diff --git "a/parse/train/HkG3e205K7/HkG3e205K7_middle.json" "b/parse/train/HkG3e205K7/HkG3e205K7_middle.json" new file mode 100644--- /dev/null +++ "b/parse/train/HkG3e205K7/HkG3e205K7_middle.json" @@ -0,0 +1,37886 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 505, + 98 + ], + "score": 1.0, + "content": "DOUBLY REPARAMETERIZED GRADIENT ESTIMATORS", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 97, + 354, + 119 + ], + "spans": [ + { + "bbox": [ + 104, + 97, + 354, + 119 + ], + "score": 1.0, + "content": "FOR MONTE CARLO OBJECTIVES", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 113, + 135, + 198, + 168 + ], + "lines": [ + { + "bbox": [ + 112, + 135, + 179, + 147 + ], + "spans": [ + { + "bbox": [ + 112, + 135, + 179, + 147 + ], + "score": 1.0, + "content": "George Tucker", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 111, + 146, + 169, + 158 + ], + "spans": [ + { + "bbox": [ + 111, + 146, + 169, + 158 + ], + "score": 1.0, + "content": "Google Brain", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 112, + 157, + 199, + 170 + ], + "spans": [ + { + "bbox": [ + 112, + 157, + 199, + 170 + ], + "score": 1.0, + "content": "gjt@google.com", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 250, + 135, + 337, + 168 + ], + "lines": [ + { + "bbox": [ + 250, + 135, + 328, + 147 + ], + "spans": [ + { + "bbox": [ + 250, + 135, + 328, + 147 + ], + "score": 1.0, + "content": "Dieterich Lawson", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 249, + 145, + 337, + 158 + ], + "spans": [ + { + "bbox": [ + 249, + 145, + 337, + 158 + ], + "score": 1.0, + "content": "New York University", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 250, + 157, + 336, + 169 + ], + "spans": [ + { + "bbox": [ + 250, + 157, + 336, + 169 + ], + "score": 1.0, + "content": "jdl404@nyu.edu", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 388, + 136, + 498, + 168 + ], + "lines": [ + { + "bbox": [ + 388, + 134, + 445, + 149 + ], + "spans": [ + { + "bbox": [ + 388, + 134, + 445, + 149 + ], + "score": 1.0, + "content": "Shixiang Gu", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 388, + 145, + 446, + 158 + ], + "spans": [ + { + "bbox": [ + 388, + 145, + 446, + 158 + ], + "score": 1.0, + "content": "Google Brain", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 388, + 156, + 499, + 170 + ], + "spans": [ + { + "bbox": [ + 388, + 156, + 499, + 170 + ], + "score": 1.0, + "content": "shanegu@google.com", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 113, + 185, + 246, + 218 + ], + "lines": [ + { + "bbox": [ + 113, + 186, + 193, + 196 + ], + "spans": [ + { + "bbox": [ + 113, + 186, + 193, + 196 + ], + "score": 1.0, + "content": "Chris J. 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These approaches maximize a variational lower bound on the in-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 304, + 469, + 316 + ], + "spans": [ + { + "bbox": [ + 142, + 304, + 469, + 316 + ], + "score": 1.0, + "content": "tractable log likelihood of the observed data. Burda et al. (2015) introduced a", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 315, + 470, + 327 + ], + "spans": [ + { + "bbox": [ + 141, + 315, + 470, + 327 + ], + "score": 1.0, + "content": "multi-sample variational bound, IWAE, that is at least as tight as the standard", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 326, + 469, + 338 + ], + "spans": [ + { + "bbox": [ + 141, + 326, + 469, + 338 + ], + "score": 1.0, + "content": "variational lower bound and becomes increasingly tight as the number of samples", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 337, + 470, + 350 + ], + "spans": [ + { + "bbox": [ + 141, + 337, + 470, + 350 + ], + "score": 1.0, + "content": "increases. Counterintuitively, the typical inference network gradient estimator for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 348, + 470, + 361 + ], + "spans": [ + { + "bbox": [ + 141, + 348, + 470, + 361 + ], + "score": 1.0, + "content": "the IWAE bound performs poorly as the number of samples increases (Rainforth", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 359, + 470, + 371 + ], + "spans": [ + { + "bbox": [ + 141, + 359, + 470, + 371 + ], + "score": 1.0, + "content": "et al., 2018; Le et al., 2018). Roeder et al. (2017) propose an improved gradient", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 142, + 370, + 469, + 381 + ], + "spans": [ + { + "bbox": [ + 142, + 370, + 469, + 381 + ], + "score": 1.0, + "content": "estimator, however, are unable to show it is unbiased. We show that it is in fact bi-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 381, + 469, + 393 + ], + "spans": [ + { + "bbox": [ + 141, + 381, + 469, + 393 + ], + "score": 1.0, + "content": "ased and that the bias can be estimated efficiently with a second application of the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 392, + 470, + 405 + ], + "spans": [ + { + "bbox": [ + 141, + 392, + 470, + 405 + ], + "score": 1.0, + "content": "reparameterization trick. The doubly reparameterized gradient (DReG) estima-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 403, + 469, + 416 + ], + "spans": [ + { + "bbox": [ + 141, + 403, + 469, + 416 + ], + "score": 1.0, + "content": "tor does not suffer as the number of samples increases, resolving the previously", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 414, + 470, + 426 + ], + "spans": [ + { + "bbox": [ + 141, + 414, + 470, + 426 + ], + "score": 1.0, + "content": "raised issues. The same idea can be used to improve many recently introduced", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 425, + 470, + 437 + ], + "spans": [ + { + "bbox": [ + 142, + 425, + 470, + 437 + ], + "score": 1.0, + "content": "training techniques for latent variable models. In particular, we show that this", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 435, + 469, + 448 + ], + "spans": [ + { + "bbox": [ + 141, + 435, + 469, + 448 + ], + "score": 1.0, + "content": "estimator reduces the variance of the IWAE gradient, the reweighted wake-sleep", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 142, + 447, + 470, + 458 + ], + "spans": [ + { + "bbox": [ + 142, + 447, + 470, + 458 + ], + "score": 1.0, + "content": "update (RWS) (Bornschein & Bengio, 2014), and the jackknife variational infer-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 457, + 469, + 470 + ], + "spans": [ + { + "bbox": [ + 141, + 457, + 469, + 470 + ], + "score": 1.0, + "content": "ence (JVI) gradient (Nowozin, 2018). Finally, we show that this computationally", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 142, + 469, + 469, + 481 + ], + "spans": [ + { + "bbox": [ + 142, + 469, + 469, + 481 + ], + "score": 1.0, + "content": "efficient, unbiased drop-in gradient estimator translates to improved performance", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 142, + 479, + 341, + 491 + ], + "spans": [ + { + "bbox": [ + 142, + 479, + 341, + 491 + ], + "score": 1.0, + "content": "for all three objectives on several modeling tasks.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 24.5, + "bbox_fs": [ + 141, + 271, + 470, + 491 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 510, + 206, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 208, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 208, + 525 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "Following the influential work by (Kingma & Welling, 2013; Rezende et al., 2014), deep generative", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "models with latent variables have been widely used to model data such as natural images (Rezende", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "& Mohamed, 2015; Kingma et al., 2016; Chen et al., 2016; Gulrajani et al., 2016), speech and music", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "time-series (Chung et al., 2015; Fraccaro et al., 2016; Krishnan et al., 2015), and video (Babaeizadeh", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "et al., 2017; Ha & Schmidhuber, 2018; Denton & Fergus, 2018). The power of these models lies", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "in combining learned nonlinear function approximators with a principled probabilistic approach,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 600, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 614 + ], + "score": 1.0, + "content": "resulting in expressive models that can capture complex distributions. Unfortunately, the nonlinear-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 609, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 625 + ], + "score": 1.0, + "content": "ities that empower these model also make marginalizing the latent variables intractable, rendering", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "direct maximum likelihood training inapplicable. Instead of directly maximizing the marginal like-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "lihood, a common approach is to maximize a tractable lower bound on the likelihood such as the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "variational evidence lower bound (ELBO) (Jordan et al., 1999; Blei et al., 2017). The tightness of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "the bound is determined by the expressiveness of the variational family. For tractability, a factorized", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 667, + 491, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 491, + 678 + ], + "score": 1.0, + "content": "variational family is commonly used, which can cause the learned model to be overly simplistic.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 534, + 506, + 678 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 683, + 502, + 705 + ], + "lines": [ + { + "bbox": [ + 105, + 682, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 505, + 695 + ], + "score": 1.0, + "content": "Burda et al. (2015) introduced a multi-sample bound, IWAE, that is at least as tight as the ELBO", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 694, + 504, + 707 + ], + "spans": [ + { + "bbox": [ + 106, + 694, + 504, + 707 + ], + "score": 1.0, + "content": "and becomes increasingly tight as the number of samples increases. Counterintuitively, although", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "the bound is tighter, Rainforth et al. (2018) theoretically and empirically showed that the standard", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "inference network gradient estimator for the IWAE bound performs poorly as the number of sam-", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "ples increases due to a diminishing signal-to-noise ratio (SNR). This motivates the search for novel", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 188, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 188, + 127 + ], + "score": 1.0, + "content": "gradient estimators.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 49.5, + "bbox_fs": [ + 105, + 682, + 505, + 707 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "the bound is tighter, Rainforth et al. (2018) theoretically and empirically showed that the standard", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "inference network gradient estimator for the IWAE bound performs poorly as the number of sam-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "ples increases due to a diminishing signal-to-noise ratio (SNR). This motivates the search for novel", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 188, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 188, + 127 + ], + "score": 1.0, + "content": "gradient estimators.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "score": 1.0, + "content": "Roeder et al. (2017) proposed a lower-variance estimator of the gradient of the IWAE bound. They", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "speculated that their estimator was unbiased, however, were unable to prove the claim. We show", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 505, + 167 + ], + "score": 1.0, + "content": "that it is in fact biased, but that it is possible to construct an unbiased estimator with a second", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "application of the reparameterization trick which we call the IWAE doubly reparameterized gradient", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "score": 1.0, + "content": "(DReG) estimator. Our estimator is an unbiased, computationally efficient drop-in replacement, and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "does not suffer as the number of samples increases, resolving the counterintuitive behavior from", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "previous work (Rainforth et al., 2018). Furthermore, our insight is applicable to alternative multi-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "sample training techniques for latent variable models: reweighted wake-sleep (RWS) (Bornschein", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 414, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 414, + 233 + ], + "score": 1.0, + "content": "& Bengio, 2014) and jackknife variational inference (JVI) (Nowozin, 2018).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 236, + 505, + 292 + ], + "lines": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "In this work, we derive DReG estimators for IWAE, RWS, and JVI and demonstrate improved", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 248, + 504, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 504, + 259 + ], + "score": 1.0, + "content": "scaling with the number of samples on a simple example. Then, we evaluate DReG estimators", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "on MNIST generative modeling, Omniglot generative modeling, and MNIST structured prediction", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 268, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 283 + ], + "score": 1.0, + "content": "tasks. In all cases, we demonstrate substantial unbiased variance reduction, which translates to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 314, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 314, + 294 + ], + "score": 1.0, + "content": "improved performance over the original estimators.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 108, + 308, + 200, + 321 + ], + "lines": [ + { + "bbox": [ + 104, + 306, + 202, + 325 + ], + "spans": [ + { + "bbox": [ + 104, + 306, + 202, + 325 + ], + "score": 1.0, + "content": "2 BACKGROUND", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 333, + 505, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 317, + 347 + ], + "score": 1.0, + "content": "Our goal is to learn a latent variable generative model", + "type": "text" + }, + { + "bbox": [ + 318, + 334, + 417, + 346 + ], + "score": 0.93, + "content": "p _ { \\theta } ( x , z ) = p _ { \\theta } ( z ) p _ { \\theta } ( x | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 333, + 444, + 347 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 445, + 336, + 452, + 344 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "are observed", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 343, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 143, + 357 + ], + "score": 1.0, + "content": "data and", + "type": "text" + }, + { + "bbox": [ + 144, + 347, + 150, + 354 + ], + "score": 0.76, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 343, + 468, + 357 + ], + "score": 1.0, + "content": "are continuous latent variables. 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Instead, we maximize a variational lower bound on", + "type": "text" + }, + { + "bbox": [ + 465, + 357, + 504, + 367 + ], + "score": 0.9, + "content": "\\log p _ { \\theta } ( x )", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 366, + 182, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 182, + 378 + ], + "score": 1.0, + "content": "such as the ELBO", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 381, + 424, + 408 + ], + "lines": [ + { + "bbox": [ + 186, + 381, + 424, + 408 + ], + "spans": [ + { + "bbox": [ + 186, + 381, + 424, + 408 + ], + "score": 0.94, + "content": "\\log p _ { \\theta } ( x ) = \\log \\mathbb { E } _ { p _ { \\theta } ( z ) } [ p _ { \\theta } ( x | z ) ] \\geq \\mathbb { E } _ { q ( z | x ) } \\left[ \\log \\frac { p _ { \\theta } ( x , z ) } { q ( z | x ) } \\right] ,", + "type": "interline_equation", + "image_path": "b166f6561ef0208c010f1736de8e24017f1b77d580992f4aee4762363365eaaa.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 381, + 424, + 394.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 186, + 394.5, + 424, + 408.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 414, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 134, + 428 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 414, + 161, + 426 + ], + "score": 0.93, + "content": "q ( z | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 413, + 505, + 428 + ], + "score": 1.0, + "content": "is a variational distribution. 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This does not", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 671, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 669, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 118, + 669, + 231, + 684 + ], + "score": 1.0, + "content": "1Meaning that we can express", + "type": "text" + }, + { + "bbox": [ + 231, + 674, + 240, + 681 + ], + "score": 0.68, + "content": "z _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 669, + 252, + 684 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 252, + 672, + 281, + 682 + ], + "score": 0.91, + "content": "z ( \\epsilon _ { i } , \\phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 669, + 309, + 684 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 310, + 674, + 316, + 680 + ], + "score": 0.73, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 669, + 483, + 684 + ], + "score": 1.0, + "content": "is a deterministic, differentiable function and", + "type": "text" + }, + { + "bbox": [ + 484, + 671, + 504, + 682 + ], + "score": 0.89, + "content": "p ( \\epsilon _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 669, + 506, + 684 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 2 + } + ] + }, + { + "bbox": [ + 105, + 681, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 180, + 693 + ], + "score": 1.0, + "content": "does not depend on", + "type": "text" + }, + { + "bbox": [ + 180, + 682, + 186, + 691 + ], + "score": 0.75, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 681, + 197, + 693 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 197, + 682, + 204, + 692 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 681, + 506, + 693 + ], + "score": 1.0, + "content": ". 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In all cases, we demonstrate substantial unbiased variance reduction, which translates to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 314, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 314, + 294 + ], + "score": 1.0, + "content": "improved performance over the original estimators.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 237, + 506, + 294 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 308, + 200, + 321 + ], + "lines": [ + { + "bbox": [ + 104, + 306, + 202, + 325 + ], + "spans": [ + { + "bbox": [ + 104, + 306, + 202, + 325 + ], + "score": 1.0, + "content": "2 BACKGROUND", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 333, + 505, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 317, + 347 + ], + "score": 1.0, + "content": "Our goal is to learn a latent variable generative model", + "type": "text" + }, + { + "bbox": [ + 318, + 334, + 417, + 346 + ], + "score": 0.93, + "content": "p _ { \\theta } ( x , z ) = p _ { \\theta } ( z ) p _ { \\theta } ( x | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 333, + 444, + 347 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 445, + 336, + 452, + 344 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "are observed", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 343, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 143, + 357 + ], + "score": 1.0, + "content": "data and", + "type": "text" + }, + { + "bbox": [ + 144, + 347, + 150, + 354 + ], + "score": 0.76, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 343, + 468, + 357 + ], + "score": 1.0, + "content": "are continuous latent variables. 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Instead, we maximize a variational lower bound on", + "type": "text" + }, + { + "bbox": [ + 465, + 357, + 504, + 367 + ], + "score": 0.9, + "content": "\\log p _ { \\theta } ( x )", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 366, + 182, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 182, + 378 + ], + "score": 1.0, + "content": "such as the ELBO", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 333, + 506, + 378 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 381, + 424, + 408 + ], + "lines": [ + { + "bbox": [ + 186, + 381, + 424, + 408 + ], + "spans": [ + { + "bbox": [ + 186, + 381, + 424, + 408 + ], + "score": 0.94, + "content": "\\log p _ { \\theta } ( x ) = \\log \\mathbb { E } _ { p _ { \\theta } ( z ) } [ p _ { \\theta } ( x | z ) ] \\geq \\mathbb { E } _ { q ( z | x ) } \\left[ \\log \\frac { p _ { \\theta } ( x , z ) } { q ( z | x ) } \\right] ,", + "type": "interline_equation", + "image_path": "b166f6561ef0208c010f1736de8e24017f1b77d580992f4aee4762363365eaaa.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 381, + 424, + 394.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 186, + 394.5, + 424, + 408.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 414, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 134, + 428 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 414, + 161, + 426 + ], + "score": 0.93, + "content": "q ( z | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 413, + 505, + 428 + ], + "score": 1.0, + "content": "is a variational distribution. 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This motivates the search for lower variance inference network gradient", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 153, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 153, + 188 + ], + "score": 1.0, + "content": "estimators.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 105, + 192, + 504, + 216 + ], + "lines": [ + { + "bbox": [ + 105, + 192, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 280, + 206 + ], + "score": 1.0, + "content": "To derive improved gradient estimators for", + "type": "text" + }, + { + "bbox": [ + 280, + 194, + 287, + 204 + ], + "score": 0.83, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 192, + 506, + 206 + ], + "score": 1.0, + "content": ", it is informative to expand the total derivative2 of the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 203, + 230, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 222, + 217 + ], + "score": 1.0, + "content": "IWAE bound with respect to", + "type": "text" + }, + { + "bbox": [ + 223, + 205, + 230, + 216 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 220, + 430, + 255 + ], + "lines": [ + { + "bbox": [ + 180, + 220, + 430, + 255 + ], + "spans": [ + { + "bbox": [ + 180, + 220, + 430, + 255 + ], + "score": 0.94, + "content": "\\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\frac { w _ { i } } { \\sum _ { j = 1 } ^ { K } w _ { j } } \\left( - \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z _ { i } | x ) + \\frac { \\partial \\log w _ { i } } { \\partial z _ { i } } \\frac { d z _ { i } } { d \\phi } \\right) \\right] .", + "type": "interline_equation", + "image_path": "e5bccda9788f0586a6e725c0bed0f007e18056e561d504e8b573fd87009fd0cb.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 180, + 220, + 430, + 231.66666666666666 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 180, + 231.66666666666666, + 430, + 243.33333333333331 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 180, + 243.33333333333331, + 430, + 254.99999999999997 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 258, + 505, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "Previously, Roeder et al. (2017) found that the first term within the parentheses of Eq. 3 can con-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 347, + 282 + ], + "score": 1.0, + "content": "tribute significant variance to the gradient estimator. When", + "type": "text" + }, + { + "bbox": [ + 347, + 270, + 376, + 280 + ], + "score": 0.89, + "content": "K = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 269, + 506, + 282 + ], + "score": 1.0, + "content": ", this term analytically vanishes", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 204, + 293 + ], + "score": 1.0, + "content": "in expectation, so when", + "type": "text" + }, + { + "bbox": [ + 204, + 281, + 234, + 291 + ], + "score": 0.9, + "content": "K > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "they suggested dropping it. Below, we abbreviate this estimator as", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 291, + 434, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 401, + 303 + ], + "score": 1.0, + "content": "STL. As we show in Section 6.1, the STL estimator introduces bias when", + "type": "text" + }, + { + "bbox": [ + 401, + 291, + 430, + 302 + ], + "score": 0.9, + "content": "K > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 291, + 434, + 303 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 107, + 318, + 455, + 332 + ], + "lines": [ + { + "bbox": [ + 104, + 317, + 455, + 334 + ], + "spans": [ + { + "bbox": [ + 104, + 317, + 455, + 334 + ], + "score": 1.0, + "content": "3 DOUBLY REPARAMETERIZED GRADIENT ESTIMATORS (DREGS)", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 343, + 506, + 366 + ], + "lines": [ + { + "bbox": [ + 106, + 344, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 505, + 355 + ], + "score": 1.0, + "content": "Our insight is that we can estimate the first term within the parentheses of Eq. 3 efficiently with a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 355, + 412, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 412, + 366 + ], + "score": 1.0, + "content": "second application of the reparameterization trick. To see this, first note that", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 143, + 370, + 466, + 406 + ], + "lines": [ + { + "bbox": [ + 143, + 370, + 466, + 406 + ], + "spans": [ + { + "bbox": [ + 143, + 370, + 466, + 406 + ], + "score": 0.94, + "content": "\\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\frac { w _ { i } } { \\sum _ { j = 1 } ^ { K } w _ { j } } \\frac { \\partial } { \\partial \\phi } \\log q ( \\boldsymbol { z } _ { i } | \\boldsymbol { x } ) \\right] = \\sum _ { i = 1 } ^ { K } \\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\frac { w _ { i } } { \\sum _ { j = 1 } ^ { K } w _ { j } } \\frac { \\partial } { \\partial \\phi } \\log q ( \\boldsymbol { z } _ { i } | \\boldsymbol { x } ) \\right] ,", + "type": "interline_equation", + "image_path": "7cfed85029e5e6bed62e4133ce9fee579e20ba92b85589bec6add3bef30bac28.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 143, + 370, + 466, + 382.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 143, + 382.0, + 466, + 394.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 143, + 394.0, + 466, + 406.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 410, + 504, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 510, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 254, + 424 + ], + "score": 1.0, + "content": "so it suffices to focus on one of the", + "type": "text" + }, + { + "bbox": [ + 254, + 411, + 265, + 421 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 410, + 480, + 424 + ], + "score": 1.0, + "content": "terms. Because the derivative is a partial derivativ", + "type": "text" + }, + { + "bbox": [ + 474, + 408, + 510, + 427 + ], + "score": 1.0, + "content": "e ∂∂ φ , it", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 131, + 437 + ], + "score": 1.0, + "content": "treats", + "type": "text" + }, + { + "bbox": [ + 131, + 424, + 185, + 436 + ], + "score": 0.93, + "content": "z _ { i } = z ( \\epsilon _ { i } , \\phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "as a constant, so we can freely change the random variable that the expectation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 435, + 193, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 146, + 448 + ], + "score": 1.0, + "content": "is over to", + "type": "text" + }, + { + "bbox": [ + 146, + 437, + 165, + 447 + ], + "score": 0.87, + "content": "z _ { 1 : K }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 435, + 193, + 448 + ], + "score": 1.0, + "content": ". Now,", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 450, + 451, + 484 + ], + "lines": [ + { + "bbox": [ + 159, + 450, + 451, + 484 + ], + "spans": [ + { + "bbox": [ + 159, + 450, + 451, + 484 + ], + "score": 0.95, + "content": "\\mathbb { E } _ { z _ { 1 : K } } \\left[ \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z _ { i } | x ) \\right] = \\mathbb { E } _ { z _ { - i } } \\mathbb { E } _ { z _ { i } } \\left[ \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z _ { i } | x ) \\right] ,", + "type": "interline_equation", + "image_path": "419c3abff00228fe517f63955ffcba2dea7c3582c8ea266544c277893217e0f7.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 159, + 450, + 451, + 461.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 159, + 461.3333333333333, + 451, + 472.66666666666663 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 159, + 472.66666666666663, + 451, + 483.99999999999994 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 133, + 500 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 489, + 213, + 500 + ], + "score": 0.89, + "content": "z _ { - i } = z _ { 1 : i - 1 , i + 1 : K }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 487, + 265, + 500 + ], + "score": 1.0, + "content": "is the set of", + "type": "text" + }, + { + "bbox": [ + 265, + 489, + 285, + 499 + ], + "score": 0.89, + "content": "z _ { 1 : K }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 487, + 320, + 500 + ], + "score": 1.0, + "content": "without", + "type": "text" + }, + { + "bbox": [ + 320, + 489, + 329, + 498 + ], + "score": 0.82, + "content": "z _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 487, + 506, + 500 + ], + "score": 1.0, + "content": ". 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Now, we can", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "use the following well-known equivalence between the REINFORCE gradient and the reparameter-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 524, + 336, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 336, + 536 + ], + "score": 1.0, + "content": "ization trick gradient (See Appendix 8.1 for a derivation)", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5 + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 539, + 421, + 567 + ], + "lines": [ + { + "bbox": [ + 189, + 539, + 421, + 567 + ], + "spans": [ + { + "bbox": [ + 189, + 539, + 421, + 567 + ], + "score": 0.93, + "content": "\\mathbb { E } _ { q _ { \\phi } ( z | x ) } \\left[ f ( z ) \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z | x ) \\right] = \\mathbb { E } _ { \\epsilon } \\left[ \\frac { \\partial f ( z ) } { \\partial z } \\frac { \\partial z ( \\epsilon , \\phi ) } { \\partial \\phi } \\right] .", + "type": "interline_equation", + "image_path": "ec4c2e7bb6d92791b9f6d56d4a94db62f9ca44c0a6ef984a67c76c4d95a50d79.jpg" + } + ] + } + ], + "index": 34.5, + "virtual_lines": [ + { + "bbox": [ + 189, + 539, + 421, + 553.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 189, + 553.0, + 421, + 567.0 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 570, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 570, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 200, + 584 + ], + "score": 1.0, + "content": "This holds even when", + "type": "text" + }, + { + "bbox": [ + 200, + 571, + 208, + 583 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 570, + 260, + 584 + ], + "score": 1.0, + "content": "depends on", + "type": "text" + }, + { + "bbox": [ + 260, + 572, + 267, + 583 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 570, + 505, + 584 + ], + "score": 1.0, + "content": ". 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Applying the identity from Eq. 5 to the right hand side of Eq. 4 gives", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 609, + 487, + 678 + ], + "lines": [ + { + "bbox": [ + 111, + 609, + 487, + 678 + ], + "spans": [ + { + "bbox": [ + 111, + 609, + 487, + 678 + ], + "score": 0.93, + "content": "\\begin{array} { r l r } & { } & { \\mathbb { E } _ { z _ { i } } \\left[ \\displaystyle \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z _ { i } | x ) \\right] = \\mathbb { E } _ { \\epsilon _ { i } } \\left[ \\displaystyle \\frac { \\partial } { \\partial z _ { i } } \\left( \\displaystyle \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\right) \\displaystyle \\frac { \\partial z _ { i } } { \\partial \\phi } \\right] } \\\\ & { } & { = \\mathbb { E } _ { \\epsilon _ { i } } \\left[ \\left( \\displaystyle \\frac { 1 } { \\sum _ { j } w _ { j } } - \\displaystyle \\frac { w _ { i } } { ( \\sum _ { j } w _ { j } ) ^ { 2 } } \\right) \\displaystyle \\frac { \\partial w _ { i } } { \\partial z _ { i } } \\displaystyle \\frac { \\partial z _ { i } } { \\partial \\phi } \\right] = \\mathbb { E } _ { \\epsilon _ { i } } \\left[ \\left( \\displaystyle \\frac { w _ { i } } { \\sum _ { j } w _ { j } } - \\displaystyle \\frac { w _ { i } ^ { 2 } } { ( \\sum _ { j } w _ { j } ) ^ { 2 } } \\right) \\displaystyle \\frac { \\partial \\log w _ { i } } { \\partial z _ { i } } \\displaystyle \\frac { \\partial z _ { i } } { \\partial \\phi } \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "d76fd48f87e3223c81da29946dcf5a875dccfa646cf5369fb933f40960516950.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 111, + 609, + 487, + 632.0 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 111, + 632.0, + 487, + 655.0 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 111, + 655.0, + 487, + 678.0 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 679, + 506, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 469, + 693 + ], + "score": 1.0, + "content": "This last expression can be efficiently estimated with a single Monte Carlo sample. 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The total derivative accounts for both sources of", + "type": "text" + } + ] + }, + { + "bbox": [ + 104, + 718, + 352, + 735 + ], + "spans": [ + { + "bbox": [ + 104, + 718, + 258, + 735 + ], + "score": 1.0, + "content": "dependence and the partial derivative ∂∂φ", + "type": "text" + }, + { + "bbox": [ + 254, + 720, + 291, + 733 + ], + "score": 1.0, + "content": "considers", + "type": "text" + }, + { + "bbox": [ + 292, + 724, + 300, + 731 + ], + "score": 0.83, + "content": "z _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 720, + 352, + 733 + ], + "score": 1.0, + "content": "as a constant.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 106, + 83, + 505, + 106 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 188 + ], + "lines": [ + { + "bbox": [ + 105, + 110, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 267, + 124 + ], + "score": 1.0, + "content": "Because the IWAE bound converges to", + "type": "text" + }, + { + "bbox": [ + 267, + 110, + 306, + 122 + ], + "score": 0.92, + "content": "\\log p _ { \\theta } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 110, + 322, + 124 + ], + "score": 1.0, + "content": "(as", + "type": "text" + }, + { + "bbox": [ + 323, + 111, + 361, + 121 + ], + "score": 0.87, + "content": "K \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 110, + 420, + 124 + ], + "score": 1.0, + "content": ") regardless of", + "type": "text" + }, + { + "bbox": [ + 421, + 112, + 431, + 122 + ], + "score": 0.82, + "content": "q _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 110, + 435, + 124 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 436, + 111, + 444, + 122 + ], + "score": 0.81, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 110, + 505, + 124 + ], + "score": 1.0, + "content": "’s affect on the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 120, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 204, + 135 + ], + "score": 1.0, + "content": "bound must diminish as", + "type": "text" + }, + { + "bbox": [ + 204, + 122, + 215, + 131 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 120, + 506, + 135 + ], + "score": 1.0, + "content": "increases. 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(2017) found that the first term within the parentheses of Eq. 3 can con-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 347, + 282 + ], + "score": 1.0, + "content": "tribute significant variance to the gradient estimator. 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To see this, first note that", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 106, + 344, + 505, + 366 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 143, + 370, + 466, + 406 + ], + "lines": [ + { + "bbox": [ + 143, + 370, + 466, + 406 + ], + "spans": [ + { + "bbox": [ + 143, + 370, + 466, + 406 + ], + "score": 0.94, + "content": "\\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\frac { w _ { i } } { \\sum _ { j = 1 } ^ { K } w _ { j } } \\frac { \\partial } { \\partial \\phi } \\log q ( \\boldsymbol { z } _ { i } | \\boldsymbol { x } ) \\right] = \\sum _ { i = 1 } ^ { K } \\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\frac { w _ { i } } { \\sum _ { j = 1 } ^ { K } w _ { j } } \\frac { \\partial } { \\partial \\phi } \\log q ( \\boldsymbol { z } _ { i } | \\boldsymbol { x } ) \\right] ,", + "type": "interline_equation", + "image_path": "7cfed85029e5e6bed62e4133ce9fee579e20ba92b85589bec6add3bef30bac28.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 143, + 370, + 466, + 382.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 143, + 382.0, + 466, + 394.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 143, + 394.0, + 466, + 406.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 410, + 504, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 510, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 254, + 424 + ], + "score": 1.0, + "content": "so it suffices to focus on one of the", + "type": "text" + }, + { + "bbox": [ + 254, + 411, + 265, + 421 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 410, + 480, + 424 + ], + "score": 1.0, + "content": "terms. Because the derivative is a partial derivativ", + "type": "text" + }, + { + "bbox": [ + 474, + 408, + 510, + 427 + ], + "score": 1.0, + "content": "e ∂∂ φ , it", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 131, + 437 + ], + "score": 1.0, + "content": "treats", + "type": "text" + }, + { + "bbox": [ + 131, + 424, + 185, + 436 + ], + "score": 0.93, + "content": "z _ { i } = z ( \\epsilon _ { i } , \\phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "as a constant, so we can freely change the random variable that the expectation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 435, + 193, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 146, + 448 + ], + "score": 1.0, + "content": "is over to", + "type": "text" + }, + { + "bbox": [ + 146, + 437, + 165, + 447 + ], + "score": 0.87, + "content": "z _ { 1 : K }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 435, + 193, + 448 + ], + "score": 1.0, + "content": ". Now,", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 408, + 510, + 448 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 450, + 451, + 484 + ], + "lines": [ + { + "bbox": [ + 159, + 450, + 451, + 484 + ], + "spans": [ + { + "bbox": [ + 159, + 450, + 451, + 484 + ], + "score": 0.95, + "content": "\\mathbb { E } _ { z _ { 1 : K } } \\left[ \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z _ { i } | x ) \\right] = \\mathbb { E } _ { z _ { - i } } \\mathbb { E } _ { z _ { i } } \\left[ \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z _ { i } | x ) \\right] ,", + "type": "interline_equation", + "image_path": "419c3abff00228fe517f63955ffcba2dea7c3582c8ea266544c277893217e0f7.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 159, + 450, + 451, + 461.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 159, + 461.3333333333333, + 451, + 472.66666666666663 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 159, + 472.66666666666663, + 451, + 483.99999999999994 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 133, + 500 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 489, + 213, + 500 + ], + "score": 0.89, + "content": "z _ { - i } = z _ { 1 : i - 1 , i + 1 : K }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 487, + 265, + 500 + ], + "score": 1.0, + "content": "is the set of", + "type": "text" + }, + { + "bbox": [ + 265, + 489, + 285, + 499 + ], + "score": 0.89, + "content": "z _ { 1 : K }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 487, + 320, + 500 + ], + "score": 1.0, + "content": "without", + "type": "text" + }, + { + "bbox": [ + 320, + 489, + 329, + 498 + ], + "score": 0.82, + "content": "z _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 487, + 506, + 500 + ], + "score": 1.0, + "content": ". The inner expectation resembles a REIN-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 498, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 352, + 511 + ], + "score": 1.0, + "content": "FORCE gradient term (Williams, 1992), where we interpret", + "type": "text" + }, + { + "bbox": [ + 352, + 498, + 378, + 514 + ], + "score": 0.93, + "content": "\\frac { w _ { i } } { \\sum _ { j } w _ { j } }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "as the “reward”. Now, we can", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "use the following well-known equivalence between the REINFORCE gradient and the reparameter-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 524, + 336, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 336, + 536 + ], + "score": 1.0, + "content": "ization trick gradient (See Appendix 8.1 for a derivation)", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5, + "bbox_fs": [ + 106, + 487, + 506, + 536 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 539, + 421, + 567 + ], + "lines": [ + { + "bbox": [ + 189, + 539, + 421, + 567 + ], + "spans": [ + { + "bbox": [ + 189, + 539, + 421, + 567 + ], + "score": 0.93, + "content": "\\mathbb { E } _ { q _ { \\phi } ( z | x ) } \\left[ f ( z ) \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z | x ) \\right] = \\mathbb { E } _ { \\epsilon } \\left[ \\frac { \\partial f ( z ) } { \\partial z } \\frac { \\partial z ( \\epsilon , \\phi ) } { \\partial \\phi } \\right] .", + "type": "interline_equation", + "image_path": "ec4c2e7bb6d92791b9f6d56d4a94db62f9ca44c0a6ef984a67c76c4d95a50d79.jpg" + } + ] + } + ], + "index": 34.5, + "virtual_lines": [ + { + "bbox": [ + 189, + 539, + 421, + 553.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 189, + 553.0, + 421, + 567.0 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 570, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 570, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 200, + 584 + ], + "score": 1.0, + "content": "This holds even when", + "type": "text" + }, + { + "bbox": [ + 200, + 571, + 208, + 583 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 570, + 260, + 584 + ], + "score": 1.0, + "content": "depends on", + "type": "text" + }, + { + "bbox": [ + 260, + 572, + 267, + 583 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 570, + 505, + 584 + ], + "score": 1.0, + "content": ". Typically, the reparameterization gradient estimator has", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "lower variance than the REINFORCE gradient estimator because it directly takes advantage of the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 592, + 448, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 159, + 606 + ], + "score": 1.0, + "content": "derivative of", + "type": "text" + }, + { + "bbox": [ + 159, + 594, + 166, + 605 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 592, + 448, + 606 + ], + "score": 1.0, + "content": ". Applying the identity from Eq. 5 to the right hand side of Eq. 4 gives", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 570, + 505, + 606 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 609, + 487, + 678 + ], + "lines": [ + { + "bbox": [ + 111, + 609, + 487, + 678 + ], + "spans": [ + { + "bbox": [ + 111, + 609, + 487, + 678 + ], + "score": 0.93, + "content": "\\begin{array} { r l r } & { } & { \\mathbb { E } _ { z _ { i } } \\left[ \\displaystyle \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z _ { i } | x ) \\right] = \\mathbb { E } _ { \\epsilon _ { i } } \\left[ \\displaystyle \\frac { \\partial } { \\partial z _ { i } } \\left( \\displaystyle \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\right) \\displaystyle \\frac { \\partial z _ { i } } { \\partial \\phi } \\right] } \\\\ & { } & { = \\mathbb { E } _ { \\epsilon _ { i } } \\left[ \\left( \\displaystyle \\frac { 1 } { \\sum _ { j } w _ { j } } - \\displaystyle \\frac { w _ { i } } { ( \\sum _ { j } w _ { j } ) ^ { 2 } } \\right) \\displaystyle \\frac { \\partial w _ { i } } { \\partial z _ { i } } \\displaystyle \\frac { \\partial z _ { i } } { \\partial \\phi } \\right] = \\mathbb { E } _ { \\epsilon _ { i } } \\left[ \\left( \\displaystyle \\frac { w _ { i } } { \\sum _ { j } w _ { j } } - \\displaystyle \\frac { w _ { i } ^ { 2 } } { ( \\sum _ { j } w _ { j } ) ^ { 2 } } \\right) \\displaystyle \\frac { \\partial \\log w _ { i } } { \\partial z _ { i } } \\displaystyle \\frac { \\partial z _ { i } } { \\partial \\phi } \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "d76fd48f87e3223c81da29946dcf5a875dccfa646cf5369fb933f40960516950.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 111, + 609, + 487, + 632.0 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 111, + 632.0, + 487, + 655.0 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 111, + 655.0, + 487, + 678.0 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 679, + 506, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 469, + 693 + ], + "score": 1.0, + "content": "This last expression can be efficiently estimated with a single Monte Carlo sample. When", + "type": "text" + }, + { + "bbox": [ + 470, + 681, + 479, + 691 + ], + "score": 0.85, + "content": "z _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 678, + 505, + 693 + ], + "score": 1.0, + "content": "is not", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 689, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 705 + ], + "score": 1.0, + "content": "reparameterizable (e.g., the models in (Mnih & Rezende, 2016)), we can use a control variate (e.g.,", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 678, + 506, + 705 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 506, + 118 + ], + "lines": [ + { + "bbox": [ + 108, + 80, + 507, + 97 + ], + "spans": [ + { + "bbox": [ + 108, + 81, + 182, + 97 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { 1 } { K } \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z _ { i } | \\boldsymbol { x } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 80, + 276, + 96 + ], + "score": 1.0, + "content": ". In both cases, when", + "type": "text" + }, + { + "bbox": [ + 276, + 83, + 309, + 93 + ], + "score": 0.91, + "content": "K = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 80, + 507, + 96 + ], + "score": 1.0, + "content": ", this term vanishes exactly and we recover the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 95, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 401, + 108 + ], + "score": 1.0, + "content": "estimator proposed in (Roeder et al., 2017) for the ELBO. However, when", + "type": "text" + }, + { + "bbox": [ + 401, + 95, + 430, + 106 + ], + "score": 0.89, + "content": "K > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 95, + 506, + 108 + ], + "score": 1.0, + "content": ", there is no reason", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 106, + 282, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 282, + 118 + ], + "score": 1.0, + "content": "to believe this term will analytically vanish.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 122, + 443, + 135 + ], + "lines": [ + { + "bbox": [ + 105, + 122, + 444, + 136 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 444, + 136 + ], + "score": 1.0, + "content": "Substituting Eq. 6 into Eq. 3, we obtain a simplification due to cancellation of terms", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 139, + 454, + 180 + ], + "lines": [ + { + "bbox": [ + 156, + 139, + 454, + 180 + ], + "spans": [ + { + "bbox": [ + 156, + 139, + 454, + 180 + ], + "score": 0.95, + "content": "\\nabla _ { \\phi } \\mathbb { E } _ { z _ { 1 : K } } \\left[ \\log \\left( \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } w _ { i } \\right) \\right] = \\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\left( \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\right) ^ { 2 } \\frac { \\partial \\log w _ { i } } { \\partial z _ { i } } \\frac { \\partial z _ { i } } { \\partial \\phi } \\right] .", + "type": "interline_equation", + "image_path": "ec1bc79fa341f57839e0d4b87743c681c198c46ce9c970aa533b0c383fcf2f1a.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 156, + 139, + 454, + 152.66666666666666 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 156, + 152.66666666666666, + 454, + 166.33333333333331 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 156, + 166.33333333333331, + 454, + 179.99999999999997 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 183, + 506, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 184, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 505, + 196 + ], + "score": 1.0, + "content": "We call the algorithm that uses the single sample Monte Carlo estimator of this expression for the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "score": 1.0, + "content": "inference network gradient the IWAE doubly reparameterized gradient estimator (IWAE-DReG).", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 205, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 279, + 219 + ], + "score": 1.0, + "content": "This estimator has the property that when", + "type": "text" + }, + { + "bbox": [ + 279, + 206, + 306, + 218 + ], + "score": 0.93, + "content": "q ( z | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 205, + 373, + 219 + ], + "score": 1.0, + "content": "is optimal (i.e.,", + "type": "text" + }, + { + "bbox": [ + 373, + 206, + 444, + 218 + ], + "score": 0.93, + "content": "q ( z | x ) = p ( z | x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 205, + 506, + 219 + ], + "score": 1.0, + "content": ", the estimator", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 506, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 229 + ], + "score": 1.0, + "content": "vanishes exactly and has zero variance, whereas this does not hold for the standard IWAE gradient", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 506, + 240 + ], + "score": 1.0, + "content": "estimator. We provide an asymptotic analysis of the IWAE-DReG estimator in Appendix 8.2. The", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "conclusion of that analysis is that, in contrast to the standard IWAE gradient estimator, the SNR of√", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 375, + 264 + ], + "score": 1.0, + "content": "the IWAE-DReG estimator exhibits the same scaling behaviour of", + "type": "text" + }, + { + "bbox": [ + 375, + 250, + 409, + 263 + ], + "score": 0.93, + "content": "\\mathcal { O } ( \\sqrt { K } )", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "for both the generation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 262, + 329, + 275 + ], + "spans": [ + { + "bbox": [ + 104, + 262, + 312, + 275 + ], + "score": 1.0, + "content": "and inference network gradients (i.e., improving in", + "type": "text" + }, + { + "bbox": [ + 312, + 263, + 322, + 272 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 262, + 329, + 275 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 107, + 290, + 325, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 325, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 325, + 305 + ], + "score": 1.0, + "content": "4 ALTERNATIVE TRAINING ALGORITHMS", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 314, + 504, + 338 + ], + "lines": [ + { + "bbox": [ + 105, + 314, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 328 + ], + "score": 1.0, + "content": "Now, we review alternative training algorithms for deep generative models and derive their doubly", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 326, + 211, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 211, + 338 + ], + "score": 1.0, + "content": "reparameterized versions.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 107, + 350, + 282, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 349, + 283, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 283, + 364 + ], + "score": 1.0, + "content": "4.1 REWEIGHTED WAKE SLEEP (RWS)", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 371, + 504, + 395 + ], + "lines": [ + { + "bbox": [ + 106, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "Bornschein & Bengio (2014) introduced RWS, an alternative multi-sample update for latent variable", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 383, + 483, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 483, + 396 + ], + "score": 1.0, + "content": "models that uses importance sampling. Computing the gradient of the log marginal likelihood", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 399, + 501, + 428 + ], + "lines": [ + { + "bbox": [ + 111, + 399, + 501, + 428 + ], + "spans": [ + { + "bbox": [ + 111, + 399, + 501, + 428 + ], + "score": 0.91, + "content": "\\nabla _ { \\theta } \\log p _ { \\theta } ( x ) = \\frac { \\nabla _ { \\theta } \\int _ { z } p _ { \\theta } ( x , z ) d z } { p _ { \\theta } ( x ) } = \\frac { \\int _ { z } p _ { \\theta } ( x , z ) \\nabla _ { \\theta } \\log p _ { \\theta } ( x , z ) d z } { p _ { \\theta } ( x ) } = \\mathbb { E } _ { p _ { \\theta } ( z | x ) } \\left[ \\nabla _ { \\theta } \\log p _ { \\theta } ( x , z ) \\right] ,", + "type": "interline_equation", + "image_path": "394d28a7dd6ec876ffa2ac39a9191b46dd55b10e0a2964d8aaa4e1f7cd17314d.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 111, + 399, + 501, + 408.6666666666667 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 111, + 408.6666666666667, + 501, + 418.33333333333337 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 111, + 418.33333333333337, + 501, + 428.00000000000006 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 433, + 503, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 200, + 446 + ], + "score": 1.0, + "content": "requires samples from", + "type": "text" + }, + { + "bbox": [ + 200, + 433, + 232, + 446 + ], + "score": 0.92, + "content": "p _ { \\theta } ( z | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 433, + 505, + 446 + ], + "score": 1.0, + "content": ", which is generally intractable. We can approximate the gradient", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 443, + 325, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 325, + 457 + ], + "score": 1.0, + "content": "with a self-normalized importance sampling estimator", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 461, + 439, + 496 + ], + "lines": [ + { + "bbox": [ + 171, + 461, + 439, + 496 + ], + "spans": [ + { + "bbox": [ + 171, + 461, + 439, + 496 + ], + "score": 0.94, + "content": "\\mathbb { E } _ { p _ { \\theta } ( z | x ) } \\left[ \\nabla _ { \\theta } \\log p _ { \\theta } ( x , z ) \\right] \\approx \\mathbb { E } _ { z _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\nabla _ { \\theta } \\log p _ { \\theta } ( x , z _ { i } ) \\right] ,", + "type": "interline_equation", + "image_path": "128e9baf4b4a0c9d8de829e1722a45678f0f66785a520fa9c76dc5f01de0f750.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 171, + 461, + 439, + 472.6666666666667 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 171, + 472.6666666666667, + 439, + 484.33333333333337 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 171, + 484.33333333333337, + 439, + 496.00000000000006 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 134, + 515 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 501, + 219, + 514 + ], + "score": 0.91, + "content": "\\begin{array} { r } { z _ { 1 : K } \\sim \\prod _ { i } q _ { \\phi } ( z _ { i } | x ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 500, + 495, + 515 + ], + "score": 1.0, + "content": ". 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However, a downside is that the updates for", + "type": "text" + }, + { + "bbox": [ + 353, + 712, + 360, + 721 + ], + "score": 0.81, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 710, + 378, + 722 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 379, + 712, + 384, + 721 + ], + "score": 0.81, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "are not gradients of a unified", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 720, + 359, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 359, + 733 + ], + "score": 1.0, + "content": "objective, so could potentially lead to instability or divergence.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "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": "4", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 506, + 118 + ], + "lines": [ + { + "bbox": [ + 108, + 80, + 507, + 97 + ], + "spans": [ + { + "bbox": [ + 108, + 81, + 182, + 97 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { 1 } { K } \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z _ { i } | \\boldsymbol { x } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 80, + 276, + 96 + ], + "score": 1.0, + "content": ". In both cases, when", + "type": "text" + }, + { + "bbox": [ + 276, + 83, + 309, + 93 + ], + "score": 0.91, + "content": "K = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 80, + 507, + 96 + ], + "score": 1.0, + "content": ", this term vanishes exactly and we recover the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 95, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 401, + 108 + ], + "score": 1.0, + "content": "estimator proposed in (Roeder et al., 2017) for the ELBO. However, when", + "type": "text" + }, + { + "bbox": [ + 401, + 95, + 430, + 106 + ], + "score": 0.89, + "content": "K > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 95, + 506, + 108 + ], + "score": 1.0, + "content": ", there is no reason", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 106, + 282, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 282, + 118 + ], + "score": 1.0, + "content": "to believe this term will analytically vanish.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 80, + 507, + 118 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 122, + 443, + 135 + ], + "lines": [ + { + "bbox": [ + 105, + 122, + 444, + 136 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 444, + 136 + ], + "score": 1.0, + "content": "Substituting Eq. 6 into Eq. 3, we obtain a simplification due to cancellation of terms", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 122, + 444, + 136 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 139, + 454, + 180 + ], + "lines": [ + { + "bbox": [ + 156, + 139, + 454, + 180 + ], + "spans": [ + { + "bbox": [ + 156, + 139, + 454, + 180 + ], + "score": 0.95, + "content": "\\nabla _ { \\phi } \\mathbb { E } _ { z _ { 1 : K } } \\left[ \\log \\left( \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } w _ { i } \\right) \\right] = \\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\left( \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\right) ^ { 2 } \\frac { \\partial \\log w _ { i } } { \\partial z _ { i } } \\frac { \\partial z _ { i } } { \\partial \\phi } \\right] .", + "type": "interline_equation", + "image_path": "ec1bc79fa341f57839e0d4b87743c681c198c46ce9c970aa533b0c383fcf2f1a.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 156, + 139, + 454, + 152.66666666666666 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 156, + 152.66666666666666, + 454, + 166.33333333333331 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 156, + 166.33333333333331, + 454, + 179.99999999999997 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 183, + 506, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 184, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 505, + 196 + ], + "score": 1.0, + "content": "We call the algorithm that uses the single sample Monte Carlo estimator of this expression for the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 505, + 208 + ], + "score": 1.0, + "content": "inference network gradient the IWAE doubly reparameterized gradient estimator (IWAE-DReG).", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 205, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 279, + 219 + ], + "score": 1.0, + "content": "This estimator has the property that when", + "type": "text" + }, + { + "bbox": [ + 279, + 206, + 306, + 218 + ], + "score": 0.93, + "content": "q ( z | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 205, + 373, + 219 + ], + "score": 1.0, + "content": "is optimal (i.e.,", + "type": "text" + }, + { + "bbox": [ + 373, + 206, + 444, + 218 + ], + "score": 0.93, + "content": "q ( z | x ) = p ( z | x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 205, + 506, + 219 + ], + "score": 1.0, + "content": ", the estimator", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 506, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 229 + ], + "score": 1.0, + "content": "vanishes exactly and has zero variance, whereas this does not hold for the standard IWAE gradient", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 506, + 240 + ], + "score": 1.0, + "content": "estimator. We provide an asymptotic analysis of the IWAE-DReG estimator in Appendix 8.2. The", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "conclusion of that analysis is that, in contrast to the standard IWAE gradient estimator, the SNR of√", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 375, + 264 + ], + "score": 1.0, + "content": "the IWAE-DReG estimator exhibits the same scaling behaviour of", + "type": "text" + }, + { + "bbox": [ + 375, + 250, + 409, + 263 + ], + "score": 0.93, + "content": "\\mathcal { O } ( \\sqrt { K } )", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "for both the generation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 262, + 329, + 275 + ], + "spans": [ + { + "bbox": [ + 104, + 262, + 312, + 275 + ], + "score": 1.0, + "content": "and inference network gradients (i.e., improving in", + "type": "text" + }, + { + "bbox": [ + 312, + 263, + 322, + 272 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 262, + 329, + 275 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5, + "bbox_fs": [ + 104, + 184, + 506, + 275 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 290, + 325, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 289, + 325, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 325, + 305 + ], + "score": 1.0, + "content": "4 ALTERNATIVE TRAINING ALGORITHMS", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 314, + 504, + 338 + ], + "lines": [ + { + "bbox": [ + 105, + 314, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 328 + ], + "score": 1.0, + "content": "Now, we review alternative training algorithms for deep generative models and derive their doubly", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 326, + 211, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 211, + 338 + ], + "score": 1.0, + "content": "reparameterized versions.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 314, + 505, + 338 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 350, + 282, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 349, + 283, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 283, + 364 + ], + "score": 1.0, + "content": "4.1 REWEIGHTED WAKE SLEEP (RWS)", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 371, + 504, + 395 + ], + "lines": [ + { + "bbox": [ + 106, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "Bornschein & Bengio (2014) introduced RWS, an alternative multi-sample update for latent variable", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 383, + 483, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 483, + 396 + ], + "score": 1.0, + "content": "models that uses importance sampling. Computing the gradient of the log marginal likelihood", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 371, + 506, + 396 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 399, + 501, + 428 + ], + "lines": [ + { + "bbox": [ + 111, + 399, + 501, + 428 + ], + "spans": [ + { + "bbox": [ + 111, + 399, + 501, + 428 + ], + "score": 0.91, + "content": "\\nabla _ { \\theta } \\log p _ { \\theta } ( x ) = \\frac { \\nabla _ { \\theta } \\int _ { z } p _ { \\theta } ( x , z ) d z } { p _ { \\theta } ( x ) } = \\frac { \\int _ { z } p _ { \\theta } ( x , z ) \\nabla _ { \\theta } \\log p _ { \\theta } ( x , z ) d z } { p _ { \\theta } ( x ) } = \\mathbb { E } _ { p _ { \\theta } ( z | x ) } \\left[ \\nabla _ { \\theta } \\log p _ { \\theta } ( x , z ) \\right] ,", + "type": "interline_equation", + "image_path": "394d28a7dd6ec876ffa2ac39a9191b46dd55b10e0a2964d8aaa4e1f7cd17314d.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 111, + 399, + 501, + 408.6666666666667 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 111, + 408.6666666666667, + 501, + 418.33333333333337 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 111, + 418.33333333333337, + 501, + 428.00000000000006 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 433, + 503, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 200, + 446 + ], + "score": 1.0, + "content": "requires samples from", + "type": "text" + }, + { + "bbox": [ + 200, + 433, + 232, + 446 + ], + "score": 0.92, + "content": "p _ { \\theta } ( z | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 433, + 505, + 446 + ], + "score": 1.0, + "content": ", which is generally intractable. We can approximate the gradient", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 443, + 325, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 325, + 457 + ], + "score": 1.0, + "content": "with a self-normalized importance sampling estimator", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 433, + 505, + 457 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 461, + 439, + 496 + ], + "lines": [ + { + "bbox": [ + 171, + 461, + 439, + 496 + ], + "spans": [ + { + "bbox": [ + 171, + 461, + 439, + 496 + ], + "score": 0.94, + "content": "\\mathbb { E } _ { p _ { \\theta } ( z | x ) } \\left[ \\nabla _ { \\theta } \\log p _ { \\theta } ( x , z ) \\right] \\approx \\mathbb { E } _ { z _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\nabla _ { \\theta } \\log p _ { \\theta } ( x , z _ { i } ) \\right] ,", + "type": "interline_equation", + "image_path": "128e9baf4b4a0c9d8de829e1722a45678f0f66785a520fa9c76dc5f01de0f750.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 171, + 461, + 439, + 472.6666666666667 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 171, + 472.6666666666667, + 439, + 484.33333333333337 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 171, + 484.33333333333337, + 439, + 496.00000000000006 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 134, + 515 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 501, + 219, + 514 + ], + "score": 0.91, + "content": "\\begin{array} { r } { z _ { 1 : K } \\sim \\prod _ { i } q _ { \\phi } ( z _ { i } | x ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 500, + 495, + 515 + ], + "score": 1.0, + "content": ". Interestingly, this is precisely the same as the IWAE gradient of", + "type": "text" + }, + { + "bbox": [ + 495, + 502, + 501, + 511 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 500, + 505, + 515 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 203, + 524 + ], + "score": 1.0, + "content": "so the RWS update for", + "type": "text" + }, + { + "bbox": [ + 203, + 513, + 210, + 522 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 512, + 495, + 524 + ], + "score": 1.0, + "content": "can be interpreted as maximizing the IWAE lower bound in terms of", + "type": "text" + }, + { + "bbox": [ + 495, + 513, + 501, + 522 + ], + "score": 0.79, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 512, + 505, + 524 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 284, + 536 + ], + "score": 1.0, + "content": "Instead of optimizing a joint objective for", + "type": "text" + }, + { + "bbox": [ + 284, + 525, + 291, + 534 + ], + "score": 0.81, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 523, + 311, + 536 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 311, + 525, + 317, + 535 + ], + "score": 0.76, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 523, + 505, + 536 + ], + "score": 1.0, + "content": ", RWS optimizes a separate objective for the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "inference network. (Bornschein & Bengio, 2014) propose a “wake” update and a “sleep” update for", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 544, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 558 + ], + "score": 1.0, + "content": "the inference network. Le et al. 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Their positive results motivate further exploration", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 222, + 339, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 339, + 233 + ], + "score": 1.0, + "content": "of convex combinations of IWAE-DReG and RWS-DReG", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 236, + 433, + 270 + ], + "lines": [ + { + "bbox": [ + 177, + 236, + 433, + 270 + ], + "spans": [ + { + "bbox": [ + 177, + 236, + 433, + 270 + ], + "score": 0.94, + "content": "\\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\left( \\alpha \\frac { w _ { i } } { \\sum _ { j } w _ { j } } + ( 1 - 2 \\alpha ) \\frac { w _ { i } ^ { 2 } } { ( \\sum _ { j } w _ { j } ) ^ { 2 } } \\right) \\frac { \\partial \\log w _ { i } } { \\partial z _ { i } } \\frac { \\partial z _ { i } } { \\partial \\phi } \\right] .", + "type": "interline_equation", + "image_path": "c354b914bd827b1d14f8e9a923ec9bbfb221c7580e7c4784f99e15a4744e457b.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 177, + 236, + 433, + 247.33333333333334 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 177, + 247.33333333333334, + 433, + 258.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 177, + 258.6666666666667, + 433, + 270.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 273, + 505, + 306 + ], + "lines": [ + { + "bbox": [ + 106, + 272, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 286 + ], + "score": 1.0, + "content": "We refer to the algorithm that uses the single sample Monte Carlo estimator of this expression as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 146, + 295 + ], + "score": 0.79, + "content": "\\mathrm { D R e G } ( \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 284, + 179, + 296 + ], + "score": 1.0, + "content": ". When", + "type": "text" + }, + { + "bbox": [ + 180, + 285, + 208, + 294 + ], + "score": 0.91, + "content": "\\alpha = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 284, + 354, + 296 + ], + "score": 1.0, + "content": ", this reduces to RWS-DReG, when", + "type": "text" + }, + { + "bbox": [ + 354, + 285, + 383, + 294 + ], + "score": 0.9, + "content": "\\alpha = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 284, + 505, + 296 + ], + "score": 1.0, + "content": ", this reduces to IWAE-DReG", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 295, + 258, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 147, + 307 + ], + "score": 1.0, + "content": "and when", + "type": "text" + }, + { + "bbox": [ + 147, + 295, + 181, + 306 + ], + "score": 0.89, + "content": "\\alpha = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 295, + 258, + 307 + ], + "score": 1.0, + "content": ", this reduces STL.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 107, + 319, + 319, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 320, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 320, + 334 + ], + "score": 1.0, + "content": "4.2 JACKKNIFE VARIATIONAL INFERENCE (JVI)", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 339, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 338, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 354 + ], + "score": 1.0, + "content": "Alternatively, Nowozin (2018) reinterprets the IWAE lower bound as a biased estimator for the log", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "marginal likelihood. He analyzes the bias and introduces a novel family of estimators, Jackknife", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 363, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 374 + ], + "score": 1.0, + "content": "Variational Inference (JVI), which trade off reduction in bias for increased variance. This additional", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "flexibility comes at the cost of no longer being a stochastic lower bound on the log marginal like-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "lihood. The first-order JVI has significantly reduced bias compared to IWAE, which empirically", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 396, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 407 + ], + "score": 1.0, + "content": "results in a better estimate of the log marginal likelihood with fewer samples (Nowozin, 2018). For", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 407, + 315, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 315, + 418 + ], + "score": 1.0, + "content": "simplicity, we focus on the first-order JVI estimator", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 421, + 465, + 461 + ], + "lines": [ + { + "bbox": [ + 145, + 421, + 465, + 461 + ], + "spans": [ + { + "bbox": [ + 145, + 421, + 465, + 461 + ], + "score": 0.93, + "content": "K \\times \\mathbb { E } _ { z _ { 1 } \\cdot K } \\left[ \\log \\left( \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } w _ { i } \\right) \\right] - \\frac { K - 1 } { K } \\sum _ { i = 1 } ^ { K } \\mathbb { E } _ { z _ { - i } } \\left[ \\log \\left( \\frac { 1 } { K - 1 } \\sum _ { j \\neq i } w _ { j } \\right) \\right] .", + "type": "interline_equation", + "image_path": "57da664026def4d2b565f31da4604d58d912ce6bfbde6af1ca1c694ee74b7955.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 145, + 421, + 465, + 434.3333333333333 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 145, + 434.3333333333333, + 465, + 447.66666666666663 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 145, + 447.66666666666663, + 465, + 460.99999999999994 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 464, + 401, + 476 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 402, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 402, + 477 + ], + "score": 1.0, + "content": "It is straightforward to apply our approach to higher order JVI estimators.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 488, + 421, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 422, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 422, + 502 + ], + "score": 1.0, + "content": "DOUBLY REPARAMETERIZED JACKKNIFE VARIATIONAL INFERENCE (JVI)", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 508, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 507, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 285, + 522 + ], + "score": 1.0, + "content": "The JVI estimator is a linear combination of", + "type": "text" + }, + { + "bbox": [ + 286, + 510, + 296, + 519 + ], + "score": 0.84, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 507, + 314, + 522 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 314, + 509, + 342, + 520 + ], + "score": 0.87, + "content": "K - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 507, + 505, + 522 + ], + "score": 1.0, + "content": "sample IWAE estimators, so we can use", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 520, + 380, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 380, + 532 + ], + "score": 1.0, + "content": "the doubly reparameterized gradient estimator (Eq. 7) for each term.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "title", + "bbox": [ + 108, + 547, + 211, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 213, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 213, + 562 + ], + "score": 1.0, + "content": "5 RELATED WORK", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 571, + 505, + 639 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "Mnih & Rezende (2016) introduced a generalized framework of Monte Carlo objectives (MCO). The", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "log of an unbiased marginal likelihood estimator is a lower bound on the log marginal likelihood by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "Jensen’s inequality. In this view, the ELBO can be seen as the MCO corresponding to a single im-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 104, + 604, + 339, + 618 + ], + "score": 1.0, + "content": "portance sample estimator of the marginal likelihood with", + "type": "text" + }, + { + "bbox": [ + 339, + 606, + 349, + 617 + ], + "score": 0.83, + "content": "q _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "as the proposal distribution. Similarly,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 209, + 628 + ], + "score": 1.0, + "content": "IWAE corresponds to the", + "type": "text" + }, + { + "bbox": [ + 210, + 616, + 220, + 626 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "-sample estimator. Maddison et al. (2017) show that the tightness of an", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 627, + 482, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 482, + 639 + ], + "score": 1.0, + "content": "MCO is directly related to the variance of the underlying estimator of the marginal likelihood.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "However, Rainforth et al. (2018) point out issues with gradient estimators of multi-sample lower", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 655, + 504, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 504, + 666 + ], + "score": 1.0, + "content": "bounds. In particular, they show that although the IWAE bound is tighter, the standard IWAE gradi-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "ent estimator’s SNR scales poorly with large numbers of samples, leading to degraded performance.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "Le et al. (2018) experimentally investigate this phenomenon and provide empirical evidence of this", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "degradation across multiple tasks. They find that RWS (Bornschein & Bengio, 2014) does not suffer", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "from this issue and find that it can outperform models trained with the IWAE bound. We conclude", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "that it is not sufficient to just tighten the bound; it is important to understand the gradient estimators", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 720, + 221, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 221, + 732 + ], + "score": 1.0, + "content": "of the tighter bound as well.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 358, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 359, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 359, + 95 + ], + "score": 1.0, + "content": "DOUBLY REPARAMETERIZED REWEIGHTED WAKE UPDATE", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 108, + 102, + 441, + 115 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 441, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 441, + 117 + ], + "score": 1.0, + "content": "The wake update gradient for the inference network (Eq. 8) can be reparameterized", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 106, + 102, + 441, + 117 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 114, + 118, + 497, + 153 + ], + "lines": [ + { + "bbox": [ + 114, + 118, + 497, + 153 + ], + "spans": [ + { + "bbox": [ + 114, + 118, + 497, + 153 + ], + "score": 0.94, + "content": "- \\mathbb { E } _ { z _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z _ { i } | x ) \\right] = \\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\left( \\frac { w _ { i } ^ { 2 } } { ( \\sum _ { j } w _ { j } ) ^ { 2 } } - \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\right) \\frac { \\partial \\log w _ { i } } { \\partial z _ { i } } \\frac { \\partial z _ { i } } { \\partial \\phi } \\right] .", + "type": "interline_equation", + "image_path": "ca69f97770ef62597f819b8dcb3822bb4a8c561547768c46c29a2a98f6718c19.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 114, + 118, + 497, + 129.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 114, + 129.66666666666666, + 497, + 141.33333333333331 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 114, + 141.33333333333331, + 497, + 152.99999999999997 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 161, + 504, + 184 + ], + "lines": [ + { + "bbox": [ + 106, + 160, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 174 + ], + "score": 1.0, + "content": "We call the algorithm that uses the single sample Monte Carlo estimator of this expression as the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 172, + 317, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 317, + 185 + ], + "score": 1.0, + "content": "wake update for the inference network RWS-DReG.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 106, + 160, + 505, + 185 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 189, + 505, + 233 + ], + "lines": [ + { + "bbox": [ + 105, + 190, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 505, + 201 + ], + "score": 1.0, + "content": "Interestingly, the inference network gradient estimator from (Roeder et al., 2017) can be seen as", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 201, + 504, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 504, + 212 + ], + "score": 1.0, + "content": "the sum of the IWAE gradient estimator and the wake update of the inference network (as the wake", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 212, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 223 + ], + "score": 1.0, + "content": "update minimizes, we add the negative of Eq. 9). Their positive results motivate further exploration", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 222, + 339, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 339, + 233 + ], + "score": 1.0, + "content": "of convex combinations of IWAE-DReG and RWS-DReG", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 190, + 505, + 233 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 236, + 433, + 270 + ], + "lines": [ + { + "bbox": [ + 177, + 236, + 433, + 270 + ], + "spans": [ + { + "bbox": [ + 177, + 236, + 433, + 270 + ], + "score": 0.94, + "content": "\\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\left( \\alpha \\frac { w _ { i } } { \\sum _ { j } w _ { j } } + ( 1 - 2 \\alpha ) \\frac { w _ { i } ^ { 2 } } { ( \\sum _ { j } w _ { j } ) ^ { 2 } } \\right) \\frac { \\partial \\log w _ { i } } { \\partial z _ { i } } \\frac { \\partial z _ { i } } { \\partial \\phi } \\right] .", + "type": "interline_equation", + "image_path": "c354b914bd827b1d14f8e9a923ec9bbfb221c7580e7c4784f99e15a4744e457b.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 177, + 236, + 433, + 247.33333333333334 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 177, + 247.33333333333334, + 433, + 258.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 177, + 258.6666666666667, + 433, + 270.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 273, + 505, + 306 + ], + "lines": [ + { + "bbox": [ + 106, + 272, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 286 + ], + "score": 1.0, + "content": "We refer to the algorithm that uses the single sample Monte Carlo estimator of this expression as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 146, + 295 + ], + "score": 0.79, + "content": "\\mathrm { D R e G } ( \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 284, + 179, + 296 + ], + "score": 1.0, + "content": ". When", + "type": "text" + }, + { + "bbox": [ + 180, + 285, + 208, + 294 + ], + "score": 0.91, + "content": "\\alpha = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 284, + 354, + 296 + ], + "score": 1.0, + "content": ", this reduces to RWS-DReG, when", + "type": "text" + }, + { + "bbox": [ + 354, + 285, + 383, + 294 + ], + "score": 0.9, + "content": "\\alpha = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 284, + 505, + 296 + ], + "score": 1.0, + "content": ", this reduces to IWAE-DReG", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 295, + 258, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 147, + 307 + ], + "score": 1.0, + "content": "and when", + "type": "text" + }, + { + "bbox": [ + 147, + 295, + 181, + 306 + ], + "score": 0.89, + "content": "\\alpha = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 295, + 258, + 307 + ], + "score": 1.0, + "content": ", this reduces STL.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 272, + 505, + 307 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 319, + 319, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 317, + 320, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 320, + 334 + ], + "score": 1.0, + "content": "4.2 JACKKNIFE VARIATIONAL INFERENCE (JVI)", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 339, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 338, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 354 + ], + "score": 1.0, + "content": "Alternatively, Nowozin (2018) reinterprets the IWAE lower bound as a biased estimator for the log", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "marginal likelihood. He analyzes the bias and introduces a novel family of estimators, Jackknife", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 363, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 374 + ], + "score": 1.0, + "content": "Variational Inference (JVI), which trade off reduction in bias for increased variance. This additional", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "flexibility comes at the cost of no longer being a stochastic lower bound on the log marginal like-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "lihood. The first-order JVI has significantly reduced bias compared to IWAE, which empirically", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 396, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 407 + ], + "score": 1.0, + "content": "results in a better estimate of the log marginal likelihood with fewer samples (Nowozin, 2018). For", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 407, + 315, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 315, + 418 + ], + "score": 1.0, + "content": "simplicity, we focus on the first-order JVI estimator", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 338, + 506, + 418 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 421, + 465, + 461 + ], + "lines": [ + { + "bbox": [ + 145, + 421, + 465, + 461 + ], + "spans": [ + { + "bbox": [ + 145, + 421, + 465, + 461 + ], + "score": 0.93, + "content": "K \\times \\mathbb { E } _ { z _ { 1 } \\cdot K } \\left[ \\log \\left( \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } w _ { i } \\right) \\right] - \\frac { K - 1 } { K } \\sum _ { i = 1 } ^ { K } \\mathbb { E } _ { z _ { - i } } \\left[ \\log \\left( \\frac { 1 } { K - 1 } \\sum _ { j \\neq i } w _ { j } \\right) \\right] .", + "type": "interline_equation", + "image_path": "57da664026def4d2b565f31da4604d58d912ce6bfbde6af1ca1c694ee74b7955.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 145, + 421, + 465, + 434.3333333333333 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 145, + 434.3333333333333, + 465, + 447.66666666666663 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 145, + 447.66666666666663, + 465, + 460.99999999999994 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 464, + 401, + 476 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 402, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 402, + 477 + ], + "score": 1.0, + "content": "It is straightforward to apply our approach to higher order JVI estimators.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 463, + 402, + 477 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 488, + 421, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 422, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 422, + 502 + ], + "score": 1.0, + "content": "DOUBLY REPARAMETERIZED JACKKNIFE VARIATIONAL INFERENCE (JVI)", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 486, + 422, + 502 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 508, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 507, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 285, + 522 + ], + "score": 1.0, + "content": "The JVI estimator is a linear combination of", + "type": "text" + }, + { + "bbox": [ + 286, + 510, + 296, + 519 + ], + "score": 0.84, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 507, + 314, + 522 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 314, + 509, + 342, + 520 + ], + "score": 0.87, + "content": "K - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 507, + 505, + 522 + ], + "score": 1.0, + "content": "sample IWAE estimators, so we can use", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 520, + 380, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 380, + 532 + ], + "score": 1.0, + "content": "the doubly reparameterized gradient estimator (Eq. 7) for each term.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 106, + 507, + 505, + 532 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 547, + 211, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 213, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 213, + 562 + ], + "score": 1.0, + "content": "5 RELATED WORK", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 571, + 505, + 639 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "Mnih & Rezende (2016) introduced a generalized framework of Monte Carlo objectives (MCO). The", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "log of an unbiased marginal likelihood estimator is a lower bound on the log marginal likelihood by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "Jensen’s inequality. In this view, the ELBO can be seen as the MCO corresponding to a single im-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 104, + 604, + 339, + 618 + ], + "score": 1.0, + "content": "portance sample estimator of the marginal likelihood with", + "type": "text" + }, + { + "bbox": [ + 339, + 606, + 349, + 617 + ], + "score": 0.83, + "content": "q _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "as the proposal distribution. Similarly,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 209, + 628 + ], + "score": 1.0, + "content": "IWAE corresponds to the", + "type": "text" + }, + { + "bbox": [ + 210, + 616, + 220, + 626 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "-sample estimator. Maddison et al. (2017) show that the tightness of an", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 627, + 482, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 482, + 639 + ], + "score": 1.0, + "content": "MCO is directly related to the variance of the underlying estimator of the marginal likelihood.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 571, + 506, + 639 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "However, Rainforth et al. (2018) point out issues with gradient estimators of multi-sample lower", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 655, + 504, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 504, + 666 + ], + "score": 1.0, + "content": "bounds. In particular, they show that although the IWAE bound is tighter, the standard IWAE gradi-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "ent estimator’s SNR scales poorly with large numbers of samples, leading to degraded performance.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "Le et al. (2018) experimentally investigate this phenomenon and provide empirical evidence of this", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "degradation across multiple tasks. They find that RWS (Bornschein & Bengio, 2014) does not suffer", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "from this issue and find that it can outperform models trained with the IWAE bound. We conclude", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "that it is not sufficient to just tighten the bound; it is important to understand the gradient estimators", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 720, + 221, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 221, + 732 + ], + "score": 1.0, + "content": "of the tighter bound as well.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 643, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Wake-sleep is an alternative approach to fitting deep generative models, first introduced in (Hinton", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "et al., 1995) as a method for training Hemholtz machines. It was extended to the multi-sample", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "setting by (Bornschein & Bengio, 2014) and the sequential setting in (Gu et al., 2015). It has been", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 343, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 343, + 128 + ], + "score": 1.0, + "content": "applied to generative modeling of images (Ba et al., 2015).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 107, + 142, + 200, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 201, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 201, + 157 + ], + "score": 1.0, + "content": "6 EXPERIMENTS", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 168, + 505, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 167, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 428, + 180 + ], + "score": 1.0, + "content": "To evaluate DReG estimators, we first measure variance and signal-to-noise ratio", + "type": "text" + }, + { + "bbox": [ + 428, + 167, + 459, + 179 + ], + "score": 0.5, + "content": "( \\mathrm { S N R } ) ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 167, + 505, + 180 + ], + "score": 1.0, + "content": "of gradient", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "score": 1.0, + "content": "estimators on a toy example which we can carefully control. Then, we evaluate gradient variance", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 189, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 506, + 203 + ], + "score": 1.0, + "content": "and model learning on MNIST generative modeling, Omniglot generative modeling, and MNIST", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 201, + 217, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 217, + 213 + ], + "score": 1.0, + "content": "structured prediction tasks.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "title", + "bbox": [ + 107, + 225, + 198, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 223, + 200, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 200, + 239 + ], + "score": 1.0, + "content": "6.1 TOY GAUSSIAN", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 505, + 312 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "We reimplemented the Gaussian example from (Rainforth et al., 2018). Consider the generative", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 255, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 155, + 270 + ], + "score": 1.0, + "content": "model with", + "type": "text" + }, + { + "bbox": [ + 156, + 257, + 210, + 269 + ], + "score": 0.92, + "content": "z \\sim N ( \\theta , I )", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 255, + 230, + 270 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 230, + 257, + 293, + 269 + ], + "score": 0.93, + "content": "x | z \\sim N ( z , I )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 255, + 388, + 270 + ], + "score": 1.0, + "content": "and inference network", + "type": "text" + }, + { + "bbox": [ + 389, + 256, + 501, + 270 + ], + "score": 0.9, + "content": "\\begin{array} { r } { q _ { \\phi } ( z | x ) \\sim N ( A x ^ { - } + b , \\frac { 2 } { 3 } I ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 255, + 505, + 270 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 134, + 280 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 268, + 184, + 280 + ], + "score": 0.92, + "content": "\\phi = \\{ A , b \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 267, + 505, + 280 + ], + "score": 1.0, + "content": ". As in (Rainforth et al., 2018), we sample a set of parameters for the model", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "and inference network close to the optimal parameters (perturbed by zero-mean Gaussian noise", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "with standard deviation 0.01), then estimate the gradient of the inference network parameters for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 301, + 249, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 228, + 313 + ], + "score": 1.0, + "content": "increasing number of samples", + "type": "text" + }, + { + "bbox": [ + 229, + 301, + 245, + 312 + ], + "score": 0.68, + "content": "( K )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 302, + 249, + 313 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 505, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "In addition to signal-to-noise ratio (SNR), we plot the squared bias and variance of the gradient", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 505, + 340 + ], + "score": 1.0, + "content": "estimators4 in Fig. 1. The bias is computed relative to the expected value of the IWAE gradient", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 340, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 290, + 352 + ], + "score": 1.0, + "content": "estimator. As a result, although the average of", + "type": "text" + }, + { + "bbox": [ + 290, + 340, + 300, + 349 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 340, + 506, + 352 + ], + "score": 1.0, + "content": "ELBO gradient estimators is an unbiased estimator", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "of the ELBO gradient, it is a biased gradient estimator of the IWAE objective. Importantly, SNR", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 361, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 373 + ], + "score": 1.0, + "content": "does not penalize estimators that are biased, so trivial constant estimators can have infinite SNR.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 405, + 385 + ], + "score": 1.0, + "content": "Thus, it is important to consider additional evaluation measures as well. As", + "type": "text" + }, + { + "bbox": [ + 406, + 373, + 416, + 383 + ], + "score": 0.79, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "increases, the SNR of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 383, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 395 + ], + "score": 1.0, + "content": "the IWAE-DReG estimator increases, whereas the SNR of the standard gradient estimator of IWAE", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 104, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "goes to 0, as previously reported. Furthermore, we can see the bias present in the STL estimator. As a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "check of our implementation, we verified that the observed “bias” for IWAE-DReG was statistically", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "indistinguishable from 0 with a paired t-test. For the biased estimators (e.g., STL), we could easily", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 428, + 283, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 283, + 440 + ], + "score": 1.0, + "content": "reject the null hypothesis with few samples.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21 + }, + { + "type": "image", + "bbox": [ + 108, + 450, + 502, + 541 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 450, + 502, + 541 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 450, + 502, + 541 + ], + "spans": [ + { + "bbox": [ + 108, + 450, + 502, + 541 + ], + "score": 0.966, + "type": "image", + "image_path": "4602530728769106fa89abfa2de18954281cbb1ea79b32fbc94d040f866f151a.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 108, + 450, + 502, + 480.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 108, + 480.3333333333333, + 502, + 510.66666666666663 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 108, + 510.66666666666663, + 502, + 541.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 560, + 505, + 615 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 560, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 572 + ], + "score": 1.0, + "content": "Figure 1: Signal-to-noise ratios (SNR), bias squared, and variance of gradient estimators with in-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 571, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 143, + 583 + ], + "score": 1.0, + "content": "creasing", + "type": "text" + }, + { + "bbox": [ + 143, + 572, + 154, + 581 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 571, + 505, + 583 + ], + "score": 1.0, + "content": "over 10 random trials with 1000 measurement samples per trial (mean in bold). The", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "observed “bias” for IWAE-DReG is not statistically significant under a paired t-test (as expected", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "because IWAE-DReG is unbiased). IWAE-DReG is unbiased, its SNR increases with K, and it has", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 604, + 321, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 321, + 615 + ], + "score": 1.0, + "content": "the lowest variance of the estimators considered here.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + } + ], + "index": 30.0 + }, + { + "type": "title", + "bbox": [ + 107, + 638, + 237, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 237, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 237, + 650 + ], + "score": 1.0, + "content": "6.2 GENERATIVE MODELING", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 505, + 692 + ], + "lines": [ + { + "bbox": [ + 105, + 658, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 505, + 671 + ], + "score": 1.0, + "content": "Training generative models of the binarized MNIST digits dataset is a standard benchmark task for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 671, + 504, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 504, + 681 + ], + "score": 1.0, + "content": "latent variable models. For this evaluation, we used the single latent layer architecture from (Burda", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 681, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 693 + ], + "score": 1.0, + "content": "et al., 2015). The generative model used 50 Gaussian latent variables with an isotropic prior and", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 700, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 118, + 699, + 380, + 712 + ], + "spans": [ + { + "bbox": [ + 118, + 699, + 380, + 712 + ], + "score": 1.0, + "content": "3Defined as the mean of the estimator divided by the standard deviation.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 709, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 190, + 724 + ], + "score": 1.0, + "content": "4All dimensions of", + "type": "text" + }, + { + "bbox": [ + 190, + 712, + 197, + 722 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 709, + 505, + 724 + ], + "score": 1.0, + "content": "behaved qualitatively similarly, so for clarity, we show curves for a single randomly", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 720, + 193, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 183, + 733 + ], + "score": 1.0, + "content": "chosen dimension of", + "type": "text" + }, + { + "bbox": [ + 183, + 722, + 190, + 732 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 720, + 193, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "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": [ + 107, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Wake-sleep is an alternative approach to fitting deep generative models, first introduced in (Hinton", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "et al., 1995) as a method for training Hemholtz machines. It was extended to the multi-sample", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "setting by (Bornschein & Bengio, 2014) and the sequential setting in (Gu et al., 2015). It has been", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 343, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 343, + 128 + ], + "score": 1.0, + "content": "applied to generative modeling of images (Ba et al., 2015).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 505, + 128 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 142, + 200, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 201, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 201, + 157 + ], + "score": 1.0, + "content": "6 EXPERIMENTS", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 168, + 505, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 167, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 428, + 180 + ], + "score": 1.0, + "content": "To evaluate DReG estimators, we first measure variance and signal-to-noise ratio", + "type": "text" + }, + { + "bbox": [ + 428, + 167, + 459, + 179 + ], + "score": 0.5, + "content": "( \\mathrm { S N R } ) ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 167, + 505, + 180 + ], + "score": 1.0, + "content": "of gradient", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "score": 1.0, + "content": "estimators on a toy example which we can carefully control. Then, we evaluate gradient variance", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 189, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 506, + 203 + ], + "score": 1.0, + "content": "and model learning on MNIST generative modeling, Omniglot generative modeling, and MNIST", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 201, + 217, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 217, + 213 + ], + "score": 1.0, + "content": "structured prediction tasks.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 167, + 506, + 213 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 225, + 198, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 223, + 200, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 200, + 239 + ], + "score": 1.0, + "content": "6.1 TOY GAUSSIAN", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 505, + 312 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "We reimplemented the Gaussian example from (Rainforth et al., 2018). Consider the generative", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 255, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 155, + 270 + ], + "score": 1.0, + "content": "model with", + "type": "text" + }, + { + "bbox": [ + 156, + 257, + 210, + 269 + ], + "score": 0.92, + "content": "z \\sim N ( \\theta , I )", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 255, + 230, + 270 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 230, + 257, + 293, + 269 + ], + "score": 0.93, + "content": "x | z \\sim N ( z , I )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 255, + 388, + 270 + ], + "score": 1.0, + "content": "and inference network", + "type": "text" + }, + { + "bbox": [ + 389, + 256, + 501, + 270 + ], + "score": 0.9, + "content": "\\begin{array} { r } { q _ { \\phi } ( z | x ) \\sim N ( A x ^ { - } + b , \\frac { 2 } { 3 } I ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 255, + 505, + 270 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 134, + 280 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 268, + 184, + 280 + ], + "score": 0.92, + "content": "\\phi = \\{ A , b \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 267, + 505, + 280 + ], + "score": 1.0, + "content": ". As in (Rainforth et al., 2018), we sample a set of parameters for the model", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "and inference network close to the optimal parameters (perturbed by zero-mean Gaussian noise", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "with standard deviation 0.01), then estimate the gradient of the inference network parameters for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 301, + 249, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 228, + 313 + ], + "score": 1.0, + "content": "increasing number of samples", + "type": "text" + }, + { + "bbox": [ + 229, + 301, + 245, + 312 + ], + "score": 0.68, + "content": "( K )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 302, + 249, + 313 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 245, + 506, + 313 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 505, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "In addition to signal-to-noise ratio (SNR), we plot the squared bias and variance of the gradient", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 505, + 340 + ], + "score": 1.0, + "content": "estimators4 in Fig. 1. The bias is computed relative to the expected value of the IWAE gradient", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 340, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 290, + 352 + ], + "score": 1.0, + "content": "estimator. As a result, although the average of", + "type": "text" + }, + { + "bbox": [ + 290, + 340, + 300, + 349 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 340, + 506, + 352 + ], + "score": 1.0, + "content": "ELBO gradient estimators is an unbiased estimator", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "of the ELBO gradient, it is a biased gradient estimator of the IWAE objective. Importantly, SNR", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 361, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 373 + ], + "score": 1.0, + "content": "does not penalize estimators that are biased, so trivial constant estimators can have infinite SNR.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 405, + 385 + ], + "score": 1.0, + "content": "Thus, it is important to consider additional evaluation measures as well. As", + "type": "text" + }, + { + "bbox": [ + 406, + 373, + 416, + 383 + ], + "score": 0.79, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "increases, the SNR of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 383, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 395 + ], + "score": 1.0, + "content": "the IWAE-DReG estimator increases, whereas the SNR of the standard gradient estimator of IWAE", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 104, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "goes to 0, as previously reported. Furthermore, we can see the bias present in the STL estimator. As a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "check of our implementation, we verified that the observed “bias” for IWAE-DReG was statistically", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "indistinguishable from 0 with a paired t-test. For the biased estimators (e.g., STL), we could easily", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 428, + 283, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 283, + 440 + ], + "score": 1.0, + "content": "reject the null hypothesis with few samples.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 318, + 506, + 440 + ] + }, + { + "type": "image", + "bbox": [ + 108, + 450, + 502, + 541 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 450, + 502, + 541 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 450, + 502, + 541 + ], + "spans": [ + { + "bbox": [ + 108, + 450, + 502, + 541 + ], + "score": 0.966, + "type": "image", + "image_path": "4602530728769106fa89abfa2de18954281cbb1ea79b32fbc94d040f866f151a.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 108, + 450, + 502, + 480.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 108, + 480.3333333333333, + 502, + 510.66666666666663 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 108, + 510.66666666666663, + 502, + 541.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 560, + 505, + 615 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 560, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 572 + ], + "score": 1.0, + "content": "Figure 1: Signal-to-noise ratios (SNR), bias squared, and variance of gradient estimators with in-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 571, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 143, + 583 + ], + "score": 1.0, + "content": "creasing", + "type": "text" + }, + { + "bbox": [ + 143, + 572, + 154, + 581 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 571, + 505, + 583 + ], + "score": 1.0, + "content": "over 10 random trials with 1000 measurement samples per trial (mean in bold). The", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "observed “bias” for IWAE-DReG is not statistically significant under a paired t-test (as expected", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "because IWAE-DReG is unbiased). IWAE-DReG is unbiased, its SNR increases with K, and it has", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 604, + 321, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 321, + 615 + ], + "score": 1.0, + "content": "the lowest variance of the estimators considered here.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + } + ], + "index": 30.0 + }, + { + "type": "title", + "bbox": [ + 107, + 638, + 237, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 237, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 237, + 650 + ], + "score": 1.0, + "content": "6.2 GENERATIVE MODELING", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 505, + 692 + ], + "lines": [ + { + "bbox": [ + 105, + 658, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 505, + 671 + ], + "score": 1.0, + "content": "Training generative models of the binarized MNIST digits dataset is a standard benchmark task for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 671, + 504, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 504, + 681 + ], + "score": 1.0, + "content": "latent variable models. For this evaluation, we used the single latent layer architecture from (Burda", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 681, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 693 + ], + "score": 1.0, + "content": "et al., 2015). The generative model used 50 Gaussian latent variables with an isotropic prior and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 136, + 95 + ], + "score": 1.0, + "content": "passed", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 136, + 84, + 144, + 92 + ], + "score": 0.72, + "content": "z", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 144, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "through two deterministic layers of 200 tanh units to parameterize factorized Bernoulli", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 265, + 106 + ], + "score": 1.0, + "content": "outputs. The inference network passed", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 266, + 95, + 273, + 104 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 273, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "through two deterministic layers of 200 tanh units to pa-", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 307, + 116 + ], + "score": 1.0, + "content": "rameterize a factorized Gaussian distribution over", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 307, + 106, + 314, + 114 + ], + "score": 0.67, + "content": "z", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 314, + 105, + 505, + 116 + ], + "score": 1.0, + "content": ". Because our interest was in improved gradient", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "estimators and optimization performance, we used the dynamically binarized MNIST dataset, which", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "minimally suffers from overfitting. We used the standard split of MNIST into train, validation, and", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 143, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 143, + 149 + ], + "score": 1.0, + "content": "test sets.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 658, + 505, + 693 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 136, + 95 + ], + "score": 1.0, + "content": "passed", + "type": "text" + }, + { + "bbox": [ + 136, + 84, + 144, + 92 + ], + "score": 0.72, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "through two deterministic layers of 200 tanh units to parameterize factorized Bernoulli", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 265, + 106 + ], + "score": 1.0, + "content": "outputs. The inference network passed", + "type": "text" + }, + { + "bbox": [ + 266, + 95, + 273, + 104 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "through two deterministic layers of 200 tanh units to pa-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 307, + 116 + ], + "score": 1.0, + "content": "rameterize a factorized Gaussian distribution over", + "type": "text" + }, + { + "bbox": [ + 307, + 106, + 314, + 114 + ], + "score": 0.67, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 105, + 505, + 116 + ], + "score": 1.0, + "content": ". Because our interest was in improved gradient", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "estimators and optimization performance, we used the dynamically binarized MNIST dataset, which", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "minimally suffers from overfitting. We used the standard split of MNIST into train, validation, and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 143, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 143, + 149 + ], + "score": 1.0, + "content": "test sets.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 188 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "score": 1.0, + "content": "We trained models with the IWAE gradient, the RWS wake update, and with the JVI estimator.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "In all three cases, the doubly reparameterized gradient estimator reduced variance5 and as a result", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 287, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 287, + 189 + ], + "score": 1.0, + "content": "substantially improved performance (Fig. 2).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "image", + "bbox": [ + 125, + 202, + 486, + 378 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 125, + 202, + 486, + 378 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 125, + 202, + 486, + 378 + ], + "spans": [ + { + "bbox": [ + 125, + 202, + 486, + 378 + ], + "score": 0.973, + "type": "image", + "image_path": "df131fe5998b53bb0ac88c7a2eb163c6e64208318f090fefa87a69dbc76fe0f7.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 125, + 202, + 486, + 260.6666666666667 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 125, + 260.6666666666667, + 486, + 319.33333333333337 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 125, + 319.33333333333337, + 486, + 378.00000000000006 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 396, + 505, + 474 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "Figure 2: MNIST generative modeling trained according to IWAE (left), RWS (middle), and JVI", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "(right). The top row compares the variance of the original gradient estimator (dashed) with the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 505, + 430 + ], + "score": 1.0, + "content": "variance of the doubly reparameterized gradient estimator (solid). The bottom row compares test", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "performance. The left and middle plots show the IWAE (stochastic) lower bound on the test set.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 441, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 107, + 441, + 505, + 452 + ], + "score": 1.0, + "content": "The right plot shows the JVI estimator (which is not a bound) on the test set. The bold lines are the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 452, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 464 + ], + "score": 1.0, + "content": "average over three trials, and individual trials are displayed as semi-transparent). 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Inter-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "estingly, the biased gradient estimators STL and RWS-DReG perform best on this task with RWS-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "DReG slightly outperforming STL. As observed in (Le et al., 2018), RWS increasingly outperforms", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 146, + 532 + ], + "score": 1.0, + "content": "IWAE as", + "type": "text" + }, + { + "bbox": [ + 146, + 520, + 157, + 530 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "increases. 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In this task, our", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 267, + 654 + ], + "score": 1.0, + "content": "goal is to model a complex observation", + "type": "text" + }, + { + "bbox": [ + 267, + 644, + 274, + 652 + ], + "score": 0.7, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 641, + 339, + 654 + ], + "score": 1.0, + "content": "given a context", + "type": "text" + }, + { + "bbox": [ + 339, + 644, + 345, + 652 + ], + "score": 0.72, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "(i.e., model the conditional distribution", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 652, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 107, + 653, + 136, + 665 + ], + "score": 0.9, + "content": "p ( x | c ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 652, + 329, + 666 + ], + "score": 1.0, + "content": ". 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Inter-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "estingly, the biased gradient estimators STL and RWS-DReG perform best on this task with RWS-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 505, + 521 + ], + "score": 1.0, + "content": "DReG slightly outperforming STL. As observed in (Le et al., 2018), RWS increasingly outperforms", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 146, + 532 + ], + "score": 1.0, + "content": "IWAE as", + "type": "text" + }, + { + "bbox": [ + 146, + 520, + 157, + 530 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "increases. Finally, we experimented with convex combinations of IWAE-DReG and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "RWS-DReG (right Fig. 3). On this dataset, convex combinations that heavily weighted RWS-DReG", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 542, + 420, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 420, + 554 + ], + "score": 1.0, + "content": "had the best performance. 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In this task, our", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 267, + 654 + ], + "score": 1.0, + "content": "goal is to model a complex observation", + "type": "text" + }, + { + "bbox": [ + 267, + 644, + 274, + 652 + ], + "score": 0.7, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 641, + 339, + 654 + ], + "score": 1.0, + "content": "given a context", + "type": "text" + }, + { + "bbox": [ + 339, + 644, + 345, + 652 + ], + "score": 0.72, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "(i.e., model the conditional distribution", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 652, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 107, + 653, + 136, + 665 + ], + "score": 0.9, + "content": "p ( x | c ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 652, + 329, + 666 + ], + "score": 1.0, + "content": ". We can use a conditional latent variable model", + "type": "text" + }, + { + "bbox": [ + 329, + 653, + 452, + 665 + ], + "score": 0.92, + "content": "p _ { \\theta } ( x , z | c ) = p _ { \\theta } ( x | z , c ) p _ { \\theta } ( z | c )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 652, + 506, + 666 + ], + "score": 1.0, + "content": ", however, as", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 663, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 505, + 677 + ], + "score": 1.0, + "content": "before, computing the marginal likelihood is generally intractable. It is straightforward to adapt the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 675, + 369, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 369, + 688 + ], + "score": 1.0, + "content": "bounds and techniques from the previous section to this problem.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 631, + 506, + 688 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 126, + 81, + 486, + 168 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 126, + 81, + 486, + 168 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 126, + 81, + 486, + 168 + ], + "spans": [ + { + "bbox": [ + 126, + 81, + 486, + 168 + ], + "score": 0.964, + "type": "image", + "image_path": "91eefa28a63730121c99b026911c4b3915048b1370526f3bf876517154d33a0f.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 126, + 81, + 486, + 110.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 126, + 110.0, + 486, + 139.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 126, + 139.0, + 486, + 168.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 186, + 506, + 263 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 186, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 199 + ], + "score": 1.0, + "content": "Figure 3: Log-likelihood lower bounds for generative modeling on MNIST. The left and middle", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 197, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 356, + 210 + ], + "score": 1.0, + "content": "plots compare performance with different number of samples", + "type": "text" + }, + { + "bbox": [ + 356, + 197, + 391, + 208 + ], + "score": 0.77, + "content": "K = 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 197, + 506, + 210 + ], + "score": 1.0, + "content": ", 256. For clarity the legend", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 208, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 221 + ], + "score": 1.0, + "content": "is shared between the plots. The bold lines are the average over three trials, and individual trials", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "are displayed as semi-transparent). The right plot compares performance as the convex combination", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 229, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 505, + 242 + ], + "score": 1.0, + "content": "between IWAE-DReG and RWS-DReG is varied (Eq. 10). To highlight differences, we plot the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 241, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 506, + 252 + ], + "score": 1.0, + "content": "difference between the test IWAE bound and the test IWAE bound IWAE-DReG achieved at that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 251, + 129, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 129, + 267 + ], + "score": 1.0, + "content": "step.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 285, + 505, + 352 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 505, + 297 + ], + "score": 1.0, + "content": "To evaluate our method in this context, we use the standard task of modeling the bottom half of a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 297, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 308 + ], + "score": 1.0, + "content": "binarized MNIST digit from the top half. 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The conditional prior feeds", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 107, + 321, + 113, + 329 + ], + "score": 0.62, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "to two deterministic layers of 200 tanh units to parameterize a factorized Gaussian distribution", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 126, + 342 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 127, + 331, + 133, + 339 + ], + "score": 0.72, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 329, + 289, + 342 + ], + "score": 1.0, + "content": ". To model the conditional distribution", + "type": "text" + }, + { + "bbox": [ + 289, + 330, + 329, + 342 + ], + "score": 0.93, + "content": "p _ { \\theta } ( x | c , z )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 329, + 397, + 342 + ], + "score": 1.0, + "content": ", we concatenate", + "type": "text" + }, + { + "bbox": [ + 397, + 331, + 404, + 339 + ], + "score": 0.74, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 329, + 425, + 342 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 425, + 331, + 432, + 340 + ], + "score": 0.73, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "and feed it to two", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 340, + 438, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 438, + 353 + ], + "score": 1.0, + "content": "deterministic layers of 200 tanh units to parameterize factorized Bernoulli outputs.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 357, + 505, + 412 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "As in the previous tasks, the doubly reparameterized gradient estimator improves across all three", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 368, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 506, + 381 + ], + "score": 1.0, + "content": "updates (IWAE, RWS, and JVI; Appendix Fig. 7). However, on this task, the biased estimators", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "(STL and RWS) underperform unbiased IWAE gradient estimators (Fig. 4). In particular, RWS", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 389, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 104, + 389, + 506, + 404 + ], + "score": 1.0, + "content": "becomes unstable later in training. 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The left plot uses", + "type": "text" + }, + { + "bbox": [ + 483, + 531, + 505, + 541 + ], + "score": 0.84, + "content": "K =", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 252, + 554 + ], + "score": 1.0, + "content": "64 samples and the right plot uses", + "type": "text" + }, + { + "bbox": [ + 252, + 542, + 295, + 552 + ], + "score": 0.89, + "content": "K \\ : = \\ : 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "samples. For clarity the legend is shared between", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "the plots. The bold lines are the average over three trials, and individual trials are displayed as", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "semi-transparent). The right plot compares performance as the convex combination between IWAE-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "DReG and RWS-DReG is varied (Eq. 10). To highlight differences, we plot the difference between", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 584, + 437, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 437, + 598 + ], + "score": 1.0, + "content": "the test IWAE bound and the test IWAE bound IWAE-DReG achieved at that step.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + } + ], + "index": 24.25 + }, + { + "type": "title", + "bbox": [ + 108, + 622, + 190, + 635 + ], + "lines": [ + { + "bbox": [ + 105, + 622, + 192, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 192, + 638 + ], + "score": 1.0, + "content": "7 DISCUSSION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 648, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "In this work, we introduce doubly reparameterized estimators for the updates in IWAE, RWS, and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "JVI. We demonstrate that across tasks they provide unbiased variance reduction, which leads to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "improved performance. Furthermore, DReG estimators have the same computational cost as the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "original estimators. As a result, we recommend that DReG estimators be used instead of the typical", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 693, + 187, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 187, + 705 + ], + "score": 1.0, + "content": "gradient estimators.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 107, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 107, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "Variational Sequential Monte Carlo (Maddison et al., 2017; Naesseth et al., 2018; Le et al., 2018)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "and Neural Adapative Sequential Monte Carlo (Gu et al., 2015) extend IWAE and RWS to sequential", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 126, + 81, + 486, + 168 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 126, + 81, + 486, + 168 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 126, + 81, + 486, + 168 + ], + "spans": [ + { + "bbox": [ + 126, + 81, + 486, + 168 + ], + "score": 0.964, + "type": "image", + "image_path": "91eefa28a63730121c99b026911c4b3915048b1370526f3bf876517154d33a0f.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 126, + 81, + 486, + 110.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 126, + 110.0, + 486, + 139.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 126, + 139.0, + 486, + 168.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 186, + 506, + 263 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 186, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 199 + ], + "score": 1.0, + "content": "Figure 3: Log-likelihood lower bounds for generative modeling on MNIST. The left and middle", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 197, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 356, + 210 + ], + "score": 1.0, + "content": "plots compare performance with different number of samples", + "type": "text" + }, + { + "bbox": [ + 356, + 197, + 391, + 208 + ], + "score": 0.77, + "content": "K = 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 197, + 506, + 210 + ], + "score": 1.0, + "content": ", 256. For clarity the legend", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 208, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 221 + ], + "score": 1.0, + "content": "is shared between the plots. The bold lines are the average over three trials, and individual trials", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "are displayed as semi-transparent). The right plot compares performance as the convex combination", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 229, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 505, + 242 + ], + "score": 1.0, + "content": "between IWAE-DReG and RWS-DReG is varied (Eq. 10). To highlight differences, we plot the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 241, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 506, + 252 + ], + "score": 1.0, + "content": "difference between the test IWAE bound and the test IWAE bound IWAE-DReG achieved at that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 251, + 129, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 129, + 267 + ], + "score": 1.0, + "content": "step.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 285, + 505, + 352 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 505, + 297 + ], + "score": 1.0, + "content": "To evaluate our method in this context, we use the standard task of modeling the bottom half of a", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 297, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 308 + ], + "score": 1.0, + "content": "binarized MNIST digit from the top half. We use a similar architecture, but now learn a conditional", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 177, + 320 + ], + "score": 1.0, + "content": "prior distribution", + "type": "text" + }, + { + "bbox": [ + 178, + 308, + 208, + 320 + ], + "score": 0.93, + "content": "p _ { \\theta } ( z | c )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 307, + 237, + 320 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 238, + 309, + 244, + 317 + ], + "score": 0.69, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "is the top half of the MNIST digit. The conditional prior feeds", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 107, + 321, + 113, + 329 + ], + "score": 0.62, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "to two deterministic layers of 200 tanh units to parameterize a factorized Gaussian distribution", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 126, + 342 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 127, + 331, + 133, + 339 + ], + "score": 0.72, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 329, + 289, + 342 + ], + "score": 1.0, + "content": ". To model the conditional distribution", + "type": "text" + }, + { + "bbox": [ + 289, + 330, + 329, + 342 + ], + "score": 0.93, + "content": "p _ { \\theta } ( x | c , z )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 329, + 397, + 342 + ], + "score": 1.0, + "content": ", we concatenate", + "type": "text" + }, + { + "bbox": [ + 397, + 331, + 404, + 339 + ], + "score": 0.74, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 329, + 425, + 342 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 425, + 331, + 432, + 340 + ], + "score": 0.73, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "and feed it to two", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 340, + 438, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 438, + 353 + ], + "score": 1.0, + "content": "deterministic layers of 200 tanh units to parameterize factorized Bernoulli outputs.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 285, + 505, + 353 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 357, + 505, + 412 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "As in the previous tasks, the doubly reparameterized gradient estimator improves across all three", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 368, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 506, + 381 + ], + "score": 1.0, + "content": "updates (IWAE, RWS, and JVI; Appendix Fig. 7). However, on this task, the biased estimators", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "(STL and RWS) underperform unbiased IWAE gradient estimators (Fig. 4). In particular, RWS", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 389, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 104, + 389, + 506, + 404 + ], + "score": 1.0, + "content": "becomes unstable later in training. We suspect that this is because RWS does not directly optimize", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 401, + 197, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 197, + 415 + ], + "score": 1.0, + "content": "a consistent objective.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 358, + 506, + 415 + ] + }, + { + "type": "image", + "bbox": [ + 125, + 426, + 485, + 513 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 125, + 426, + 485, + 513 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 125, + 426, + 485, + 513 + ], + "spans": [ + { + "bbox": [ + 125, + 426, + 485, + 513 + ], + "score": 0.963, + "type": "image", + "image_path": "57312dcd20d125385cf7e6526c7c22ebd0eaef48131ffdc8b992f9c2c7c72fc0.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 125, + 426, + 485, + 455.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 125, + 455.0, + 485, + 484.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 125, + 484.0, + 485, + 513.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 531, + 506, + 597 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 483, + 543 + ], + "score": 1.0, + "content": "Figure 4: Log-likelihood lower bounds for structured prediction on MNIST. The left plot uses", + "type": "text" + }, + { + "bbox": [ + 483, + 531, + 505, + 541 + ], + "score": 0.84, + "content": "K =", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 252, + 554 + ], + "score": 1.0, + "content": "64 samples and the right plot uses", + "type": "text" + }, + { + "bbox": [ + 252, + 542, + 295, + 552 + ], + "score": 0.89, + "content": "K \\ : = \\ : 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "samples. For clarity the legend is shared between", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "the plots. The bold lines are the average over three trials, and individual trials are displayed as", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "semi-transparent). The right plot compares performance as the convex combination between IWAE-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "DReG and RWS-DReG is varied (Eq. 10). To highlight differences, we plot the difference between", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 584, + 437, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 437, + 598 + ], + "score": 1.0, + "content": "the test IWAE bound and the test IWAE bound IWAE-DReG achieved at that step.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + } + ], + "index": 24.25 + }, + { + "type": "title", + "bbox": [ + 108, + 622, + 190, + 635 + ], + "lines": [ + { + "bbox": [ + 105, + 622, + 192, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 192, + 638 + ], + "score": 1.0, + "content": "7 DISCUSSION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 648, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "In this work, we introduce doubly reparameterized estimators for the updates in IWAE, RWS, and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "JVI. We demonstrate that across tasks they provide unbiased variance reduction, which leads to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "improved performance. Furthermore, DReG estimators have the same computational cost as the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "original estimators. As a result, we recommend that DReG estimators be used instead of the typical", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 693, + 187, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 187, + 705 + ], + "score": 1.0, + "content": "gradient estimators.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 649, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 107, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 107, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "Variational Sequential Monte Carlo (Maddison et al., 2017; Naesseth et al., 2018; Le et al., 2018)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "and Neural Adapative Sequential Monte Carlo (Gu et al., 2015) extend IWAE and RWS to sequential", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "latent variable models, respectively. It would be interesting to develop DReG estimators for these", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 187, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 187, + 105 + ], + "score": 1.0, + "content": "approaches as well.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 710, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "latent variable models, respectively. It would be interesting to develop DReG estimators for these", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 187, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 187, + 105 + ], + "score": 1.0, + "content": "approaches as well.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 108, + 110, + 504, + 144 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 505, + 122 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 505, + 122 + ], + "score": 1.0, + "content": "We found that a convex combination of IWAE-DReG and RWS-DReG performed best, however, the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 122, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 505, + 134 + ], + "score": 1.0, + "content": "weighting was task dependent. In future work, we intend to apply ideas from (Baydin et al., 2017)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 329, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 329, + 145 + ], + "score": 1.0, + "content": "to automatically adapt the weighting based on the data.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 149, + 504, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "score": 1.0, + "content": "Finally, the form of the IWAE-DReG estimator (Eq. 7) is surprisingly simple and suggests that there", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 160, + 380, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 380, + 173 + ], + "score": 1.0, + "content": "may be a more direct derivation that is applicable to general MCOs.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 108, + 188, + 218, + 200 + ], + "lines": [ + { + "bbox": [ + 107, + 188, + 219, + 202 + ], + "spans": [ + { + "bbox": [ + 107, + 188, + 219, + 202 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 105, + 212, + 504, + 235 + ], + "lines": [ + { + "bbox": [ + 106, + 212, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 225 + ], + "score": 1.0, + "content": "We thank Ben Poole and Diederik P. Kingma for helpful discussion and comments on drafts of this", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 223, + 443, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 443, + 237 + ], + "score": 1.0, + "content": "paper. We thank Sergey Levine and Jascha Sohl-Dickstein for insightful discussion.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "title", + "bbox": [ + 107, + 251, + 175, + 263 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 176, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 176, + 264 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 108, + 269, + 504, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 267, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 506, + 284 + ], + "score": 1.0, + "content": "Jimmy Ba, Ruslan R Salakhutdinov, Roger B Grosse, and Brendan J Frey. 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The left plot uses", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 107, + 540, + 141, + 550 + ], + "score": 0.9, + "content": "K = 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 538, + 266, + 552 + ], + "score": 1.0, + "content": "samples and the right plot uses", + "type": "text" + }, + { + "bbox": [ + 266, + 540, + 306, + 550 + ], + "score": 0.89, + "content": "K = 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "samples. For clarity the legend is shared between", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "the plots. The bold lines are the average over three trials, and individual trials are displayed as", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "semi-transparent). The right plot compares performance as the convex combination between IWAE-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "DReG and RWS-DReG is varied. 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We plot", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 470, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 482 + ], + "score": 1.0, + "content": "the trace of the variance of the doubly reparameterized gradient estimator relative to the original", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "gradient estimator for IWAE (left), RWS (middle), and JVI (right) as the number of samples (K) is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 492, + 136, + 502 + ], + "spans": [ + { + "bbox": [ + 104, + 492, + 136, + 502 + ], + "score": 1.0, + "content": "varied.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + } + ], + "index": 11.75 + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 362, + 535 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 363, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 250, + 536 + ], + "score": 1.0, + "content": "for a reparameterizable distribution", + "type": "text" + }, + { + "bbox": [ + 250, + 523, + 273, + 536 + ], + "score": 0.92, + "content": "q _ { \\phi } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 522, + 363, + 536 + ], + "score": 1.0, + "content": ". 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The bottom row compares test performance.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "The left and middle plots show the IWAE (stochastic) lower bound on the test set. The right plot", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 319, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 505, + 332 + ], + "score": 1.0, + "content": "shows the JVI estimator (which is not a bound) on the test set. The bold lines are the average over", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 330, + 487, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 449, + 343 + ], + "score": 1.0, + "content": "three trials, and individual trials are displayed as semi-transparent). All methods used", + "type": "text" + }, + { + "bbox": [ + 450, + 331, + 483, + 341 + ], + "score": 0.9, + "content": "K = 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 330, + 487, + 343 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "image", + "bbox": [ + 111, + 354, + 498, + 441 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 354, + 498, + 441 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 111, + 354, + 498, + 441 + ], + "spans": [ + { + "bbox": [ + 111, + 354, + 498, + 441 + ], + "score": 0.964, + "type": "image", + "image_path": "39fd367425ce5cdcaf0a8bab14584ea5412d42a363430f34083726be95e22c55.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 111, + 354, + 498, + 383.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 111, + 383.0, + 498, + 412.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 111, + 412.0, + 498, + 441.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 458, + 506, + 503 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 459, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 470 + ], + "score": 1.0, + "content": "Figure 8: Variance of the gradient estimators on the MNIST generative modeling task. We plot", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 470, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 482 + ], + "score": 1.0, + "content": "the trace of the variance of the doubly reparameterized gradient estimator relative to the original", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "gradient estimator for IWAE (left), RWS (middle), and JVI (right) as the number of samples (K) is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 492, + 136, + 502 + ], + "spans": [ + { + "bbox": [ + 104, + 492, + 136, + 502 + ], + "score": 1.0, + "content": "varied.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + } + ], + "index": 11.75 + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 362, + 535 + ], + "lines": [ + { + "bbox": [ + 105, + 522, + 363, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 250, + 536 + ], + "score": 1.0, + "content": "for a reparameterizable distribution", + "type": "text" + }, + { + "bbox": [ + 250, + 523, + 273, + 536 + ], + "score": 0.92, + "content": "q _ { \\phi } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 522, + 363, + 536 + ], + "score": 1.0, + "content": ". To see this, note that", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 522, + 363, + 536 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 115, + 541, + 496, + 626 + ], + "lines": [ + { + "bbox": [ + 115, + 541, + 496, + 626 + ], + "spans": [ + { + "bbox": [ + 115, + 541, + 496, + 626 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\displaystyle \\frac { d } { d \\phi } \\int _ { z } q _ { \\phi } ( z ) f ( z , \\phi ) d z = \\int _ { z } \\frac { \\partial } { \\partial \\phi } q _ { \\phi } ( z ) f ( z , \\phi ) d z = \\int _ { z } f ( z , \\phi ) \\frac { \\partial } { \\partial \\phi } q _ { \\phi } ( z ) + q _ { \\phi } ( z ) \\frac { \\partial } { \\partial \\phi } f ( z , \\phi ) d z } \\\\ & { \\quad \\quad \\quad \\quad = \\int _ { z } f ( z , \\phi ) q _ { \\phi } ( z ) \\frac { \\partial \\log q _ { \\phi } ( z ) } { \\partial \\phi } d z + \\mathbb { E } _ { q _ { \\phi } ( z ) } \\left[ \\frac { \\partial f ( z , \\phi ) } { \\partial \\phi } \\right] } \\\\ & { \\quad \\quad \\quad = \\mathbb { E } _ { q _ { \\phi } ( z ) } \\left[ f ( z , \\phi ) \\frac { \\partial \\log q _ { \\phi } ( z ) } { \\partial \\phi } \\right] + \\mathbb { E } _ { q _ { \\phi } ( z ) } \\left[ \\frac { \\partial f ( z , \\phi ) } { \\partial \\phi } \\right] , } \\end{array}", + "type": "interline_equation", + "image_path": "4c423f2d20ec4f8ea693be4851a2c983319dabb689e10698500a4f325712605d.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 115, + 541, + 496, + 569.3333333333334 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 115, + 569.3333333333334, + 496, + 597.6666666666667 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 115, + 597.6666666666667, + 496, + 626.0000000000001 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 629, + 308, + 641 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 308, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 308, + 642 + ], + "score": 1.0, + "content": "via the REINFORCE gradient. On the other hand,", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 628, + 308, + 642 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 116, + 647, + 496, + 734 + ], + "lines": [ + { + "bbox": [ + 116, + 647, + 496, + 734 + ], + "spans": [ + { + "bbox": [ + 116, + 647, + 496, + 734 + ], + "score": 0.95, + "content": "\\begin{array} { l } { \\displaystyle \\frac { d } { d \\phi } \\int _ { z } q _ { \\phi } ( z ) f ( z , \\phi ) d z = \\frac { d } { d \\phi } \\mathbb { E } _ { q _ { \\phi } ( z ) } \\left[ f ( z , \\phi ) \\right] = \\frac { d } { d \\phi } \\mathbb { E } _ { \\epsilon } \\left[ f ( z ( \\epsilon , \\phi ) , \\phi ) \\right] = \\mathbb { E } _ { \\epsilon } \\left[ \\frac { d } { d \\phi } f ( z ( \\epsilon , \\phi ) , \\phi ) \\right] } \\\\ { = \\mathbb { E } _ { \\epsilon } \\left[ \\frac { \\partial f ( z , \\phi ) } { \\partial z } \\frac { \\partial z ( \\epsilon , \\phi ) } { \\partial \\phi } \\right] + \\mathbb { E } _ { \\epsilon } \\left[ \\frac { \\partial f ( z , \\phi ) } { \\partial \\phi } \\Big | _ { z = z ( \\epsilon , \\phi ) } \\right] } \\\\ { = \\mathbb { E } _ { \\epsilon } \\left[ \\frac { \\partial f ( z , \\phi ) } { \\partial z } \\frac { \\partial z ( \\epsilon , \\phi ) } { \\partial \\phi } \\right] + \\mathbb { E } _ { q _ { \\phi } ( z ) } \\left[ \\frac { \\partial f ( z , \\phi ) } { \\partial \\phi } \\right] , } \\end{array}", + "type": "interline_equation", + "image_path": "cb6984b844cb2ed51b88b049e3ad00081da251901be497873530b869b80bb83b.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 116, + 647, + 496, + 676.0 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 116, + 676.0, + 496, + 705.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 116, + 705.0, + 496, + 734.0 + ], + "spans": [], + "index": 23 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 332, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 332, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 332, + 95 + ], + "score": 1.0, + "content": "via the reparameterization trick. Thus, we conclude that", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 97, + 505, + 140 + ], + "lines": [ + { + "bbox": [ + 111, + 97, + 505, + 125 + ], + "spans": [ + { + "bbox": [ + 111, + 97, + 505, + 125 + ], + "score": 0.89, + "content": "\\Xi _ { q _ { \\phi } ( z ) } \\left[ f ( z , \\phi ) \\frac { \\partial \\log q _ { \\phi } ( z ) } { \\partial \\phi } \\right] + \\mathbb { E } _ { q _ { \\phi } ( z ) } \\left[ \\frac { \\partial f ( z , \\phi ) } { \\partial \\phi } \\right] = \\mathbb { E } _ { \\epsilon } \\left[ \\frac { \\partial f ( z , \\phi ) } { \\partial z } \\frac { \\partial z ( \\epsilon , \\phi ) } { \\partial \\phi } \\right] + \\mathbb { E } _ { q _ { \\phi } ( z ) } \\left[ \\frac { \\partial f ( z , \\phi ) } { \\partial \\phi } \\right] ,", + "type": "inline_equation", + "image_path": "135bc9f8f2e13160a39bb1caae7ab506032fc48009d68e3c0f9bc76391d41d43.jpg" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 126, + 237, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 237, + 141 + ], + "score": 1.0, + "content": "from which the identity follows.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 107, + 153, + 234, + 164 + ], + "lines": [ + { + "bbox": [ + 106, + 153, + 235, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 235, + 165 + ], + "score": 1.0, + "content": "8.2 ASYMPTOTIC ANALYSIS", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 173, + 505, + 248 + ], + "lines": [ + { + "bbox": [ + 106, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "At a high level, Rainforth et al. (2018) show that the expected value of the IWAE gradient of the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 183, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 288, + 198 + ], + "score": 1.0, + "content": "inference network collapses to zero with rate √", + "type": "text" + }, + { + "bbox": [ + 289, + 184, + 309, + 196 + ], + "score": 0.9, + "content": "1 / K", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 183, + 506, + 198 + ], + "score": 1.0, + "content": ", while its standard deviation is only shrinking at", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 195, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 104, + 196, + 142, + 210 + ], + "score": 1.0, + "content": "a rate of", + "type": "text" + }, + { + "bbox": [ + 142, + 195, + 171, + 209 + ], + "score": 0.93, + "content": "1 / \\sqrt { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 196, + 505, + 210 + ], + "score": 1.0, + "content": ". This is the essence of the problem that results in the SNR (expectation divided by", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 208, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 104, + 209, + 409, + 223 + ], + "score": 1.0, + "content": "standard deviation) of the inference network gradients going to zero at a rate √", + "type": "text" + }, + { + "bbox": [ + 409, + 208, + 506, + 222 + ], + "score": 0.95, + "content": "\\mathcal { O } ( ( 1 / K ) / ( 1 / \\sqrt { K } ) ) \\stackrel { - } { = }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 221, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 107, + 221, + 151, + 235 + ], + "score": 0.93, + "content": "\\mathcal { O } ( 1 / \\sqrt { K } )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 222, + 218, + 235 + ], + "score": 1.0, + "content": ", worsening with", + "type": "text" + }, + { + "bbox": [ + 218, + 223, + 229, + 232 + ], + "score": 0.77, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 222, + 505, + 235 + ], + "score": 1.0, + "content": ". In contrast, Rainforth et al. (2018) show that the generation network", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 234, + 306, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 189, + 248 + ], + "score": 1.0, + "content": "gradients scales like", + "type": "text" + }, + { + "bbox": [ + 189, + 234, + 223, + 248 + ], + "score": 0.93, + "content": "\\mathcal { O } ( \\sqrt { K } )", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 235, + 291, + 248 + ], + "score": 1.0, + "content": ", improving with", + "type": "text" + }, + { + "bbox": [ + 292, + 236, + 302, + 245 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 235, + 306, + 248 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 252, + 505, + 320 + ], + "lines": [ + { + "bbox": [ + 105, + 251, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 265 + ], + "score": 1.0, + "content": "Because the IWAE-DReG estimator is unbiased, we cannot hope to change the scaling of the ex-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 172, + 275 + ], + "score": 1.0, + "content": "pected value in", + "type": "text" + }, + { + "bbox": [ + 172, + 264, + 183, + 273 + ], + "score": 0.79, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 263, + 505, + 275 + ], + "score": 1.0, + "content": ", but we can hope to change the scaling of the variance. In particular, in this", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 275, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 505, + 286 + ], + "score": 1.0, + "content": "subsection, we provide an informal argument, via the delta method, that the standard deviation of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 284, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 104, + 284, + 209, + 300 + ], + "score": 1.0, + "content": "IWAE-DReG scales like", + "type": "text" + }, + { + "bbox": [ + 209, + 285, + 238, + 297 + ], + "score": 0.91, + "content": "K ^ { - 3 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 284, + 398, + 300 + ], + "score": 1.0, + "content": ", which results in an overall scaling of", + "type": "text" + }, + { + "bbox": [ + 398, + 285, + 433, + 298 + ], + "score": 0.93, + "content": "\\mathcal { O } ( \\sqrt { K } )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 284, + 506, + 300 + ], + "score": 1.0, + "content": "for the inference", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 296, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 295, + 309 + ], + "score": 1.0, + "content": "network gradient’s SNR (i.e., increasing with", + "type": "text" + }, + { + "bbox": [ + 295, + 298, + 306, + 307 + ], + "score": 0.75, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 296, + 506, + 309 + ], + "score": 1.0, + "content": "). Thus, the SNR of the IWAE-DReG estimator", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 308, + 383, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 194, + 321 + ], + "score": 1.0, + "content": "improves similarly in", + "type": "text" + }, + { + "bbox": [ + 194, + 309, + 204, + 318 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 308, + 383, + 321 + ], + "score": 1.0, + "content": "for both inference and generation networks.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 502, + 348 + ], + "lines": [ + { + "bbox": [ + 105, + 323, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 361, + 339 + ], + "score": 1.0, + "content": "We will appeal to the delta method on a two-variable function", + "type": "text" + }, + { + "bbox": [ + 361, + 325, + 413, + 337 + ], + "score": 0.91, + "content": "g : \\mathbb { R } ^ { 2 } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 323, + 505, + 339 + ], + "score": 1.0, + "content": ". Define the following", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 335, + 417, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 213, + 348 + ], + "score": 1.0, + "content": "notation for the partials of", + "type": "text" + }, + { + "bbox": [ + 213, + 338, + 219, + 348 + ], + "score": 0.83, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 335, + 391, + 348 + ], + "score": 1.0, + "content": "evaluated at the mean of random variables", + "type": "text" + }, + { + "bbox": [ + 391, + 337, + 413, + 348 + ], + "score": 0.89, + "content": "X , Y", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 335, + 417, + 348 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 224, + 351, + 387, + 381 + ], + "lines": [ + { + "bbox": [ + 224, + 351, + 387, + 381 + ], + "spans": [ + { + "bbox": [ + 224, + 351, + 387, + 381 + ], + "score": 0.94, + "content": "g _ { x } ( X , Y ) = \\left. { \\frac { \\partial g ( x , y ) } { \\partial x } } \\right| _ { ( x , y ) = ( \\operatorname { \\mathbb { E } } ( X ) , \\operatorname { \\mathbb { E } } ( Y ) ) }", + "type": "interline_equation", + "image_path": "0550594ef77e41cdf0fe02a2709e60ecc6c162e3e5214baacdf2e5d5f524f94f.jpg" + } + ] + } + ], + "index": 18.5, + "virtual_lines": [ + { + "bbox": [ + 224, + 351, + 387, + 366.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 224, + 366.0, + 387, + 381.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 384, + 491, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 492, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 250, + 400 + ], + "score": 1.0, + "content": "The delta method approximation of", + "type": "text" + }, + { + "bbox": [ + 251, + 384, + 308, + 397 + ], + "score": 0.93, + "content": "\\operatorname { V a r } ( g ( X , Y ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 381, + 492, + 400 + ], + "score": 1.0, + "content": "is given by (Section 5.5 of Casella & Berger),", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 401, + 493, + 415 + ], + "lines": [ + { + "bbox": [ + 118, + 401, + 493, + 415 + ], + "spans": [ + { + "bbox": [ + 118, + 401, + 493, + 415 + ], + "score": 0.9, + "content": "\\mathrm { V a r } ( g ( X , Y ) ) \\approx g _ { x } ( X , Y ) ^ { 2 } \\mathrm { V a r } ( X ) + 2 g _ { x } ( X , Y ) g _ { y } ( X , Y ) \\mathrm { C o v } ( X , Y ) + g _ { y } ( X , Y ) ^ { 2 } \\mathrm { V a r } ( Y )", + "type": "interline_equation", + "image_path": "b3b10e38c9369fa68d8f5f1c312fea2ad3f4e359acac9ae5c882918c57faa511.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 118, + 401, + 493, + 415 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 419, + 504, + 468 + ], + "lines": [ + { + "bbox": [ + 163, + 417, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 163, + 417, + 168, + 454 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 230, + 417, + 252, + 454 + ], + "score": 1.0, + "content": "f gen, and", + "type": "text" + }, + { + "bbox": [ + 300, + 420, + 307, + 430 + ], + "score": 0.83, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 417, + 339, + 454 + ], + "score": 1.0, + "content": "s a si. Let", + "type": "text" + }, + { + "bbox": [ + 417, + 417, + 443, + 454 + ], + "score": 1.0, + "content": "aramet, then", + "type": "text" + }, + { + "bbox": [ + 479, + 420, + 505, + 430 + ], + "score": 0.84, + "content": "u _ { i } \\ =", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 430, + 478, + 446 + ], + "spans": [ + { + "bbox": [ + 107, + 430, + 162, + 446 + ], + "score": 0.89, + "content": "w _ { i } ^ { 2 } \\frac { \\partial \\log { w _ { i } } } { \\partial z _ { i } } \\frac { \\partial z _ { i } } { \\partial \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 430, + 230, + 445 + ], + "score": 0.89, + "content": "\\textstyle X = \\sum _ { i = 1 } ^ { K } u _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 430, + 315, + 445 + ], + "score": 0.93, + "content": "\\textstyle Y = \\sum _ { i = 1 } ^ { K } w _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 431, + 417, + 444 + ], + "score": 0.93, + "content": "g ( X , Y ) = X / Y ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 432, + 478, + 444 + ], + "score": 0.93, + "content": "g ( X , Y )", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 443, + 504, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 392, + 458 + ], + "score": 1.0, + "content": "IWAE-DReG estimator whose variance we seek to understand. Letting", + "type": "text" + }, + { + "bbox": [ + 392, + 445, + 439, + 457 + ], + "score": 0.93, + "content": "Z = \\mathbb { E } ( w _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 443, + 457, + 458 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 458, + 445, + 504, + 457 + ], + "score": 0.93, + "content": "U = \\mathbb { E } ( u _ { i } )", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 456, + 260, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 260, + 469 + ], + "score": 1.0, + "content": "we get in this case after cancellations,", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 471, + 437, + 497 + ], + "lines": [ + { + "bbox": [ + 173, + 471, + 437, + 497 + ], + "spans": [ + { + "bbox": [ + 173, + 471, + 437, + 497 + ], + "score": 0.91, + "content": "\\mathrm { V a r } ( g ( X , Y ) ) \\approx \\frac { 1 } { Z ^ { 4 } } \\frac { \\mathrm { V a r } ( X ) } { K ^ { 4 } } - \\frac { 4 U } { Z ^ { 5 } } \\frac { \\mathrm { C o v } ( X , Y ) } { K ^ { 4 } } + \\frac { 4 U ^ { 2 } } { Z ^ { 6 } } \\frac { \\mathrm { V a r } ( Y ) } { K ^ { 4 } }", + "type": "interline_equation", + "image_path": "5c80068d419799d37733f6db8ae9022d943350366082e7dad348c6623c447a29.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 173, + 471, + 437, + 497 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 505, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 504, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 143, + 513 + ], + "score": 1.0, + "content": "Because", + "type": "text" + }, + { + "bbox": [ + 144, + 502, + 155, + 511 + ], + "score": 0.86, + "content": "w _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 498, + 312, + 513 + ], + "score": 1.0, + "content": "are all mutually independent, we get", + "type": "text" + }, + { + "bbox": [ + 313, + 500, + 406, + 512 + ], + "score": 0.93, + "content": "\\operatorname { V a r } ( Y ) = K \\operatorname { V a r } ( w _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 498, + 470, + 513 + ], + "score": 1.0, + "content": ". Similarly for", + "type": "text" + }, + { + "bbox": [ + 470, + 500, + 504, + 512 + ], + "score": 0.9, + "content": "\\operatorname { V a r } ( X )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 124, + 524 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 125, + 513, + 135, + 522 + ], + "score": 0.82, + "content": "u _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 510, + 196, + 524 + ], + "score": 1.0, + "content": ". Because the", + "type": "text" + }, + { + "bbox": [ + 196, + 513, + 208, + 522 + ], + "score": 0.83, + "content": "w _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 510, + 228, + 524 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 229, + 513, + 239, + 522 + ], + "score": 0.83, + "content": "u _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 510, + 435, + 524 + ], + "score": 1.0, + "content": "are identically distributed and independent for", + "type": "text" + }, + { + "bbox": [ + 436, + 512, + 463, + 523 + ], + "score": 0.9, + "content": "i \\neq j", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 510, + 506, + 524 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 520, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 227, + 534 + ], + "score": 0.92, + "content": "\\operatorname { C o v } ( X , Y ) = K \\operatorname { C o v } ( w _ { i } , u _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 520, + 349, + 535 + ], + "score": 1.0, + "content": ". All together we can see that", + "type": "text" + }, + { + "bbox": [ + 349, + 522, + 406, + 534 + ], + "score": 0.93, + "content": "\\mathrm { V a r } ( g ( \\bar { X , Y } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 520, + 453, + 535 + ], + "score": 1.0, + "content": "scales like", + "type": "text" + }, + { + "bbox": [ + 453, + 522, + 474, + 532 + ], + "score": 0.89, + "content": "\\dot { K } ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 520, + 506, + 535 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 533, + 273, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 240, + 545 + ], + "score": 1.0, + "content": "the standard deviation scales like", + "type": "text" + }, + { + "bbox": [ + 240, + 533, + 269, + 545 + ], + "score": 0.91, + "content": "K ^ { - 3 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 534, + 273, + 545 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 106, + 558, + 356, + 571 + ], + "lines": [ + { + "bbox": [ + 105, + 558, + 357, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 357, + 572 + ], + "score": 1.0, + "content": "8.3 UNIFIED SURROGATE OBJECTIVES FOR ESTIMATORS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 579, + 505, + 635 + ], + "lines": [ + { + "bbox": [ + 106, + 579, + 504, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 242, + 591 + ], + "score": 1.0, + "content": "In the main text, we assumed that", + "type": "text" + }, + { + "bbox": [ + 242, + 580, + 249, + 590 + ], + "score": 0.8, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 579, + 266, + 591 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 266, + 580, + 274, + 591 + ], + "score": 0.86, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 579, + 504, + 591 + ], + "score": 1.0, + "content": "were disjoint, however, it can be helpful to share parame-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 159, + 603 + ], + "score": 1.0, + "content": "ters between", + "type": "text" + }, + { + "bbox": [ + 159, + 593, + 166, + 603 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 591, + 183, + 603 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 184, + 592, + 190, + 603 + ], + "score": 0.78, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "(e.g., (Fraccaro et al., 2016)). With the IWAE bound, we differentiate a single", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 241, + 614 + ], + "score": 1.0, + "content": "objective with respect to both the", + "type": "text" + }, + { + "bbox": [ + 241, + 603, + 248, + 613 + ], + "score": 0.82, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 601, + 266, + 614 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 266, + 604, + 272, + 613 + ], + "score": 0.79, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "parameters. Thus it is straightforward to adapt IWAE and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "score": 1.0, + "content": "IWAE-DReG to the shared parameter setting. In this section, we discuss how to deal with shared", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 625, + 188, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 188, + 635 + ], + "score": 1.0, + "content": "parameters in RWS.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 640, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 182, + 654 + ], + "score": 1.0, + "content": "Suppose that both", + "type": "text" + }, + { + "bbox": [ + 182, + 642, + 189, + 652 + ], + "score": 0.82, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 640, + 208, + 654 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 208, + 643, + 215, + 652 + ], + "score": 0.81, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 640, + 304, + 654 + ], + "score": 1.0, + "content": "are parameterized by", + "type": "text" + }, + { + "bbox": [ + 304, + 641, + 311, + 650 + ], + "score": 0.77, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 640, + 484, + 654 + ], + "score": 1.0, + "content": ". If we denote the unshared parameters of", + "type": "text" + }, + { + "bbox": [ + 484, + 643, + 491, + 652 + ], + "score": 0.81, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 640, + 505, + 654 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 651, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 107, + 652, + 114, + 663 + ], + "score": 0.82, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 651, + 322, + 664 + ], + "score": 1.0, + "content": ", then we can restrict the RWS wake update to only", + "type": "text" + }, + { + "bbox": [ + 322, + 652, + 329, + 663 + ], + "score": 0.83, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 651, + 506, + 664 + ], + "score": 1.0, + "content": ". 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Letting", + "type": "text" + }, + { + "bbox": [ + 392, + 445, + 439, + 457 + ], + "score": 0.93, + "content": "Z = \\mathbb { E } ( w _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 443, + 457, + 458 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 458, + 445, + 504, + 457 + ], + "score": 0.93, + "content": "U = \\mathbb { E } ( u _ { i } )", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 456, + 260, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 260, + 469 + ], + "score": 1.0, + "content": "we get in this case after cancellations,", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 417, + 505, + 469 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 471, + 437, + 497 + ], + "lines": [ + { + "bbox": [ + 173, + 471, + 437, + 497 + ], + "spans": [ + { + "bbox": [ + 173, + 471, + 437, + 497 + ], + "score": 0.91, + "content": "\\mathrm { V a r } ( g ( X , Y ) ) \\approx \\frac { 1 } { Z ^ { 4 } } \\frac { \\mathrm { V a r } ( X ) } { K ^ { 4 } } - \\frac { 4 U } { Z ^ { 5 } } \\frac { \\mathrm { C o v } ( X , Y ) } { K ^ { 4 } } + \\frac { 4 U ^ { 2 } } { Z ^ { 6 } } \\frac { \\mathrm { V a r } ( Y ) } { K ^ { 4 } }", + "type": "interline_equation", + "image_path": "5c80068d419799d37733f6db8ae9022d943350366082e7dad348c6623c447a29.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 173, + 471, + 437, + 497 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 505, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 504, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 143, + 513 + ], + "score": 1.0, + "content": "Because", + "type": "text" + }, + { + "bbox": [ + 144, + 502, + 155, + 511 + ], + "score": 0.86, + "content": "w _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 498, + 312, + 513 + ], + "score": 1.0, + "content": "are all mutually independent, we get", + "type": "text" + }, + { + "bbox": [ + 313, + 500, + 406, + 512 + ], + "score": 0.93, + "content": "\\operatorname { V a r } ( Y ) = K \\operatorname { V a r } ( w _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 498, + 470, + 513 + ], + "score": 1.0, + "content": ". Similarly for", + "type": "text" + }, + { + "bbox": [ + 470, + 500, + 504, + 512 + ], + "score": 0.9, + "content": "\\operatorname { V a r } ( X )", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 124, + 524 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 125, + 513, + 135, + 522 + ], + "score": 0.82, + "content": "u _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 510, + 196, + 524 + ], + "score": 1.0, + "content": ". Because the", + "type": "text" + }, + { + "bbox": [ + 196, + 513, + 208, + 522 + ], + "score": 0.83, + "content": "w _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 510, + 228, + 524 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 229, + 513, + 239, + 522 + ], + "score": 0.83, + "content": "u _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 510, + 435, + 524 + ], + "score": 1.0, + "content": "are identically distributed and independent for", + "type": "text" + }, + { + "bbox": [ + 436, + 512, + 463, + 523 + ], + "score": 0.9, + "content": "i \\neq j", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 510, + 506, + 524 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 520, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 227, + 534 + ], + "score": 0.92, + "content": "\\operatorname { C o v } ( X , Y ) = K \\operatorname { C o v } ( w _ { i } , u _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 520, + 349, + 535 + ], + "score": 1.0, + "content": ". All together we can see that", + "type": "text" + }, + { + "bbox": [ + 349, + 522, + 406, + 534 + ], + "score": 0.93, + "content": "\\mathrm { V a r } ( g ( \\bar { X , Y } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 520, + 453, + 535 + ], + "score": 1.0, + "content": "scales like", + "type": "text" + }, + { + "bbox": [ + 453, + 522, + 474, + 532 + ], + "score": 0.89, + "content": "\\dot { K } ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 520, + 506, + 535 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 533, + 273, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 240, + 545 + ], + "score": 1.0, + "content": "the standard deviation scales like", + "type": "text" + }, + { + "bbox": [ + 240, + 533, + 269, + 545 + ], + "score": 0.91, + "content": "K ^ { - 3 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 534, + 273, + 545 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 498, + 506, + 545 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 558, + 356, + 571 + ], + "lines": [ + { + "bbox": [ + 105, + 558, + 357, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 357, + 572 + ], + "score": 1.0, + "content": "8.3 UNIFIED SURROGATE OBJECTIVES FOR ESTIMATORS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 579, + 505, + 635 + ], + "lines": [ + { + "bbox": [ + 106, + 579, + 504, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 242, + 591 + ], + "score": 1.0, + "content": "In the main text, we assumed that", + "type": "text" + }, + { + "bbox": [ + 242, + 580, + 249, + 590 + ], + "score": 0.8, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 579, + 266, + 591 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 266, + 580, + 274, + 591 + ], + "score": 0.86, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 579, + 504, + 591 + ], + "score": 1.0, + "content": "were disjoint, however, it can be helpful to share parame-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 159, + 603 + ], + "score": 1.0, + "content": "ters between", + "type": "text" + }, + { + "bbox": [ + 159, + 593, + 166, + 603 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 591, + 183, + 603 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 184, + 592, + 190, + 603 + ], + "score": 0.78, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "(e.g., (Fraccaro et al., 2016)). With the IWAE bound, we differentiate a single", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 241, + 614 + ], + "score": 1.0, + "content": "objective with respect to both the", + "type": "text" + }, + { + "bbox": [ + 241, + 603, + 248, + 613 + ], + "score": 0.82, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 601, + 266, + 614 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 266, + 604, + 272, + 613 + ], + "score": 0.79, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "parameters. Thus it is straightforward to adapt IWAE and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "score": 1.0, + "content": "IWAE-DReG to the shared parameter setting. In this section, we discuss how to deal with shared", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 625, + 188, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 188, + 635 + ], + "score": 1.0, + "content": "parameters in RWS.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 579, + 506, + 635 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 640, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 182, + 654 + ], + "score": 1.0, + "content": "Suppose that both", + "type": "text" + }, + { + "bbox": [ + 182, + 642, + 189, + 652 + ], + "score": 0.82, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 640, + 208, + 654 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 208, + 643, + 215, + 652 + ], + "score": 0.81, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 640, + 304, + 654 + ], + "score": 1.0, + "content": "are parameterized by", + "type": "text" + }, + { + "bbox": [ + 304, + 641, + 311, + 650 + ], + "score": 0.77, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 640, + 484, + 654 + ], + "score": 1.0, + "content": ". If we denote the unshared parameters of", + "type": "text" + }, + { + "bbox": [ + 484, + 643, + 491, + 652 + ], + "score": 0.81, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 640, + 505, + 654 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 651, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 107, + 652, + 114, + 663 + ], + "score": 0.82, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 651, + 322, + 664 + ], + "score": 1.0, + "content": ", then we can restrict the RWS wake update to only", + "type": "text" + }, + { + "bbox": [ + 322, + 652, + 329, + 663 + ], + "score": 0.83, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 651, + 506, + 664 + ], + "score": 1.0, + "content": ". 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