation", + "image_path": "01589a065d21d81fafaf4c122c7537746528ccc3172c3a93611a75f1ce056d6a.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 263, + 372, + 347, + 388 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 259, + 390, + 351, + 406 + ], + "lines": [ + { + "bbox": [ + 259, + 390, + 351, + 406 + ], + "spans": [ + { + "bbox": [ + 259, + 390, + 351, + 406 + ], + "score": 0.85, + "content": "\\widetilde Y _ { u } ( t + 1 ) = g ( \\widetilde Y _ { u } ( t ) ) .", + "type": "interline_equation", + "image_path": "ce3e5f30df241bc7588b4371d14629e7c41e23165272c2b7dae8f3ee5418d8d8.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 259, + 390, + 351, + 406 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 407, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 315, + 420 + ], + "score": 1.0, + "content": "The functional relationship is given as follows: for", + "type": "text" + }, + { + "bbox": [ + 315, + 408, + 364, + 419 + ], + "score": 0.91, + "content": "1 \\leq i \\leq N", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "indexing tuples on the description", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 384, + 431 + ], + "score": 1.0, + "content": "tape, while processing that tuple, the UTM is in a state distribution", + "type": "text" + }, + { + "bbox": [ + 385, + 419, + 477, + 431 + ], + "score": 0.92, + "content": "\\lambda _ { i } \\cdot \\bar { q } + ( 1 - \\lambda _ { i } ) \\cdot \\neg \\bar { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 419, + 505, + 431 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 107, + 430, + 124, + 442 + ], + "score": 0.44, + "content": "\\bar { q } \\in", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 430, + 229, + 443 + ], + "score": 1.0, + "content": "{copySymbol, copyState,", + "type": "text" + }, + { + "bbox": [ + 230, + 430, + 268, + 442 + ], + "score": 0.3, + "content": "\\mathrm { c o p y D i r } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 430, + 506, + 443 + ], + "score": 1.0, + "content": ". 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We define the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 485, + 392, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 318, + 500 + ], + "score": 1.0, + "content": "conditionally independent joint distribution between", + "type": "text" + }, + { + "bbox": [ + 318, + 485, + 379, + 499 + ], + "score": 0.92, + "content": "\\{ \\widetilde { Y } _ { 0 , t - 1 } , \\widetilde { S } _ { t - 1 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 485, + 392, + 500 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 472, + 505, + 500 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 502, + 429, + 549 + ], + "lines": [ + { + "bbox": [ + 182, + 502, + 429, + 549 + ], + "spans": [ + { + "bbox": [ + 182, + 502, + 429, + 549 + ], + "score": 0.93, + "content": "\\begin{array} { l } { { \\lambda _ { i } = \\displaystyle \\sum _ { \\sigma \\in \\Sigma } \\delta _ { \\theta ( i ) _ { 1 } = \\sigma } P ( \\widetilde { Y } _ { 0 , t - 1 } = \\sigma ) \\cdot \\sum _ { q \\in Q } \\delta _ { \\theta ( i ) _ { 2 } = q } P ( \\widetilde { S } _ { t - 1 } = q ) \\hfill } } \\\\ { { \\quad = P ( \\widetilde { Y } _ { 0 , t - 1 } = \\theta ( i ) _ { 1 } ) \\cdot P ( \\widetilde { S } _ { t - 1 } = \\theta ( i ) _ { 2 } ) . } } \\end{array}", + "type": "interline_equation", + "image_path": "00f2cdcbaf17862a9524c183d3cd6929539574a482f723be628aae9bcc2006d0.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 182, + 502, + 429, + 517.6666666666666 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 182, + 517.6666666666666, + 429, + 533.3333333333333 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 182, + 533.3333333333333, + 429, + 548.9999999999999 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 556, + 506, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 504, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 305, + 570 + ], + "score": 1.0, + "content": "We then calculate a recursive set of equations for", + "type": "text" + }, + { + "bbox": [ + 305, + 557, + 352, + 568 + ], + "score": 0.93, + "content": "0 \\leq j \\leq N", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 554, + 450, + 570 + ], + "score": 1.0, + "content": "describing distributions", + "type": "text" + }, + { + "bbox": [ + 450, + 556, + 504, + 569 + ], + "score": 0.93, + "content": "P ( \\hat { s } _ { j } ) , P ( \\hat { q } _ { j } )", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 568, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 123, + 584 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 568, + 149, + 582 + ], + "score": 0.93, + "content": "P ( \\hat { d } _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 568, + 430, + 584 + ], + "score": 1.0, + "content": "on the staging tape after processing all tuples up to and including tuple", + "type": "text" + }, + { + "bbox": [ + 430, + 571, + 436, + 582 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 568, + 506, + 584 + ], + "score": 1.0, + "content": ". These are given", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 582, + 274, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 118, + 596 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 582, + 255, + 596 + ], + "score": 0.92, + "content": "P ( \\hat { s } _ { 0 } ) = P ( \\hat { q } _ { 0 } ) = P ( \\hat { d } _ { 0 } ) = 1 \\cdot X", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 583, + 274, + 596 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 554, + 506, + 596 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 108, + 600, + 504, + 688 + ], + "lines": [ + { + "bbox": [ + 124, + 600, + 504, + 688 + ], + "spans": [ + { + "bbox": [ + 124, + 600, + 504, + 688 + ], + "score": 0.66, + "content": "\\begin{array} { r l } & { \\displaystyle { P ( \\hat { s } _ { i } ) = \\sum _ { \\sigma \\in \\Sigma } \\{ \\lambda _ { i } \\cdot P ( s _ { i } ^ { \\prime } = \\sigma ) + ( 1 - \\lambda _ { i } ) \\cdot P ( \\hat { s } _ { i - 1 } = \\sigma ) \\} \\cdot \\sigma + ( 1 - \\lambda _ { i } ) \\cdot P ( \\hat { s } _ { i - 1 } = X ) \\cdot X } } \\\\ & { \\displaystyle { P ( \\hat { q } _ { i } ) = \\sum _ { \\ q \\in Q } \\{ \\lambda _ { i } \\cdot P ( q _ { i } ^ { \\prime } = q ) + ( 1 - \\lambda _ { i } ) \\cdot P ( \\hat { q } _ { i - 1 } = q ) \\} \\cdot q + ( 1 - \\lambda _ { i } ) \\cdot P ( \\hat { q } _ { i - 1 } = X ) \\cdot X } } \\\\ & { \\displaystyle { \\hat { l } _ { i } ) = \\sum _ { \\alpha \\in \\{ L , R , S \\} } \\{ \\lambda _ { i } \\cdot P ( d _ { i } = a ) + ( 1 - \\lambda _ { i } ) \\cdot P ( \\hat { d } _ { i - 1 } = a ) \\} \\cdot a + ( 1 - \\lambda _ { i } ) \\cdot P ( \\hat { d } _ { i - 1 } = X ) \\cdot X . } } \\end{array}", + "type": "interline_equation", + "image_path": "1c4ab3562612c52181d5fb39c49de516678335b4023913e0cae6179ec06de62f.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 108, + 600, + 504, + 629.3333333333334 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 108, + 629.3333333333334, + 504, + 658.6666666666667 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 108, + 658.6666666666667, + 504, + 688.0000000000001 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 688, + 470, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 471, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 122, + 704 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 689, + 327, + 703 + ], + "score": 0.8, + "content": "A _ { \\sigma } = P ( \\widehat { s } _ { N } = X ) \\cdot P ( \\widetilde { Y } _ { 0 , t - 1 } = \\sigma ) + P ( \\widehat { s } _ { N } = \\sigma ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 688, + 471, + 704 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 484, + 334, + 505, + 345 + ], + "score": 0.84, + "content": "\\lambda =", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 343, + 231, + 357 + ], + "spans": [ + { + "bbox": [ + 107, + 344, + 188, + 357 + ], + "score": 0.92, + "content": "( \\lambda _ { 1 } , \\dots , \\lambda _ { N } ) \\in \\mathbb { R } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 343, + 231, + 357 + ], + "score": 1.0, + "content": ". 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Input: range of β's, set of training sets T each of size n, approximate samples {w1,..,WR} from pβ(w|Dn) for each training set Dn and each β
for training set Dn ∈ T do
for β in range of β's do
ples from pβ(w|Dn)
end for
Perform generalised least squares to fit X in Equation (7),call result λ(Dn)
end for
Output: ∑Dn∈T λ(Dn)
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HyperparameterdetectAparityCheck
Dataset size (n)200100
Minimum sequence length (a)41
Maximum sequence length (b)7/8/9/105/6/7
Number of samples (R)20.0002.000
Number of burn-in steps1,000500
Number of datasets (|T|)43
Target accept probability0.80.8
Concentration (α)1.01.0
Chain temperature (T)log(500)/log(1000)log(300)
Number of timesteps (t)1042
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