diff --git "a/parse/train/S1EwLkW0W/S1EwLkW0W_middle.json" "b/parse/train/S1EwLkW0W/S1EwLkW0W_middle.json" new file mode 100644--- /dev/null +++ "b/parse/train/S1EwLkW0W/S1EwLkW0W_middle.json" @@ -0,0 +1,50497 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 475, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 477, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 477, + 97 + ], + "score": 1.0, + "content": "DISSECTING ADAM: THE SIGN, MAGNITUDE AND", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 97, + 403, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 97, + 403, + 117 + ], + "score": 1.0, + "content": "VARIANCE OF STOCHASTIC GRADIENTS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 136, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 277, + 186, + 335, + 200 + ], + "spans": [ + { + "bbox": [ + 277, + 186, + 335, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 142, + 213, + 468, + 302 + ], + "lines": [ + { + "bbox": [ + 142, + 213, + 470, + 227 + ], + "spans": [ + { + "bbox": [ + 142, + 213, + 470, + 227 + ], + "score": 1.0, + "content": "The ADAM optimizer is exceedingly popular in the deep learning community. 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We will denote by", + "type": "text" + }, + { + "bbox": [ + 403, + 518, + 491, + 532 + ], + "score": 0.95, + "content": "\\sigma ( \\theta ) _ { i } ^ { 2 } : = { \\bf v a r } [ g ( \\theta ) _ { i } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 518, + 505, + 532 + ], + "score": 1.0, + "content": "its", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 530, + 208, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 208, + 542 + ], + "score": 1.0, + "content": "element-wise variances.1", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 547, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 561 + ], + "score": 1.0, + "content": "The basic stochastic optimizer is stochastic gradient descent (SGD, Robbins & Monro, 1951) and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 558, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 572 + ], + "score": 1.0, + "content": "its momentum variants (Polyak, 1964; Nesterov, 1983). 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We disentangle these two aspects", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 269, + 470, + 281 + ], + "spans": [ + { + "bbox": [ + 142, + 269, + 470, + 281 + ], + "score": 1.0, + "content": "and analyze them in isolation, shedding light on ADAM’s inner workings. Trans-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 280, + 469, + 292 + ], + "spans": [ + { + "bbox": [ + 141, + 280, + 469, + 292 + ], + "score": 1.0, + "content": "ferring the “variance adaptation” to momentum-SGD gives rise to a novel method,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 290, + 424, + 302 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 424, + 302 + ], + "score": 1.0, + "content": "completing the practitioner’s toolbox for problems where ADAM fails.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5, + "bbox_fs": [ + 141, + 213, + 470, + 302 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 327, + 206, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 208, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 208, + 343 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 104, + 354, + 498, + 366 + ], + "lines": [ + { + "bbox": [ + 106, + 353, + 498, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 498, + 367 + ], + "score": 1.0, + "content": "Many prominent machine learning models pose empirical risk minimization problems of the form", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 353, + 498, + 367 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 373, + 461, + 408 + ], + "lines": [ + { + "bbox": [ + 149, + 373, + 461, + 408 + ], + "spans": [ + { + "bbox": [ + 149, + 373, + 461, + 408 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { \\theta \\in \\mathbb { R } ^ { d } } \\mathcal { L } ( \\theta ) = \\frac { 1 } { M } \\sum _ { k = 1 } ^ { M } \\ell ( \\theta ; x _ { k } ) , \\quad \\mathrm { w i t h ~ g r a d i e n t } \\quad \\nabla \\mathcal { L } ( \\theta ) = \\frac { 1 } { M } \\sum _ { k = 1 } ^ { M } \\nabla \\ell ( \\theta ; x _ { k } ) ,", + "type": "interline_equation", + "image_path": "2827f491b949607875f342141ffb066fe6a6658505b873fdfb21b89d89dfe206.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 149, + 373, + 461, + 384.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 149, + 384.6666666666667, + 461, + 396.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 149, + 396.33333333333337, + 461, + 408.00000000000006 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 416, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 415, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 133, + 431 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 416, + 163, + 428 + ], + "score": 0.92, + "content": "\\theta \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 415, + 265, + 431 + ], + "score": 1.0, + "content": "is a vector of parameters,", + "type": "text" + }, + { + "bbox": [ + 265, + 417, + 322, + 429 + ], + "score": 0.93, + "content": "\\{ x _ { 1 } , \\ldots , x _ { M } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 415, + 395, + 431 + ], + "score": 1.0, + "content": "a training set, and", + "type": "text" + }, + { + "bbox": [ + 395, + 417, + 423, + 429 + ], + "score": 0.93, + "content": "\\ell ( \\theta ; x )", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 415, + 506, + 431 + ], + "score": 1.0, + "content": "is a loss quantifying", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 255, + 441 + ], + "score": 1.0, + "content": "the performance of parameter vector", + "type": "text" + }, + { + "bbox": [ + 255, + 429, + 262, + 438 + ], + "score": 0.81, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 428, + 311, + 441 + ], + "score": 1.0, + "content": "on example", + "type": "text" + }, + { + "bbox": [ + 312, + 431, + 318, + 438 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 428, + 506, + 441 + ], + "score": 1.0, + "content": ". Computing the exact gradient in each step of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 438, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 367, + 452 + ], + "score": 1.0, + "content": "an iterative optimization algorithm becomes inefficient for large", + "type": "text" + }, + { + "bbox": [ + 367, + 440, + 379, + 449 + ], + "score": 0.8, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 438, + 505, + 452 + ], + "score": 1.0, + "content": ". Instead, we construct a mini-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 449, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 131, + 463 + ], + "score": 1.0, + "content": "batch", + "type": "text" + }, + { + "bbox": [ + 131, + 450, + 204, + 462 + ], + "score": 0.93, + "content": "B \\subset \\{ 1 , \\ldots , M \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 449, + 218, + 463 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 218, + 450, + 261, + 462 + ], + "score": 0.93, + "content": "| B | \\ll M", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 449, + 506, + 463 + ], + "score": 1.0, + "content": "data points sampled uniformly and independently from the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 461, + 351, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 351, + 474 + ], + "score": 1.0, + "content": "training set and compute an approximate stochastic gradient", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 415, + 506, + 474 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 480, + 360, + 510 + ], + "lines": [ + { + "bbox": [ + 250, + 480, + 360, + 510 + ], + "spans": [ + { + "bbox": [ + 250, + 480, + 360, + 510 + ], + "score": 0.95, + "content": "g ( \\theta ) = \\frac { 1 } { | \\boldsymbol { \\mathcal { B } } | } \\sum _ { \\boldsymbol { k } \\in \\boldsymbol { B } } \\nabla \\ell ( \\theta ; x _ { k } ) ,", + "type": "interline_equation", + "image_path": "0e483b3452a8d15297f6af43a367d40e816216fa0769026f29b71ee5c64b3d78.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 250, + 480, + 360, + 495.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 250, + 495.0, + 360, + 510.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 518, + 504, + 542 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 237, + 532 + ], + "score": 1.0, + "content": "which is an unbiased estimate,", + "type": "text" + }, + { + "bbox": [ + 237, + 519, + 315, + 531 + ], + "score": 0.93, + "content": "\\mathbf { E } [ g ( \\theta ) ] ~ = ~ \\nabla \\mathcal { L } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 518, + 403, + 532 + ], + "score": 1.0, + "content": ". We will denote by", + "type": "text" + }, + { + "bbox": [ + 403, + 518, + 491, + 532 + ], + "score": 0.95, + "content": "\\sigma ( \\theta ) _ { i } ^ { 2 } : = { \\bf v a r } [ g ( \\theta ) _ { i } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 518, + 505, + 532 + ], + "score": 1.0, + "content": "its", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 530, + 208, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 208, + 542 + ], + "score": 1.0, + "content": "element-wise variances.1", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 518, + 505, + 542 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 547, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 561 + ], + "score": 1.0, + "content": "The basic stochastic optimizer is stochastic gradient descent (SGD, Robbins & Monro, 1951) and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 558, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 572 + ], + "score": 1.0, + "content": "its momentum variants (Polyak, 1964; Nesterov, 1983). A number of methods, widely-used in the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "deep learning community, choose per-element update magnitudes based on the history of stochastic", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 580, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 506, + 594 + ], + "score": 1.0, + "content": "gradient observations. Among these are ADAGRAD (Duchi et al., 2011), RMSPROP (Tieleman &", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 591, + 411, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 411, + 604 + ], + "score": 1.0, + "content": "Hinton, 2012), ADADELTA (Zeiler, 2012) and ADAM (Kingma & Ba, 2015).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 547, + 506, + 604 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 620, + 248, + 631 + ], + "lines": [ + { + "bbox": [ + 106, + 619, + 249, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 249, + 632 + ], + "score": 1.0, + "content": "1.1 A CLOSER LOOK AT ADAM", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 641, + 505, + 675 + ], + "lines": [ + { + "bbox": [ + 106, + 641, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 506, + 655 + ], + "score": 1.0, + "content": "We start out from a reinterpretation of the widely-used ADAM optimizer. 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The left panel", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 235, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 104, + 235, + 204, + 250 + ], + "score": 1.0, + "content": "shows the true gradient", + "type": "text" + }, + { + "bbox": [ + 204, + 237, + 258, + 249 + ], + "score": 0.93, + "content": "\\nabla { \\mathcal { L } } = ( 2 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 235, + 459, + 250 + ], + "score": 1.0, + "content": "and stochastic gradients scattered around it with", + "type": "text" + }, + { + "bbox": [ + 459, + 236, + 505, + 249 + ], + "score": 0.92, + "content": "( \\sigma _ { 1 } , \\bar { \\sigma _ { 2 } } ) =", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 423, + 261 + ], + "score": 1.0, + "content": "(1, 1.5). In the right panel, we employ a variance adaptation (to be derived in", + "type": "text" + }, + { + "bbox": [ + 423, + 248, + 444, + 259 + ], + "score": 0.63, + "content": "\\ S 3 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "that scales the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 257, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 107, + 260, + 111, + 268 + ], + "score": 0.78, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 111, + 257, + 181, + 272 + ], + "score": 1.0, + "content": "-th coordinate by", + "type": "text" + }, + { + "bbox": [ + 181, + 258, + 226, + 270 + ], + "score": 0.92, + "content": "( 1 \\dot { + } \\eta _ { i } ^ { 2 } ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 257, + 311, + 272 + ], + "score": 1.0, + "content": ". In this example, the", + "type": "text" + }, + { + "bbox": [ + 312, + 259, + 322, + 270 + ], + "score": 0.89, + "content": "\\theta _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 257, + 506, + 272 + ], + "score": 1.0, + "content": "-coordinate has much higher relative variance", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 108, + 268, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 108, + 269, + 155, + 281 + ], + "score": 0.88, + "content": "( \\eta _ { 2 } ^ { 2 } = 2 . 2 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 268, + 191, + 282 + ], + "score": 1.0, + "content": "than the", + "type": "text" + }, + { + "bbox": [ + 192, + 270, + 202, + 281 + ], + "score": 0.87, + "content": "\\theta _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 268, + 251, + 282 + ], + "score": 1.0, + "content": "-coordinate", + "type": "text" + }, + { + "bbox": [ + 251, + 269, + 298, + 281 + ], + "score": 0.89, + "content": "( \\eta _ { 1 } ^ { 2 } = 0 . 2 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 268, + 506, + 282 + ], + "score": 1.0, + "content": "and is thus shortened. This reduces the variance of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 280, + 471, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 471, + 293 + ], + "score": 1.0, + "content": "the update direction at the expense of biasing it away from the true gradient in expectation.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 285, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 287, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 287, + 328 + ], + "score": 1.0, + "content": "tic gradients and their element-wise square2,", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 330, + 439, + 365 + ], + "lines": [ + { + "bbox": [ + 171, + 331, + 439, + 365 + ], + "spans": [ + { + "bbox": [ + 171, + 331, + 439, + 365 + ], + "score": 0.89, + "content": "\\begin{array} { r l } & { \\tilde { m } _ { t } = \\beta _ { 1 } \\tilde { m } _ { t - 1 } + ( 1 - \\beta _ { 1 } ) g _ { t } , \\qquad m _ { t } = ( 1 - \\beta _ { 1 } ^ { t } ) ^ { - 1 } \\tilde { m } _ { t } , } \\\\ & { \\tilde { v } _ { t } = \\beta _ { 2 } \\tilde { v } _ { t - 1 } + ( 1 - \\beta _ { 2 } ) g _ { t } ^ { 2 } , \\qquad v _ { t } = ( 1 - \\beta _ { 2 } ^ { t } ) ^ { - 1 } \\tilde { v } _ { t } . } \\end{array}", + "type": "interline_equation", + "image_path": "40acc415dc093489e11b98ca5fa224743a665dbc37328222f6b4a96dabc1459e.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 171, + 330, + 439, + 341.6666666666667 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 171, + 341.6666666666667, + 439, + 353.33333333333337 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 171, + 353.33333333333337, + 439, + 365.00000000000006 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 369, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 131, + 383 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 131, + 371, + 144, + 380 + ], + "score": 0.84, + "content": "m _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 367, + 163, + 383 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 163, + 371, + 172, + 380 + ], + "score": 0.86, + "content": "v _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 367, + 505, + 383 + ], + "score": 1.0, + "content": "are “bias-corrected” versions of the exponential moving averages to obtain convex", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 380, + 394, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 394, + 393 + ], + "score": 1.0, + "content": "combinations of past observed (squared) gradients. 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Ignoring", + "type": "text" + }, + { + "bbox": [ + 451, + 432, + 456, + 439 + ], + "score": 0.75, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 429, + 504, + 442 + ], + "score": 1.0, + "content": "and assum-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 440, + 389, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 122, + 452 + ], + "score": 1.0, + "content": "ing", + "type": "text" + }, + { + "bbox": [ + 122, + 440, + 164, + 453 + ], + "score": 0.93, + "content": "| m _ { t , i } | > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 440, + 389, + 452 + ], + "score": 1.0, + "content": "for the moment, we can rewrite the update direction as3", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 459, + 415, + 495 + ], + "lines": [ + { + "bbox": [ + 197, + 459, + 415, + 495 + ], + "spans": [ + { + "bbox": [ + 197, + 459, + 415, + 495 + ], + "score": 0.94, + "content": "{ \\frac { m _ { t } } { \\sqrt { v _ { t } } } } = { \\frac { \\mathrm { s i g n } ( m _ { t } ) | m _ { t } | } { \\sqrt { v _ { t } } } } = { \\frac { \\mathrm { s i g n } ( m _ { t } ) } { \\sqrt { \\frac { v _ { t } } { m _ { t } ^ { 2 } } } } } = { \\frac { \\mathrm { s i g n } ( m _ { t } ) } { \\sqrt { 1 + { \\frac { v _ { t } - m _ { t } ^ { 2 } } { m _ { t } ^ { 2 } } } } } } .", + "type": "interline_equation", + "image_path": "f2ff03bacef7b932faaaf43778b06143be45035df7d9b28eda66e8af8b171b43.jpg" + } + ] + } + ], + "index": 18.5, + "virtual_lines": [ + { + "bbox": [ + 197, + 459, + 415, + 477.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 197, + 477.0, + 415, + 495.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 131, + 514 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 502, + 144, + 512 + ], + "score": 0.84, + "content": "m _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 500, + 162, + 514 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 162, + 503, + 172, + 512 + ], + "score": 0.83, + "content": "v _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 500, + 440, + 514 + ], + "score": 1.0, + "content": "approximate the first and second moment of the stochastic gradient", + "type": "text" + }, + { + "bbox": [ + 440, + 503, + 449, + 512 + ], + "score": 0.83, + "content": "g _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 500, + 505, + 514 + ], + "score": 1.0, + "content": ", respectively,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 107, + 512, + 141, + 524 + ], + "score": 0.9, + "content": "v _ { t } - m _ { t } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "can be seen as an estimate of element-wise stochastic gradient variances. The division by", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 383, + 535 + ], + "score": 1.0, + "content": "the non-central second moment effectively removes the magnitude of", + "type": "text" + }, + { + "bbox": [ + 383, + 524, + 396, + 534 + ], + "score": 0.86, + "content": "m _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "; it only appears in the ratio", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 533, + 493, + 547 + ], + "spans": [ + { + "bbox": [ + 107, + 533, + 167, + 546 + ], + "score": 0.92, + "content": "( v _ { t } - m _ { t } ^ { 2 } ) / m _ { t } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 533, + 493, + 547 + ], + "score": 1.0, + "content": ". 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Fig. 1 shows a sketch of this variance adaptation.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 680, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 117, + 677, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 117, + 677, + 415, + 694 + ], + "score": 1.0, + "content": "2 Notation: Divisions, squares, etc. on vectors are to be understood element-wise.", + "type": "text" + }, + { + "bbox": [ + 415, + 682, + 424, + 690 + ], + "score": 0.79, + "content": "\\odot", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 677, + 506, + 694 + ], + "score": 1.0, + "content": "denotes element-wise", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 688, + 507, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 245, + 704 + ], + "score": 1.0, + "content": "multiplication. 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This reduces the variance of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 280, + 471, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 471, + 293 + ], + "score": 1.0, + "content": "the update direction at the expense of biasing it away from the true gradient in expectation.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 285, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 312, + 287, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 287, + 328 + ], + "score": 1.0, + "content": "tic gradients and their element-wise square2,", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 312, + 287, + 328 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 330, + 439, + 365 + ], + "lines": [ + { + "bbox": [ + 171, + 331, + 439, + 365 + ], + "spans": [ + { + "bbox": [ + 171, + 331, + 439, + 365 + ], + "score": 0.89, + "content": "\\begin{array} { r l } & { \\tilde { m } _ { t } = \\beta _ { 1 } \\tilde { m } _ { t - 1 } + ( 1 - \\beta _ { 1 } ) g _ { t } , \\qquad m _ { t } = ( 1 - \\beta _ { 1 } ^ { t } ) ^ { - 1 } \\tilde { m } _ { t } , } \\\\ & { \\tilde { v } _ { t } = \\beta _ { 2 } \\tilde { v } _ { t - 1 } + ( 1 - \\beta _ { 2 } ) g _ { t } ^ { 2 } , \\qquad v _ { t } = ( 1 - \\beta _ { 2 } ^ { t } ) ^ { - 1 } \\tilde { v } _ { t } . } \\end{array}", + "type": "interline_equation", + "image_path": "40acc415dc093489e11b98ca5fa224743a665dbc37328222f6b4a96dabc1459e.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 171, + 330, + 439, + 341.6666666666667 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 171, + 341.6666666666667, + 439, + 353.33333333333337 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 171, + 353.33333333333337, + 439, + 365.00000000000006 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 369, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 131, + 383 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 131, + 371, + 144, + 380 + ], + "score": 0.84, + "content": "m _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 367, + 163, + 383 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 163, + 371, + 172, + 380 + ], + "score": 0.86, + "content": "v _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 367, + 505, + 383 + ], + "score": 1.0, + "content": "are “bias-corrected” versions of the exponential moving averages to obtain convex", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 380, + 394, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 394, + 393 + ], + "score": 1.0, + "content": "combinations of past observed (squared) gradients. 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Ignoring", + "type": "text" + }, + { + "bbox": [ + 451, + 432, + 456, + 439 + ], + "score": 0.75, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 429, + 504, + 442 + ], + "score": 1.0, + "content": "and assum-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 440, + 389, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 122, + 452 + ], + "score": 1.0, + "content": "ing", + "type": "text" + }, + { + "bbox": [ + 122, + 440, + 164, + 453 + ], + "score": 0.93, + "content": "| m _ { t , i } | > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 440, + 389, + 452 + ], + "score": 1.0, + "content": "for the moment, we can rewrite the update direction as3", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 429, + 504, + 453 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 197, + 459, + 415, + 495 + ], + "lines": [ + { + "bbox": [ + 197, + 459, + 415, + 495 + ], + "spans": [ + { + "bbox": [ + 197, + 459, + 415, + 495 + ], + "score": 0.94, + "content": "{ \\frac { m _ { t } } { \\sqrt { v _ { t } } } } = { \\frac { \\mathrm { s i g n } ( m _ { t } ) | m _ { t } | } { \\sqrt { v _ { t } } } } = { \\frac { \\mathrm { s i g n } ( m _ { t } ) } { \\sqrt { \\frac { v _ { t } } { m _ { t } ^ { 2 } } } } } = { \\frac { \\mathrm { s i g n } ( m _ { t } ) } { \\sqrt { 1 + { \\frac { v _ { t } - m _ { t } ^ { 2 } } { m _ { t } ^ { 2 } } } } } } .", + "type": "interline_equation", + "image_path": "f2ff03bacef7b932faaaf43778b06143be45035df7d9b28eda66e8af8b171b43.jpg" + } + ] + } + ], + "index": 18.5, + "virtual_lines": [ + { + "bbox": [ + 197, + 459, + 415, + 477.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 197, + 477.0, + 415, + 495.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 131, + 514 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 502, + 144, + 512 + ], + "score": 0.84, + "content": "m _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 500, + 162, + 514 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 162, + 503, + 172, + 512 + ], + "score": 0.83, + "content": "v _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 500, + 440, + 514 + ], + "score": 1.0, + "content": "approximate the first and second moment of the stochastic gradient", + "type": "text" + }, + { + "bbox": [ + 440, + 503, + 449, + 512 + ], + "score": 0.83, + "content": "g _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 500, + 505, + 514 + ], + "score": 1.0, + "content": ", respectively,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 107, + 512, + 141, + 524 + ], + "score": 0.9, + "content": "v _ { t } - m _ { t } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "can be seen as an estimate of element-wise stochastic gradient variances. The division by", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 383, + 535 + ], + "score": 1.0, + "content": "the non-central second moment effectively removes the magnitude of", + "type": "text" + }, + { + "bbox": [ + 383, + 524, + 396, + 534 + ], + "score": 0.86, + "content": "m _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "; it only appears in the ratio", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 533, + 493, + 547 + ], + "spans": [ + { + "bbox": [ + 107, + 533, + 167, + 546 + ], + "score": 0.92, + "content": "( v _ { t } - m _ { t } ^ { 2 } ) / m _ { t } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 533, + 493, + 547 + ], + "score": 1.0, + "content": ". 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Fig. 1 shows a sketch of this variance adaptation.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 139, + 635, + 506, + 666 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 118, + 106, + 494, + 188 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 193, + 89, + 417, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 192, + 87, + 418, + 102 + ], + "spans": [ + { + "bbox": [ + 192, + 87, + 418, + 102 + ], + "score": 1.0, + "content": "Table 1: The methods under consideration in this paper.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 118, + 106, + 494, + 188 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 118, + 106, + 494, + 188 + ], + "spans": [ + { + "bbox": [ + 118, + 106, + 494, + 188 + ], + "score": 0.976, + "html": "
Sign + MagnitudeSign
Not Variance-AdaptedSGDSSD "Stochastic Sign Descent"
Variance-Adapted
SVAG “Stochastic Variance-Adapted Gradient"ADAM
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The purpose of this paper is to disentangle these", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 262, + 399, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 399, + 274 + ], + "score": 1.0, + "content": "two intertwined aspects in order to discuss and analyze them in isolation.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 279, + 505, + 400 + ], + "lines": [ + { + "bbox": [ + 106, + 279, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 505, + 292 + ], + "score": 1.0, + "content": "This perspective naturally suggests two alternative methods by incorporating one of the aspects but", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "not the other (see Table 1). Taking the sign of the stochastic gradient (or momentum term) without", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "score": 1.0, + "content": "any further modification gives rise to “Stochastic Sign Descent” (SSD). On the other hand, “Stochas-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 311, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 325 + ], + "score": 1.0, + "content": "tic Variance-Adapted Gradient” (SVAG) applies element-wise variance adaptation factors directly on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "the stochastic gradient (or momentum term) instead of on its sign. We proceed as follows: In Sec-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 333, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 347 + ], + "score": 1.0, + "content": "tion 2, we investigate the sign aspect. In the simplified setting of stochastic quadratic problems, we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "derive conditions under which the element-wise sign of a stochastic gradient can be a better update", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 355, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 369 + ], + "score": 1.0, + "content": "direction than the stochastic gradient itself. Section 3 discusses the variance adaptation. We present", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "score": 1.0, + "content": "a principled derivation of “optimal” element-wise variance adaptation factors for a stochastic gra-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "dient as well as its sign. Subsequently, we incorporate momentum and briefly discuss the practical", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 389, + 461, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 461, + 402 + ], + "score": 1.0, + "content": "estimation of stochastic gradient variance. Section 4 presents some experimental results.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 414, + 202, + 426 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 204, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 204, + 428 + ], + "score": 1.0, + "content": "1.3 RELATED WORK", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 436, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 505, + 448 + ], + "score": 1.0, + "content": "The idea of using the sign of the gradient as the principal source of the optimizer update has already", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "received some attention in the literature. The RPROP algorithm (Riedmiller & Braun, 1993) ignores", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 458, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 506, + 470 + ], + "score": 1.0, + "content": "the magnitude of the gradient and dynamically adapts the per-element magnitude of the update based", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 469, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 480 + ], + "score": 1.0, + "content": "on observed sign changes. With the goal of reducing communication cost in distributed training of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 479, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 492 + ], + "score": 1.0, + "content": "neural networks, Seide et al. (2014) empirically investigate the use of the sign of stochastic gradients.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "score": 1.0, + "content": "Regarding the variance adaptation, Schaul et al. (2013) derive element-wise step sizes for stochastic", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 502, + 501, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 501, + 514 + ], + "score": 1.0, + "content": "gradient descent that have (among other factors) a dependency on the stochastic gradient variance.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 108, + 527, + 303, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 526, + 304, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 304, + 540 + ], + "score": 1.0, + "content": "1.4 THE SIGN OF A STOCHASTIC GRADIENT", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 548, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "We briefly establish a fact that will be used throughout the paper. The sign of a stochastic gradient", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 558, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 107, + 559, + 181, + 571 + ], + "score": 0.92, + "content": "s ( \\theta ) = \\mathrm { s i g n } ( g ( \\theta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 558, + 505, + 572 + ], + "score": 1.0, + "content": "estimates the sign of the true gradient. Its distribution (and thus the quality of this", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 568, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 104, + 568, + 343, + 584 + ], + "score": 1.0, + "content": "estimate) is fully characterized by the success probabilities", + "type": "text" + }, + { + "bbox": [ + 344, + 570, + 474, + 582 + ], + "score": 0.9, + "content": "\\rho _ { i } : = \\mathbf { P } \\left[ s ( \\theta ) _ { i } = \\mathrm { s i g n } ( \\nabla \\mathcal { L } ( \\theta ) _ { i } ) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 568, + 506, + 584 + ], + "score": 1.0, + "content": ". These", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 580, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 378, + 594 + ], + "score": 1.0, + "content": "depend on the distribution of the stochastic gradient. If we assume", + "type": "text" + }, + { + "bbox": [ + 379, + 582, + 397, + 593 + ], + "score": 0.91, + "content": "g ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 580, + 506, + 594 + ], + "score": 1.0, + "content": "to be Gaussian—which is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 592, + 424, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 424, + 606 + ], + "score": 1.0, + "content": "strongly supported by a Central Limit Theorem argument on Eq. (2)—we have", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 609, + 428, + 638 + ], + "lines": [ + { + "bbox": [ + 182, + 609, + 428, + 638 + ], + "spans": [ + { + "bbox": [ + 182, + 609, + 428, + 638 + ], + "score": 0.93, + "content": "\\rho _ { i } : = \\mathbf { P } \\left[ s ( \\theta ) _ { i } = \\operatorname { s i g n } ( \\nabla \\mathcal { L } ( \\theta ) _ { i } ) \\right] = \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\operatorname { e r f } \\left( \\frac { | \\nabla \\mathcal { L } ( \\theta ) _ { i } | } { \\sqrt { 2 } \\sigma ( \\theta ) _ { i } } \\right) ,", + "type": "interline_equation", + "image_path": "43d5335feda9b547eee32115e8fcc4fa2340cc2babbc47982e4f1274ea9b3db0.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 182, + 609, + 428, + 618.6666666666666 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 182, + 618.6666666666666, + 428, + 628.3333333333333 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 182, + 628.3333333333333, + 428, + 637.9999999999999 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 442, + 657 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 441, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 122, + 659 + ], + "score": 1.0, + "content": "see", + "type": "text" + }, + { + "bbox": [ + 122, + 645, + 141, + 656 + ], + "score": 0.82, + "content": "\\mathrm { \\ S B } . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 643, + 295, + 659 + ], + "score": 1.0, + "content": "in the supplements. Furthermore, it is", + "type": "text" + }, + { + "bbox": [ + 295, + 644, + 441, + 657 + ], + "score": 0.9, + "content": "\\mathbf { E } [ s ( \\theta ) _ { i } ] = ( 2 \\rho _ { i } - 1 ) \\operatorname { s i g n } ( \\nabla { \\mathcal { L } } ( \\theta ) _ { i } ) .", + "type": "inline_equation" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "title", + "bbox": [ + 108, + 672, + 212, + 685 + ], + "lines": [ + { + "bbox": [ + 104, + 671, + 213, + 687 + ], + "spans": [ + { + "bbox": [ + 104, + 671, + 213, + 687 + ], + "score": 1.0, + "content": "2 WHY THE SIGN?", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Can it make sense to ignore the gradient magnitude? We provide some intuition under which circum-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "score": 1.0, + "content": "stances the element-wise sign of a stochastic gradient is a better update direction than the stochastic", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "gradient itself. This question is difficult to tackle in general, which is why we restrict the problem", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 118, + 106, + 494, + 188 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 193, + 89, + 417, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 192, + 87, + 418, + 102 + ], + "spans": [ + { + "bbox": [ + 192, + 87, + 418, + 102 + ], + "score": 1.0, + "content": "Table 1: The methods under consideration in this paper.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 118, + 106, + 494, + 188 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 118, + 106, + 494, + 188 + ], + "spans": [ + { + "bbox": [ + 118, + 106, + 494, + 188 + ], + "score": 0.976, + "html": "
Sign + MagnitudeSign
Not Variance-AdaptedSGDSSD "Stochastic Sign Descent"
Variance-Adapted
SVAG “Stochastic Variance-Adapted Gradient"ADAM
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The purpose of this paper is to disentangle these", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 262, + 399, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 399, + 274 + ], + "score": 1.0, + "content": "two intertwined aspects in order to discuss and analyze them in isolation.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 228, + 505, + 274 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 279, + 505, + 400 + ], + "lines": [ + { + "bbox": [ + 106, + 279, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 505, + 292 + ], + "score": 1.0, + "content": "This perspective naturally suggests two alternative methods by incorporating one of the aspects but", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "not the other (see Table 1). Taking the sign of the stochastic gradient (or momentum term) without", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "score": 1.0, + "content": "any further modification gives rise to “Stochastic Sign Descent” (SSD). On the other hand, “Stochas-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 311, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 325 + ], + "score": 1.0, + "content": "tic Variance-Adapted Gradient” (SVAG) applies element-wise variance adaptation factors directly on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "the stochastic gradient (or momentum term) instead of on its sign. We proceed as follows: In Sec-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 333, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 347 + ], + "score": 1.0, + "content": "tion 2, we investigate the sign aspect. In the simplified setting of stochastic quadratic problems, we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "derive conditions under which the element-wise sign of a stochastic gradient can be a better update", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 355, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 369 + ], + "score": 1.0, + "content": "direction than the stochastic gradient itself. Section 3 discusses the variance adaptation. We present", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "score": 1.0, + "content": "a principled derivation of “optimal” element-wise variance adaptation factors for a stochastic gra-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "dient as well as its sign. Subsequently, we incorporate momentum and briefly discuss the practical", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 389, + 461, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 461, + 402 + ], + "score": 1.0, + "content": "estimation of stochastic gradient variance. Section 4 presents some experimental results.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 279, + 505, + 402 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 414, + 202, + 426 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 204, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 204, + 428 + ], + "score": 1.0, + "content": "1.3 RELATED WORK", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 435, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 436, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 505, + 448 + ], + "score": 1.0, + "content": "The idea of using the sign of the gradient as the principal source of the optimizer update has already", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "received some attention in the literature. The RPROP algorithm (Riedmiller & Braun, 1993) ignores", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 458, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 506, + 470 + ], + "score": 1.0, + "content": "the magnitude of the gradient and dynamically adapts the per-element magnitude of the update based", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 469, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 480 + ], + "score": 1.0, + "content": "on observed sign changes. With the goal of reducing communication cost in distributed training of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 479, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 492 + ], + "score": 1.0, + "content": "neural networks, Seide et al. (2014) empirically investigate the use of the sign of stochastic gradients.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "score": 1.0, + "content": "Regarding the variance adaptation, Schaul et al. (2013) derive element-wise step sizes for stochastic", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 502, + 501, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 501, + 514 + ], + "score": 1.0, + "content": "gradient descent that have (among other factors) a dependency on the stochastic gradient variance.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 436, + 506, + 514 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 527, + 303, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 526, + 304, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 304, + 540 + ], + "score": 1.0, + "content": "1.4 THE SIGN OF A STOCHASTIC GRADIENT", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 548, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "We briefly establish a fact that will be used throughout the paper. The sign of a stochastic gradient", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 558, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 107, + 559, + 181, + 571 + ], + "score": 0.92, + "content": "s ( \\theta ) = \\mathrm { s i g n } ( g ( \\theta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 558, + 505, + 572 + ], + "score": 1.0, + "content": "estimates the sign of the true gradient. Its distribution (and thus the quality of this", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 568, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 104, + 568, + 343, + 584 + ], + "score": 1.0, + "content": "estimate) is fully characterized by the success probabilities", + "type": "text" + }, + { + "bbox": [ + 344, + 570, + 474, + 582 + ], + "score": 0.9, + "content": "\\rho _ { i } : = \\mathbf { P } \\left[ s ( \\theta ) _ { i } = \\mathrm { s i g n } ( \\nabla \\mathcal { L } ( \\theta ) _ { i } ) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 568, + 506, + 584 + ], + "score": 1.0, + "content": ". These", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 580, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 378, + 594 + ], + "score": 1.0, + "content": "depend on the distribution of the stochastic gradient. If we assume", + "type": "text" + }, + { + "bbox": [ + 379, + 582, + 397, + 593 + ], + "score": 0.91, + "content": "g ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 580, + 506, + 594 + ], + "score": 1.0, + "content": "to be Gaussian—which is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 592, + 424, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 424, + 606 + ], + "score": 1.0, + "content": "strongly supported by a Central Limit Theorem argument on Eq. (2)—we have", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 547, + 506, + 606 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 609, + 428, + 638 + ], + "lines": [ + { + "bbox": [ + 182, + 609, + 428, + 638 + ], + "spans": [ + { + "bbox": [ + 182, + 609, + 428, + 638 + ], + "score": 0.93, + "content": "\\rho _ { i } : = \\mathbf { P } \\left[ s ( \\theta ) _ { i } = \\operatorname { s i g n } ( \\nabla \\mathcal { L } ( \\theta ) _ { i } ) \\right] = \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\operatorname { e r f } \\left( \\frac { | \\nabla \\mathcal { L } ( \\theta ) _ { i } | } { \\sqrt { 2 } \\sigma ( \\theta ) _ { i } } \\right) ,", + "type": "interline_equation", + "image_path": "43d5335feda9b547eee32115e8fcc4fa2340cc2babbc47982e4f1274ea9b3db0.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 182, + 609, + 428, + 618.6666666666666 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 182, + 618.6666666666666, + 428, + 628.3333333333333 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 182, + 628.3333333333333, + 428, + 637.9999999999999 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 442, + 657 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 441, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 122, + 659 + ], + "score": 1.0, + "content": "see", + "type": "text" + }, + { + "bbox": [ + 122, + 645, + 141, + 656 + ], + "score": 0.82, + "content": "\\mathrm { \\ S B } . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 643, + 295, + 659 + ], + "score": 1.0, + "content": "in the supplements. Furthermore, it is", + "type": "text" + }, + { + "bbox": [ + 295, + 644, + 441, + 657 + ], + "score": 0.9, + "content": "\\mathbf { E } [ s ( \\theta ) _ { i } ] = ( 2 \\rho _ { i } - 1 ) \\operatorname { s i g n } ( \\nabla { \\mathcal { L } } ( \\theta ) _ { i } ) .", + "type": "inline_equation" + } + ], + "index": 37 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 643, + 441, + 659 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 672, + 212, + 685 + ], + "lines": [ + { + "bbox": [ + 104, + 671, + 213, + 687 + ], + "spans": [ + { + "bbox": [ + 104, + 671, + 213, + 687 + ], + "score": 1.0, + "content": "2 WHY THE SIGN?", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Can it make sense to ignore the gradient magnitude? We provide some intuition under which circum-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "score": 1.0, + "content": "stances the element-wise sign of a stochastic gradient is a better update direction than the stochastic", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "gradient itself. This question is difficult to tackle in general, which is why we restrict the problem", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 699, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "class to the simple, yet insightful, case of stochastic quadratic problems, where we can investigate", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 405, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 405, + 106 + ], + "score": 1.0, + "content": "the effects of curvature properties and its interaction with stochastic noise.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 108, + 505, + 142 + ], + "lines": [ + { + "bbox": [ + 105, + 107, + 504, + 122 + ], + "spans": [ + { + "bbox": [ + 105, + 107, + 460, + 122 + ], + "score": 1.0, + "content": "Model Problem (Stochastic Quadratic Problem (QP)). Consider the loss function", + "type": "text" + }, + { + "bbox": [ + 460, + 109, + 504, + 121 + ], + "score": 0.9, + "content": "\\ell ( \\theta , x ) ~ =", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 107, + 118, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 107, + 119, + 198, + 132 + ], + "score": 0.89, + "content": "0 . 5 ( \\theta - x ) ^ { T } Q ( \\theta - x )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 118, + 366, + 133 + ], + "score": 1.0, + "content": "with a symmetric positive definite matrix", + "type": "text" + }, + { + "bbox": [ + 367, + 119, + 401, + 131 + ], + "score": 0.92, + "content": "Q \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 118, + 506, + 133 + ], + "score": 1.0, + "content": "and “data” coming from", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 131, + 260, + 143 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 170, + 143 + ], + "score": 1.0, + "content": "the distribution", + "type": "text" + }, + { + "bbox": [ + 170, + 131, + 237, + 143 + ], + "score": 0.92, + "content": "x \\sim \\dot { \\mathcal { N } } ( x ^ { * } , \\nu ^ { 2 } I )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 131, + 260, + 143 + ], + "score": 1.0, + "content": ". It is", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 148, + 425, + 174 + ], + "lines": [ + { + "bbox": [ + 184, + 148, + 425, + 174 + ], + "spans": [ + { + "bbox": [ + 184, + 148, + 425, + 174 + ], + "score": 0.91, + "content": "\\mathcal { L } ( \\theta ) : = \\mathbf { E } _ { x } [ \\ell ( \\theta , x ) ] = \\frac { 1 } { 2 } ( \\theta - x ^ { * } ) ^ { T } Q ( \\theta - x ^ { * } ) + \\frac { \\nu ^ { 2 } } { 2 } \\operatorname { t r } ( Q ) ,", + "type": "interline_equation", + "image_path": "42d9e6f824cee8a492d83fbde137dbf6af0b153b2eee7a6a9573e91302a23b5f.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 184, + 148, + 425, + 174 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 179, + 484, + 193 + ], + "lines": [ + { + "bbox": [ + 105, + 178, + 482, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 126, + 194 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 180, + 211, + 192 + ], + "score": 0.93, + "content": "\\nabla { \\mathcal { L } } ( \\theta ) = Q ( \\theta - x ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 178, + 351, + 194 + ], + "score": 1.0, + "content": ". Stochastic gradients are given by", + "type": "text" + }, + { + "bbox": [ + 351, + 180, + 482, + 192 + ], + "score": 0.91, + "content": "g ( \\theta ) = Q ( \\theta - x ) \\sim \\mathcal { N } ( x ^ { * } , \\nu ^ { 2 } I )", + "type": "inline_equation" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 108, + 205, + 254, + 217 + ], + "lines": [ + { + "bbox": [ + 106, + 205, + 255, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 255, + 218 + ], + "score": 1.0, + "content": "2.1 THEORETICAL COMPARISON", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 240 + ], + "score": 1.0, + "content": "We want to compare update directions on stochastic QPs in terms of their expected decrease in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 237, + 433, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 346, + 250 + ], + "score": 1.0, + "content": "function value from a single update step. If we update from", + "type": "text" + }, + { + "bbox": [ + 346, + 238, + 352, + 248 + ], + "score": 0.82, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 237, + 363, + 250 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 364, + 238, + 393, + 248 + ], + "score": 0.91, + "content": "\\theta + \\alpha z", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 237, + 433, + 250 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 255, + 420, + 281 + ], + "lines": [ + { + "bbox": [ + 191, + 255, + 420, + 281 + ], + "spans": [ + { + "bbox": [ + 191, + 255, + 420, + 281 + ], + "score": 0.9, + "content": "\\mathbf { E } [ \\mathcal { L } ( \\theta + \\alpha z ) ] = \\mathcal { L } ( \\theta ) + \\alpha \\nabla \\mathcal { L } ( \\theta ) ^ { T } \\mathbf { E } [ z ] + \\frac { \\alpha ^ { 2 } } { 2 } \\mathbf { E } [ z ^ { T } Q z ] .", + "type": "interline_equation", + "image_path": "cd92c2f18a4345377bd2146571dd4f6a8c6e863ac11c47e63f7ce0f7f9d0a85a.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 191, + 255, + 420, + 281 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 285, + 504, + 309 + ], + "lines": [ + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "For this comparison of update directions, we allow for the optimal step size that minimizes Eq. (10),", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 296, + 503, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 215, + 309 + ], + "score": 1.0, + "content": "which is easily found to be", + "type": "text" + }, + { + "bbox": [ + 216, + 296, + 343, + 309 + ], + "score": 0.92, + "content": "\\alpha _ { * } = - \\nabla \\mathcal { L } ( \\boldsymbol { \\theta } ) ^ { T } \\mathbf { E } [ \\boldsymbol { z } ] / \\mathbf { E } [ \\boldsymbol { z } ^ { T } Q \\boldsymbol { z } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 296, + 503, + 309 + ], + "score": 1.0, + "content": "and yields an expected improvement of", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 314, + 415, + 342 + ], + "lines": [ + { + "bbox": [ + 195, + 314, + 415, + 342 + ], + "spans": [ + { + "bbox": [ + 195, + 314, + 415, + 342 + ], + "score": 0.94, + "content": "\\mathcal { I } ( z ) : = \\left| \\mathbf { E } [ \\mathcal { L } ( \\theta + \\alpha _ { * } z ) ] - \\mathcal { L } ( \\theta ) \\right| = \\frac { ( \\nabla \\mathcal { L } ( \\theta ) ^ { T } \\mathbf { E } [ z ] ) ^ { 2 } } { 2 \\mathbf { E } [ z ^ { T } Q z ] } .", + "type": "interline_equation", + "image_path": "a2e37ef26e325b4664114ec9013bcccafab07f1293dd13eb7b5c69fd0b597d9c.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 195, + 314, + 415, + 342 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 347, + 429, + 360 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 429, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 429, + 361 + ], + "score": 1.0, + "content": "We find the following expressions/bounds for the improvement of SGD and SSD:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 122, + 366, + 471, + 406 + ], + "lines": [ + { + "bbox": [ + 122, + 366, + 471, + 406 + ], + "spans": [ + { + "bbox": [ + 122, + 366, + 471, + 406 + ], + "score": 0.93, + "content": "\\mathcal { I } ( g ) = \\frac { 1 } { 2 } \\frac { ( \\nabla \\mathcal { L } ( \\theta ) ^ { T } \\nabla \\mathcal { L } ( \\theta ) ) ^ { 2 } } { \\nabla \\mathcal { L } ( \\theta ) ^ { T } Q \\nabla \\mathcal { L } ( \\theta ) + \\nu ^ { 2 } \\sum _ { i = 1 } ^ { d } \\lambda _ { i } ^ { 3 } } , \\quad \\mathcal { Z } ( s ) \\ge \\frac { 1 } { 2 } \\frac { \\left( \\sum _ { i = 1 } ^ { d } ( 2 \\rho _ { i } - 1 ) | \\nabla \\mathcal { L } ( \\theta ) _ { i } | \\right) ^ { 2 } } { \\sum _ { i , j = 1 } ^ { d } | q _ { i j } | }", + "type": "interline_equation", + "image_path": "9f9bbe6646d39f24c62f9c4049c428317ff757154ae1217f1e7751178768d1a9.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 122, + 366, + 471, + 379.3333333333333 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 122, + 379.3333333333333, + 471, + 392.66666666666663 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 122, + 392.66666666666663, + 471, + 405.99999999999994 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 412, + 504, + 435 + ], + "lines": [ + { + "bbox": [ + 106, + 411, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 149, + 425 + ], + "score": 1.0, + "content": "where the", + "type": "text" + }, + { + "bbox": [ + 149, + 413, + 187, + 424 + ], + "score": 0.92, + "content": "\\lambda _ { i } \\in \\mathbb { R } _ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 411, + 279, + 425 + ], + "score": 1.0, + "content": "are the eigenvalues of", + "type": "text" + }, + { + "bbox": [ + 280, + 413, + 289, + 424 + ], + "score": 0.87, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 411, + 416, + 425 + ], + "score": 1.0, + "content": "with orthonormal eigenvectors", + "type": "text" + }, + { + "bbox": [ + 416, + 412, + 451, + 424 + ], + "score": 0.92, + "content": "v _ { i } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 411, + 506, + 425 + ], + "score": 1.0, + "content": ". Derivations", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 171, + 436 + ], + "score": 1.0, + "content": "can be found in", + "type": "text" + }, + { + "bbox": [ + 171, + 424, + 190, + 435 + ], + "score": 0.84, + "content": "\\mathrm { \\ S B . l }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "of the supplements. Comparing these expressions, we make two observations.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 439, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 104, + 438, + 507, + 456 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 137, + 456 + ], + "score": 1.0, + "content": "Firstly,", + "type": "text" + }, + { + "bbox": [ + 138, + 440, + 158, + 452 + ], + "score": 0.9, + "content": "\\mathcal { T } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 438, + 248, + 456 + ], + "score": 1.0, + "content": "has a dependency on", + "type": "text" + }, + { + "bbox": [ + 249, + 439, + 289, + 454 + ], + "score": 0.91, + "content": "\\textstyle \\sum _ { i , j } | q _ { i j } |", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 438, + 507, + 456 + ], + "score": 1.0, + "content": ". This quantity relates to the eigenvalues, as well as", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 451, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 257, + 466 + ], + "score": 1.0, + "content": "the orientation of the eigenbasis of", + "type": "text" + }, + { + "bbox": [ + 257, + 454, + 266, + 464 + ], + "score": 0.84, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 451, + 324, + 466 + ], + "score": 1.0, + "content": ". By writing", + "type": "text" + }, + { + "bbox": [ + 324, + 453, + 333, + 464 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 451, + 506, + 466 + ], + "score": 1.0, + "content": "in its eigendecomposition one finds that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 462, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 207, + 477 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { i , j } | q _ { i j } | \\leq \\sum _ { i } \\lambda _ { i } \\| v _ { i } \\| _ { 1 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 462, + 432, + 477 + ], + "score": 1.0, + "content": ". If the eigenvectors are perfectly axis-aligned (diagonal", + "type": "text" + }, + { + "bbox": [ + 432, + 464, + 441, + 475 + ], + "score": 0.79, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 462, + 506, + 477 + ], + "score": 1.0, + "content": "), their 1-norms", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 475, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 122, + 490 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 122, + 476, + 200, + 489 + ], + "score": 0.93, + "content": "\\mathbf { \\| } v _ { i } \\| _ { 1 } = \\| v _ { i } \\| _ { 2 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 475, + 505, + 490 + ], + "score": 1.0, + "content": ". It is intuitive that this is the best case for the intrinsically axis-aligned sign", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 487, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 317, + 502 + ], + "score": 1.0, + "content": "update. In general, the 1-norm is only bounded by", + "type": "text" + }, + { + "bbox": [ + 317, + 487, + 422, + 501 + ], + "score": 0.92, + "content": "\\| v _ { i } \\| _ { 1 } \\leq \\sqrt { d } \\| v _ { i } \\| _ { 2 } = \\sqrt { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 488, + 506, + 502 + ], + "score": 1.0, + "content": ", suggesting that the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 499, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 514 + ], + "score": 1.0, + "content": "sign update will have difficulties with arbitrarily oriented eigenbases. We can alternatively express", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 334, + 524 + ], + "score": 1.0, + "content": "this matter in terms of “diagonal dominance”. Assuming", + "type": "text" + }, + { + "bbox": [ + 334, + 511, + 343, + 523 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 510, + 412, + 524 + ], + "score": 1.0, + "content": "has a percentage", + "type": "text" + }, + { + "bbox": [ + 412, + 511, + 450, + 523 + ], + "score": 0.92, + "content": "c \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 510, + 505, + 524 + ], + "score": 1.0, + "content": "of its “mass”", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 522, + 339, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 190, + 536 + ], + "score": 1.0, + "content": "on the diagonal, i.e.,", + "type": "text" + }, + { + "bbox": [ + 190, + 522, + 281, + 536 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\sum _ { i } | \\bar { q _ { i i } } | \\geq c \\sum _ { i , j } | q _ { i j } | } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 522, + 339, + 536 + ], + "score": 1.0, + "content": ", we can write", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 541, + 456, + 581 + ], + "lines": [ + { + "bbox": [ + 154, + 541, + 456, + 581 + ], + "spans": [ + { + "bbox": [ + 154, + 541, + 456, + 581 + ], + "score": 0.95, + "content": "\\mathcal { T } ( s ) \\geq \\frac { 1 } { 2 } \\frac { \\Big ( \\sum _ { i = 1 } ^ { d } ( 2 \\rho _ { i } - 1 ) | \\nabla \\mathcal { L } ( \\theta ) _ { i } | \\Big ) ^ { 2 } } { c ^ { - 1 } \\sum _ { i = 1 } ^ { d } | q _ { i i } | } = \\frac { 1 } { 2 } \\frac { \\Big ( \\sum _ { i = 1 } ^ { d } ( 2 \\rho _ { i } - 1 ) | \\nabla \\mathcal { L } ( \\theta ) _ { i } | \\Big ) ^ { 2 } } { c ^ { - 1 } \\sum _ { i = 1 } ^ { d } \\lambda _ { i } } .", + "type": "interline_equation", + "image_path": "2f196e16828416b6df93774ec198ac522b1ff30f1cbe8602fdfe6b592a8b44e8.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 154, + 541, + 456, + 554.3333333333334 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 154, + 554.3333333333334, + 456, + 567.6666666666667 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 154, + 567.6666666666667, + 456, + 581.0000000000001 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 586, + 505, + 620 + ], + "lines": [ + { + "bbox": [ + 105, + 585, + 504, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 504, + 599 + ], + "score": 1.0, + "content": "Becker & LeCun (1988) empirically investigated the diagonal dominance of Hessians in optimiza-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "tion problems arising from neural networks and found relatively high percentages of mass on the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 608, + 387, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 158, + 621 + ], + "score": 1.0, + "content": "diagonals of", + "type": "text" + }, + { + "bbox": [ + 158, + 608, + 190, + 618 + ], + "score": 0.87, + "content": "c = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 608, + 213, + 621 + ], + "score": 1.0, + "content": "up to", + "type": "text" + }, + { + "bbox": [ + 214, + 608, + 245, + 619 + ], + "score": 0.88, + "content": "c = 0 . 6", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 608, + 387, + 621 + ], + "score": 1.0, + "content": "for the problems they investigated.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 624, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 101, + 618, + 510, + 651 + ], + "spans": [ + { + "bbox": [ + 101, + 618, + 149, + 651 + ], + "score": 1.0, + "content": "Secondly, hugely ob", + "type": "text" + }, + { + "bbox": [ + 150, + 627, + 170, + 639 + ], + "score": 0.91, + "content": "\\boldsymbol { \\mathcal { T } } ( \\boldsymbol { g } )", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 618, + 289, + 651 + ], + "score": 1.0, + "content": "contains the constant offset ive for ill-conditioned and no", + "type": "text" + }, + { + "bbox": [ + 289, + 624, + 338, + 640 + ], + "score": 0.93, + "content": "\\nu ^ { 2 } \\textstyle \\sum _ { i = 1 } ^ { d } \\lambda _ { i } ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 618, + 358, + 651 + ], + "score": 1.0, + "content": "in ts. In", + "type": "text" + }, + { + "bbox": [ + 378, + 618, + 510, + 651 + ], + "score": 1.0, + "content": "nominator, which can become, on the other hand, there is no", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 358, + 638, + 378, + 650 + ], + "spans": [ + { + "bbox": [ + 358, + 638, + 378, + 650 + ], + "score": 0.91, + "content": "\\mathcal { T } ( s )", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "such interaction between the magnitude of the noise and the eigenspectrum; the noise only manifests", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 270, + 673 + ], + "score": 1.0, + "content": "in the element-wise success probabilities", + "type": "text" + }, + { + "bbox": [ + 270, + 662, + 279, + 672 + ], + "score": 0.85, + "content": "\\rho _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 659, + 505, + 673 + ], + "score": 1.0, + "content": ", its effect in the denominator is bounded. A recent paper", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 670, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 506, + 683 + ], + "score": 1.0, + "content": "(Chaudhari et al., 2016) investigated the eigenspectrum in deep learning problems and found it to be", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 682, + 478, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 478, + 695 + ], + "score": 1.0, + "content": "very ill-conditioned with the majority of eigenvalues close to zero and a few very large ones.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "In summary, we can expect the sign update to be beneficial for noisy, ill-conditioned problems with", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "“diagonally dominant” Hessians. 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Consider the loss function", + "type": "text" + }, + { + "bbox": [ + 460, + 109, + 504, + 121 + ], + "score": 0.9, + "content": "\\ell ( \\theta , x ) ~ =", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 107, + 118, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 107, + 119, + 198, + 132 + ], + "score": 0.89, + "content": "0 . 5 ( \\theta - x ) ^ { T } Q ( \\theta - x )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 118, + 366, + 133 + ], + "score": 1.0, + "content": "with a symmetric positive definite matrix", + "type": "text" + }, + { + "bbox": [ + 367, + 119, + 401, + 131 + ], + "score": 0.92, + "content": "Q \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 118, + 506, + 133 + ], + "score": 1.0, + "content": "and “data” coming from", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 131, + 260, + 143 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 170, + 143 + ], + "score": 1.0, + "content": "the distribution", + "type": "text" + }, + { + "bbox": [ + 170, + 131, + 237, + 143 + ], + "score": 0.92, + "content": "x \\sim \\dot { \\mathcal { N } } ( x ^ { * } , \\nu ^ { 2 } I )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 131, + 260, + 143 + ], + "score": 1.0, + "content": ". 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Stochastic gradients are given by", + "type": "text" + }, + { + "bbox": [ + 351, + 180, + 482, + 192 + ], + "score": 0.91, + "content": "g ( \\theta ) = Q ( \\theta - x ) \\sim \\mathcal { N } ( x ^ { * } , \\nu ^ { 2 } I )", + "type": "inline_equation" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 178, + 482, + 194 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 205, + 254, + 217 + ], + "lines": [ + { + "bbox": [ + 106, + 205, + 255, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 255, + 218 + ], + "score": 1.0, + "content": "2.1 THEORETICAL COMPARISON", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 240 + ], + "score": 1.0, + "content": "We want to compare update directions on stochastic QPs in terms of their expected decrease in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 237, + 433, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 346, + 250 + ], + "score": 1.0, + "content": "function value from a single update step. If we update from", + "type": "text" + }, + { + "bbox": [ + 346, + 238, + 352, + 248 + ], + "score": 0.82, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 237, + 363, + 250 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 364, + 238, + 393, + 248 + ], + "score": 0.91, + "content": "\\theta + \\alpha z", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 237, + 433, + 250 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 226, + 505, + 250 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 255, + 420, + 281 + ], + "lines": [ + { + "bbox": [ + 191, + 255, + 420, + 281 + ], + "spans": [ + { + "bbox": [ + 191, + 255, + 420, + 281 + ], + "score": 0.9, + "content": "\\mathbf { E } [ \\mathcal { L } ( \\theta + \\alpha z ) ] = \\mathcal { L } ( \\theta ) + \\alpha \\nabla \\mathcal { L } ( \\theta ) ^ { T } \\mathbf { E } [ z ] + \\frac { \\alpha ^ { 2 } } { 2 } \\mathbf { E } [ z ^ { T } Q z ] .", + "type": "interline_equation", + "image_path": "cd92c2f18a4345377bd2146571dd4f6a8c6e863ac11c47e63f7ce0f7f9d0a85a.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 191, + 255, + 420, + 281 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 285, + 504, + 309 + ], + "lines": [ + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "For this comparison of update directions, we allow for the optimal step size that minimizes Eq. (10),", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 296, + 503, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 215, + 309 + ], + "score": 1.0, + "content": "which is easily found to be", + "type": "text" + }, + { + "bbox": [ + 216, + 296, + 343, + 309 + ], + "score": 0.92, + "content": "\\alpha _ { * } = - \\nabla \\mathcal { L } ( \\boldsymbol { \\theta } ) ^ { T } \\mathbf { E } [ \\boldsymbol { z } ] / \\mathbf { E } [ \\boldsymbol { z } ^ { T } Q \\boldsymbol { z } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 296, + 503, + 309 + ], + "score": 1.0, + "content": "and yields an expected improvement of", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 106, + 286, + 505, + 309 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 314, + 415, + 342 + ], + "lines": [ + { + "bbox": [ + 195, + 314, + 415, + 342 + ], + "spans": [ + { + "bbox": [ + 195, + 314, + 415, + 342 + ], + "score": 0.94, + "content": "\\mathcal { I } ( z ) : = \\left| \\mathbf { E } [ \\mathcal { L } ( \\theta + \\alpha _ { * } z ) ] - \\mathcal { L } ( \\theta ) \\right| = \\frac { ( \\nabla \\mathcal { L } ( \\theta ) ^ { T } \\mathbf { E } [ z ] ) ^ { 2 } } { 2 \\mathbf { E } [ z ^ { T } Q z ] } .", + "type": "interline_equation", + "image_path": "a2e37ef26e325b4664114ec9013bcccafab07f1293dd13eb7b5c69fd0b597d9c.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 195, + 314, + 415, + 342 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 347, + 429, + 360 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 429, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 429, + 361 + ], + "score": 1.0, + "content": "We find the following expressions/bounds for the improvement of SGD and SSD:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 347, + 429, + 361 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 122, + 366, + 471, + 406 + ], + "lines": [ + { + "bbox": [ + 122, + 366, + 471, + 406 + ], + "spans": [ + { + "bbox": [ + 122, + 366, + 471, + 406 + ], + "score": 0.93, + "content": "\\mathcal { I } ( g ) = \\frac { 1 } { 2 } \\frac { ( \\nabla \\mathcal { L } ( \\theta ) ^ { T } \\nabla \\mathcal { L } ( \\theta ) ) ^ { 2 } } { \\nabla \\mathcal { L } ( \\theta ) ^ { T } Q \\nabla \\mathcal { L } ( \\theta ) + \\nu ^ { 2 } \\sum _ { i = 1 } ^ { d } \\lambda _ { i } ^ { 3 } } , \\quad \\mathcal { Z } ( s ) \\ge \\frac { 1 } { 2 } \\frac { \\left( \\sum _ { i = 1 } ^ { d } ( 2 \\rho _ { i } - 1 ) | \\nabla \\mathcal { L } ( \\theta ) _ { i } | \\right) ^ { 2 } } { \\sum _ { i , j = 1 } ^ { d } | q _ { i j } | }", + "type": "interline_equation", + "image_path": "9f9bbe6646d39f24c62f9c4049c428317ff757154ae1217f1e7751178768d1a9.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 122, + 366, + 471, + 379.3333333333333 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 122, + 379.3333333333333, + 471, + 392.66666666666663 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 122, + 392.66666666666663, + 471, + 405.99999999999994 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 412, + 504, + 435 + ], + "lines": [ + { + "bbox": [ + 106, + 411, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 149, + 425 + ], + "score": 1.0, + "content": "where the", + "type": "text" + }, + { + "bbox": [ + 149, + 413, + 187, + 424 + ], + "score": 0.92, + "content": "\\lambda _ { i } \\in \\mathbb { R } _ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 411, + 279, + 425 + ], + "score": 1.0, + "content": "are the eigenvalues of", + "type": "text" + }, + { + "bbox": [ + 280, + 413, + 289, + 424 + ], + "score": 0.87, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 411, + 416, + 425 + ], + "score": 1.0, + "content": "with orthonormal eigenvectors", + "type": "text" + }, + { + "bbox": [ + 416, + 412, + 451, + 424 + ], + "score": 0.92, + "content": "v _ { i } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 411, + 506, + 425 + ], + "score": 1.0, + "content": ". Derivations", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 171, + 436 + ], + "score": 1.0, + "content": "can be found in", + "type": "text" + }, + { + "bbox": [ + 171, + 424, + 190, + 435 + ], + "score": 0.84, + "content": "\\mathrm { \\ S B . l }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "of the supplements. Comparing these expressions, we make two observations.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 106, + 411, + 506, + 436 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 439, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 104, + 438, + 507, + 456 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 137, + 456 + ], + "score": 1.0, + "content": "Firstly,", + "type": "text" + }, + { + "bbox": [ + 138, + 440, + 158, + 452 + ], + "score": 0.9, + "content": "\\mathcal { T } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 438, + 248, + 456 + ], + "score": 1.0, + "content": "has a dependency on", + "type": "text" + }, + { + "bbox": [ + 249, + 439, + 289, + 454 + ], + "score": 0.91, + "content": "\\textstyle \\sum _ { i , j } | q _ { i j } |", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 438, + 507, + 456 + ], + "score": 1.0, + "content": ". This quantity relates to the eigenvalues, as well as", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 451, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 257, + 466 + ], + "score": 1.0, + "content": "the orientation of the eigenbasis of", + "type": "text" + }, + { + "bbox": [ + 257, + 454, + 266, + 464 + ], + "score": 0.84, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 451, + 324, + 466 + ], + "score": 1.0, + "content": ". By writing", + "type": "text" + }, + { + "bbox": [ + 324, + 453, + 333, + 464 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 451, + 506, + 466 + ], + "score": 1.0, + "content": "in its eigendecomposition one finds that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 462, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 207, + 477 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { i , j } | q _ { i j } | \\leq \\sum _ { i } \\lambda _ { i } \\| v _ { i } \\| _ { 1 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 462, + 432, + 477 + ], + "score": 1.0, + "content": ". If the eigenvectors are perfectly axis-aligned (diagonal", + "type": "text" + }, + { + "bbox": [ + 432, + 464, + 441, + 475 + ], + "score": 0.79, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 462, + 506, + 477 + ], + "score": 1.0, + "content": "), their 1-norms", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 475, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 122, + 490 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 122, + 476, + 200, + 489 + ], + "score": 0.93, + "content": "\\mathbf { \\| } v _ { i } \\| _ { 1 } = \\| v _ { i } \\| _ { 2 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 475, + 505, + 490 + ], + "score": 1.0, + "content": ". It is intuitive that this is the best case for the intrinsically axis-aligned sign", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 487, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 317, + 502 + ], + "score": 1.0, + "content": "update. In general, the 1-norm is only bounded by", + "type": "text" + }, + { + "bbox": [ + 317, + 487, + 422, + 501 + ], + "score": 0.92, + "content": "\\| v _ { i } \\| _ { 1 } \\leq \\sqrt { d } \\| v _ { i } \\| _ { 2 } = \\sqrt { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 488, + 506, + 502 + ], + "score": 1.0, + "content": ", suggesting that the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 499, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 514 + ], + "score": 1.0, + "content": "sign update will have difficulties with arbitrarily oriented eigenbases. We can alternatively express", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 334, + 524 + ], + "score": 1.0, + "content": "this matter in terms of “diagonal dominance”. Assuming", + "type": "text" + }, + { + "bbox": [ + 334, + 511, + 343, + 523 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 510, + 412, + 524 + ], + "score": 1.0, + "content": "has a percentage", + "type": "text" + }, + { + "bbox": [ + 412, + 511, + 450, + 523 + ], + "score": 0.92, + "content": "c \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 510, + 505, + 524 + ], + "score": 1.0, + "content": "of its “mass”", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 522, + 339, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 190, + 536 + ], + "score": 1.0, + "content": "on the diagonal, i.e.,", + "type": "text" + }, + { + "bbox": [ + 190, + 522, + 281, + 536 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\sum _ { i } | \\bar { q _ { i i } } | \\geq c \\sum _ { i , j } | q _ { i j } | } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 522, + 339, + 536 + ], + "score": 1.0, + "content": ", we can write", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 438, + 507, + 536 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 541, + 456, + 581 + ], + "lines": [ + { + "bbox": [ + 154, + 541, + 456, + 581 + ], + "spans": [ + { + "bbox": [ + 154, + 541, + 456, + 581 + ], + "score": 0.95, + "content": "\\mathcal { T } ( s ) \\geq \\frac { 1 } { 2 } \\frac { \\Big ( \\sum _ { i = 1 } ^ { d } ( 2 \\rho _ { i } - 1 ) | \\nabla \\mathcal { L } ( \\theta ) _ { i } | \\Big ) ^ { 2 } } { c ^ { - 1 } \\sum _ { i = 1 } ^ { d } | q _ { i i } | } = \\frac { 1 } { 2 } \\frac { \\Big ( \\sum _ { i = 1 } ^ { d } ( 2 \\rho _ { i } - 1 ) | \\nabla \\mathcal { L } ( \\theta ) _ { i } | \\Big ) ^ { 2 } } { c ^ { - 1 } \\sum _ { i = 1 } ^ { d } \\lambda _ { i } } .", + "type": "interline_equation", + "image_path": "2f196e16828416b6df93774ec198ac522b1ff30f1cbe8602fdfe6b592a8b44e8.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 154, + 541, + 456, + 554.3333333333334 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 154, + 554.3333333333334, + 456, + 567.6666666666667 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 154, + 567.6666666666667, + 456, + 581.0000000000001 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 586, + 505, + 620 + ], + "lines": [ + { + "bbox": [ + 105, + 585, + 504, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 504, + 599 + ], + "score": 1.0, + "content": "Becker & LeCun (1988) empirically investigated the diagonal dominance of Hessians in optimiza-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "tion problems arising from neural networks and found relatively high percentages of mass on the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 608, + 387, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 158, + 621 + ], + "score": 1.0, + "content": "diagonals of", + "type": "text" + }, + { + "bbox": [ + 158, + 608, + 190, + 618 + ], + "score": 0.87, + "content": "c = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 608, + 213, + 621 + ], + "score": 1.0, + "content": "up to", + "type": "text" + }, + { + "bbox": [ + 214, + 608, + 245, + 619 + ], + "score": 0.88, + "content": "c = 0 . 6", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 608, + 387, + 621 + ], + "score": 1.0, + "content": "for the problems they investigated.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 585, + 505, + 621 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 624, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 101, + 618, + 510, + 651 + ], + "spans": [ + { + "bbox": [ + 101, + 618, + 149, + 651 + ], + "score": 1.0, + "content": "Secondly, hugely ob", + "type": "text" + }, + { + "bbox": [ + 150, + 627, + 170, + 639 + ], + "score": 0.91, + "content": "\\boldsymbol { \\mathcal { T } } ( \\boldsymbol { g } )", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 618, + 289, + 651 + ], + "score": 1.0, + "content": "contains the constant offset ive for ill-conditioned and no", + "type": "text" + }, + { + "bbox": [ + 289, + 624, + 338, + 640 + ], + "score": 0.93, + "content": "\\nu ^ { 2 } \\textstyle \\sum _ { i = 1 } ^ { d } \\lambda _ { i } ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 618, + 358, + 651 + ], + "score": 1.0, + "content": "in ts. In", + "type": "text" + }, + { + "bbox": [ + 378, + 618, + 510, + 651 + ], + "score": 1.0, + "content": "nominator, which can become, on the other hand, there is no", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 358, + 638, + 378, + 650 + ], + "spans": [ + { + "bbox": [ + 358, + 638, + 378, + 650 + ], + "score": 0.91, + "content": "\\mathcal { T } ( s )", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "such interaction between the magnitude of the noise and the eigenspectrum; the noise only manifests", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 270, + 673 + ], + "score": 1.0, + "content": "in the element-wise success probabilities", + "type": "text" + }, + { + "bbox": [ + 270, + 662, + 279, + 672 + ], + "score": 0.85, + "content": "\\rho _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 659, + 505, + 673 + ], + "score": 1.0, + "content": ", its effect in the denominator is bounded. A recent paper", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 670, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 506, + 683 + ], + "score": 1.0, + "content": "(Chaudhari et al., 2016) investigated the eigenspectrum in deep learning problems and found it to be", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 682, + 478, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 478, + 695 + ], + "score": 1.0, + "content": "very ill-conditioned with the majority of eigenvalues close to zero and a few very large ones.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 101, + 618, + 510, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "In summary, we can expect the sign update to be beneficial for noisy, ill-conditioned problems with", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "“diagonally dominant” Hessians. 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In both cases, a motivation", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 658, + 267, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 267, + 668 + ], + "score": 1.0, + "content": "arises from the following consideration:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 673, + 505, + 729 + ], + "lines": [ + { + "bbox": [ + 106, + 673, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 273, + 686 + ], + "score": 1.0, + "content": "Assume we want to update in a direction", + "type": "text" + }, + { + "bbox": [ + 273, + 673, + 305, + 685 + ], + "score": 0.9, + "content": "p \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 673, + 320, + 686 + ], + "score": 1.0, + "content": "(or", + "type": "text" + }, + { + "bbox": [ + 321, + 673, + 354, + 686 + ], + "score": 0.72, + "content": "\\mathrm { s i g n } ( p ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 673, + 505, + 686 + ], + "score": 1.0, + "content": ", but only have access to an unbiased", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 682, + 504, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 142, + 698 + ], + "score": 1.0, + "content": "estimate", + "type": "text" + }, + { + "bbox": [ + 142, + 684, + 174, + 696 + ], + "score": 0.91, + "content": "\\hat { p } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 682, + 196, + 698 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 196, + 685, + 235, + 697 + ], + "score": 0.92, + "content": "\\mathbf { E } [ \\hat { p } ] = p", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 682, + 381, + 698 + ], + "score": 1.0, + "content": ". 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(2016) by uniformly drawing", + "type": "text" + }, + { + "bbox": [ + 410, + 435, + 430, + 445 + ], + "score": 0.85, + "content": "90 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "of the eigenvalues", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 446, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 129, + 458 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 129, + 446, + 150, + 458 + ], + "score": 0.38, + "content": "[ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 446, + 169, + 458 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 170, + 446, + 189, + 456 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 446, + 250, + 458 + ], + "score": 1.0, + "content": "from [30, 60].", + "type": "text" + }, + { + "bbox": [ + 251, + 446, + 260, + 457 + ], + "score": 0.81, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 446, + 361, + 458 + ], + "score": 1.0, + "content": "is then generated by (1)", + "type": "text" + }, + { + "bbox": [ + 362, + 446, + 393, + 457 + ], + "score": 0.92, + "content": "Q = \\Lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 446, + 506, + 458 + ], + "score": 1.0, + "content": "to produce an axis-aligned", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 104, + 455, + 176, + 469 + ], + "score": 1.0, + "content": "problem and (2)", + "type": "text" + }, + { + "bbox": [ + 176, + 456, + 231, + 468 + ], + "score": 0.91, + "content": "Q \\ = \\ R \\mathring { \\Lambda } R ^ { \\hat { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 455, + 327, + 469 + ], + "score": 1.0, + "content": "with a rotation matrix", + "type": "text" + }, + { + "bbox": [ + 328, + 457, + 337, + 466 + ], + "score": 0.7, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "drawn uniformly at random (see Diaco-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "nis & Shahshahani, 1987). This makes four different matrices, which we consider at noise levels", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 107, + 478, + 179, + 491 + ], + "score": 0.92, + "content": "\\nu \\in \\{ 0 , 0 . 1 , 4 . 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 479, + 506, + 491 + ], + "score": 1.0, + "content": ". We compare SGD and SSD, both with the optimal step size as derived from", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 489, + 330, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 330, + 503 + ], + "score": 1.0, + "content": "Eq. (10), which can be computed exactly in this setting.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 391, + 506, + 503 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 573 + ], + "lines": [ + { + "bbox": [ + 106, + 506, + 504, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 504, + 519 + ], + "score": 1.0, + "content": "Figure 2 shows the results, which confirm the theoretical findings. On the well-conditioned, noise-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 531 + ], + "score": 1.0, + "content": "free problem, gradient descent vastly outperforms the sign-based method. Surprisingly, adding even", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "a little noise almost evens out the difference in performance. The orientation of the eigenbasis had", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "little effect on the performance of SSD in the well-conditioned case. On the ill-conditioned problem,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "the methods work roughly equally well when the eigenbasis is randomly rotated. As predicted, SSD", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 560, + 495, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 495, + 574 + ], + "score": 1.0, + "content": "benefits drastically from an axis-aligned eigenbasis (last row), where it clearly outperforms SGD.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 506, + 505, + 574 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 595, + 431, + 608 + ], + "lines": [ + { + "bbox": [ + 104, + 594, + 433, + 610 + ], + "spans": [ + { + "bbox": [ + 104, + 594, + 433, + 610 + ], + "score": 1.0, + "content": "3 VARIANCE-BASED ELEMENT-WISE STEP SIZE ADAPTATION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 623, + 504, + 668 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "Besides the sign direction, the other defining property of ADAM are variance-based element-wise", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 635, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 647 + ], + "score": 1.0, + "content": "step sizes. Considering the variance adaptation in isolation from the sign aspect naturally suggests", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 646, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 505, + 658 + ], + "score": 1.0, + "content": "to employ it directly on the stochastic gradient, without taking the sign. In both cases, a motivation", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 658, + 267, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 267, + 668 + ], + "score": 1.0, + "content": "arises from the following consideration:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 623, + 506, + 668 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 673, + 505, + 729 + ], + "lines": [ + { + "bbox": [ + 106, + 673, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 273, + 686 + ], + "score": 1.0, + "content": "Assume we want to update in a direction", + "type": "text" + }, + { + "bbox": [ + 273, + 673, + 305, + 685 + ], + "score": 0.9, + "content": "p \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 673, + 320, + 686 + ], + "score": 1.0, + "content": "(or", + "type": "text" + }, + { + "bbox": [ + 321, + 673, + 354, + 686 + ], + "score": 0.72, + "content": "\\mathrm { s i g n } ( p ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 673, + 505, + 686 + ], + "score": 1.0, + "content": ", but only have access to an unbiased", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 682, + 504, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 142, + 698 + ], + "score": 1.0, + "content": "estimate", + "type": "text" + }, + { + "bbox": [ + 142, + 684, + 174, + 696 + ], + "score": 0.91, + "content": "\\hat { p } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 682, + 196, + 698 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 196, + 685, + 235, + 697 + ], + "score": 0.92, + "content": "\\mathbf { E } [ \\hat { p } ] = p", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 682, + 381, + 698 + ], + "score": 1.0, + "content": ". 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One way to make “optimal” use of these factors is to choose them such as to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 706, + 505, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 706, + 505, + 720 + ], + "score": 1.0, + "content": "minimize the expected distance to the desired update direction. Using the squared Euclidean norm", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 717, + 312, + 731 + ], + "spans": [ + { + "bbox": [ + 105, + 717, + 312, + 731 + ], + "score": 1.0, + "content": "as a distance measure, we find the following result.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 673, + 506, + 731 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 446, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 447, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 169, + 96 + ], + "score": 1.0, + "content": "Lemma 1. Let", + "type": "text" + }, + { + "bbox": [ + 169, + 82, + 200, + 94 + ], + "score": 0.91, + "content": "\\hat { p } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 80, + 308, + 96 + ], + "score": 1.0, + "content": "be a random variable with", + "type": "text" + }, + { + "bbox": [ + 309, + 82, + 347, + 95 + ], + "score": 0.9, + "content": "\\mathbf { E } [ \\hat { p } ] = p", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 80, + 365, + 96 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 365, + 82, + 419, + 95 + ], + "score": 0.89, + "content": "\\mathbf { v a r } [ p _ { i } ] = \\sigma _ { i } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 80, + 447, + 96 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 100, + 442, + 128 + ], + "lines": [ + { + "bbox": [ + 169, + 100, + 442, + 128 + ], + "spans": [ + { + "bbox": [ + 169, + 100, + 442, + 128 + ], + "score": 0.92, + "content": "\\operatorname* { m i n } _ { \\gamma \\in \\mathbb { R } ^ { d } } \\mathbf { E } [ \\| \\gamma \\odot \\hat { p } - p \\| _ { 2 } ^ { 2 } ] \\quad i s ~ s o l \\nu e d b y \\quad \\gamma _ { i } = \\frac { p _ { i } ^ { 2 } } { p _ { i } ^ { 2 } + \\sigma _ { i } ^ { 2 } } = \\frac { 1 } { 1 + \\sigma _ { i } ^ { 2 } / p _ { i } ^ { 2 } }", + "type": "interline_equation", + "image_path": "61a9c05b25e25fda0d0d14f0065f3f881afcef361098b5c172c479dd2eab4031.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 169, + 100, + 442, + 128 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 133, + 124, + 144 + ], + "lines": [ + { + "bbox": [ + 105, + 133, + 125, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 125, + 145 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 142, + 440, + 163 + ], + "lines": [ + { + "bbox": [ + 169, + 142, + 440, + 163 + ], + "spans": [ + { + "bbox": [ + 169, + 142, + 440, + 163 + ], + "score": 0.84, + "content": "\\operatorname* { m i n } _ { \\gamma \\in \\mathbb { R } ^ { d } } \\mathbf { E } [ \\| \\gamma \\odot \\mathrm { s i g n } ( \\hat { p } ) - \\mathrm { s i g n } ( p ) \\| _ { 2 } ^ { 2 } ] \\quad i s s o l \\nu e d b y \\quad \\gamma _ { i } = ( 2 \\rho _ { i } - 1 ) ,", + "type": "interline_equation", + "image_path": "c312c61c9595e2f90117d21a32ca791025494adc82c8490fa8d3e5b10ea1a74b.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 169, + 142, + 440, + 163 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 252, + 179 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 253, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 132, + 181 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 166, + 248, + 178 + ], + "score": 0.89, + "content": "\\rho _ { i } = \\mathbf { P } [ \\mathrm { s i g n } ( \\hat { p } _ { i } ) = \\mathrm { s i g n } ( p _ { i } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 165, + 253, + 181 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 187, + 504, + 210 + ], + "lines": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 173, + 200 + ], + "score": 1.0, + "content": "In the sign case,", + "type": "text" + }, + { + "bbox": [ + 173, + 190, + 183, + 199 + ], + "score": 0.86, + "content": "\\gamma _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 187, + 369, + 200 + ], + "score": 1.0, + "content": "is proportional to the success probability with", + "type": "text" + }, + { + "bbox": [ + 369, + 188, + 397, + 199 + ], + "score": 0.91, + "content": "\\gamma _ { i } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "if we are certain about the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 198, + 435, + 212 + ], + "spans": [ + { + "bbox": [ + 104, + 198, + 128, + 212 + ], + "score": 1.0, + "content": "sign", + "type": "text" + }, + { + "bbox": [ + 128, + 199, + 158, + 210 + ], + "score": 0.89, + "content": "\\rho _ { i } = 1 { \\ : }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 198, + 178, + 212 + ], + "score": 1.0, + "content": ") and", + "type": "text" + }, + { + "bbox": [ + 178, + 199, + 206, + 210 + ], + "score": 0.92, + "content": "\\gamma _ { i } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 198, + 396, + 212 + ], + "score": 1.0, + "content": "if we have no information about the sign at all", + "type": "text" + }, + { + "bbox": [ + 397, + 199, + 429, + 210 + ], + "score": 0.87, + "content": "\\rho _ { i } = . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 198, + 435, + 212 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 107, + 223, + 427, + 235 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 429, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 429, + 236 + ], + "score": 1.0, + "content": "3.1 VARIANCE ADAPTATION FOR THE SIGN OF A STOCHASTIC GRADIENT", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 243, + 505, + 348 + ], + "lines": [ + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 196, + 257 + ], + "score": 1.0, + "content": "Applying Eq. (15) to", + "type": "text" + }, + { + "bbox": [ + 196, + 245, + 225, + 256 + ], + "score": 0.91, + "content": "\\hat { p } = g", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 244, + 506, + 257 + ], + "score": 1.0, + "content": ", the optimal variance adaptation factors for the sign of a stochastic", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 104, + 255, + 205, + 268 + ], + "score": 1.0, + "content": "gradient are found to be", + "type": "text" + }, + { + "bbox": [ + 205, + 256, + 260, + 267 + ], + "score": 0.91, + "content": "\\gamma _ { i } = 2 \\rho _ { i } - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 255, + 291, + 268 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 291, + 255, + 419, + 267 + ], + "score": 0.91, + "content": "\\rho _ { i } = { \\bf P } [ \\mathrm { s i g n } ( g _ { i } ) = \\mathrm { s i g n } ( \\nabla { \\mathcal { L } } _ { i } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 255, + 505, + 268 + ], + "score": 1.0, + "content": ". Recall from Eq. (8)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 505, + 279 + ], + "score": 1.0, + "content": "that, under the Gaussian assumption, the success probabilities of the sign of a stochastic gradient", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 276, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 104, + 276, + 121, + 292 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 121, + 277, + 223, + 291 + ], + "score": 0.93, + "content": "2 \\rho _ { i } - 1 = \\mathrm { e r f } [ ( \\sqrt { 2 } \\eta _ { i } ) ^ { - 1 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 276, + 399, + 292 + ], + "score": 1.0, + "content": ". ADAM uses the variance adaptation factors", + "type": "text" + }, + { + "bbox": [ + 399, + 277, + 452, + 291 + ], + "score": 0.93, + "content": "( 1 + \\eta _ { i } ^ { 2 } ) ^ { - 1 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 276, + 506, + 292 + ], + "score": 1.0, + "content": ", which turns", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 290, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 246, + 304 + ], + "score": 1.0, + "content": "out to be a close approximation of", + "type": "text" + }, + { + "bbox": [ + 246, + 290, + 304, + 303 + ], + "score": 0.91, + "content": "\\mathrm { e r f } [ ( \\sqrt { 2 } \\eta _ { i } ) ^ { - 1 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 290, + 506, + 304 + ], + "score": 1.0, + "content": ", as shown in Figure 5 in the supplements. Hence,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "ADAM can be regarded as an approximate realization of this optimal variance adaptation scheme.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "We experimented with both variants and found them to have identical effects. The small difference", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 320, + 336 + ], + "score": 1.0, + "content": "between them can be regarded as insignificant when", + "type": "text" + }, + { + "bbox": [ + 320, + 326, + 327, + 336 + ], + "score": 0.81, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "itself is subject to approximation error. We", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 334, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 158, + 349 + ], + "score": 1.0, + "content": "thus stick to", + "type": "text" + }, + { + "bbox": [ + 158, + 335, + 212, + 348 + ], + "score": 0.93, + "content": "( 1 + \\eta _ { i } ^ { 2 } ) ^ { - 1 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 334, + 505, + 349 + ], + "score": 1.0, + "content": "for accordance with ADAM and to avoid the (more costly) error function.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 107, + 361, + 362, + 373 + ], + "lines": [ + { + "bbox": [ + 105, + 359, + 362, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 362, + 375 + ], + "score": 1.0, + "content": "3.2 STOCHASTIC VARIANCE-ADAPTED GRADIENT (SVAG)", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 381, + 474, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 475, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 192, + 394 + ], + "score": 1.0, + "content": "Applying Eq. 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(2013), where it appears together with diagonal curvature", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "estimates in element-wise step sizes for SGD. We refer to this method (without curvature estimates)", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 453, + 493, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 493, + 464 + ], + "score": 1.0, + "content": "as “Stochastic Variance-Adapted Gradient” (SVAG). A momentum variant will be derived below.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 468, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 482 + ], + "score": 1.0, + "content": "Intriguingly, variance adaptation of this form guarantees convergence without manually decreasing", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 479, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 251, + 493 + ], + "score": 1.0, + "content": "the global step size. We recover the", + "type": "text" + }, + { + "bbox": [ + 252, + 480, + 282, + 492 + ], + "score": 0.94, + "content": "\\mathcal { O } ( 1 / t )", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 479, + 505, + 493 + ], + "score": 1.0, + "content": "rate of SGD for smooth, strongly convex functions. We", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 490, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 420, + 504 + ], + "score": 1.0, + "content": "emphasize that this result considers an “idealized” version of SVAG with exact", + "type": "text" + }, + { + "bbox": [ + 420, + 490, + 431, + 503 + ], + "score": 0.88, + "content": "\\eta _ { i } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 491, + 505, + 504 + ], + "score": 1.0, + "content": ". It is a motivation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "for this form of variance adaptation, not a statement about the performance with estimated variances.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 516, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 514, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 175, + 531 + ], + "score": 1.0, + "content": "Theorem 1. Let", + "type": "text" + }, + { + "bbox": [ + 175, + 517, + 182, + 528 + ], + "score": 0.76, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 514, + 195, + 531 + ], + "score": 1.0, + "content": "be", + "type": "text" + }, + { + "bbox": [ + 195, + 519, + 202, + 528 + ], + "score": 0.78, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 514, + 286, + 531 + ], + "score": 1.0, + "content": "-strongly convex and", + "type": "text" + }, + { + "bbox": [ + 286, + 517, + 294, + 527 + ], + "score": 0.78, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 514, + 406, + 531 + ], + "score": 1.0, + "content": "-smooth. Assume we update", + "type": "text" + }, + { + "bbox": [ + 406, + 516, + 501, + 529 + ], + "score": 0.9, + "content": "\\theta _ { t + 1 } = \\theta _ { t } - \\alpha ( \\gamma _ { t } \\odot g _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 514, + 505, + 531 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 527, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 133, + 542 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 529, + 142, + 540 + ], + "score": 0.82, + "content": "g _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 527, + 260, + 542 + ], + "score": 1.0, + "content": "is a stochastic gradient with", + "type": "text" + }, + { + "bbox": [ + 261, + 528, + 338, + 540 + ], + "score": 0.55, + "content": "\\mathbf { E } [ g _ { t } | \\theta _ { t } ] = \\nabla f ( \\theta _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 527, + 343, + 542 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 343, + 528, + 418, + 541 + ], + "score": 0.88, + "content": "\\mathbf { v a r } [ g _ { t , i } | \\theta _ { t } ] = \\sigma _ { t , i } ^ { 2 } .", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 527, + 506, + 542 + ], + "score": 1.0, + "content": ", variance adaptation", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 540, + 429, + 554 + ], + "spans": [ + { + "bbox": [ + 104, + 540, + 136, + 554 + ], + "score": 1.0, + "content": "factors", + "type": "text" + }, + { + "bbox": [ + 137, + 540, + 241, + 554 + ], + "score": 0.94, + "content": "\\gamma _ { t , i } = ( 1 + \\sigma _ { t , i } ^ { 2 } / \\nabla f _ { t , i } ^ { 2 } ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 540, + 263, + 554 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 263, + 541, + 301, + 553 + ], + "score": 0.91, + "content": "\\alpha = 1 / L", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 540, + 338, + 554 + ], + "score": 1.0, + "content": ". 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As noted in the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 648, + 509, + 684 + ], + "spans": [ + { + "bbox": [ + 107, + 659, + 204, + 673 + ], + "score": 0.91, + "content": "\\sigma _ { t , i } ^ { 2 } \\approx \\widehat { s } _ { t , i } = v _ { t , i } - m _ { t , i } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 648, + 509, + 684 + ], + "score": 1.0, + "content": "ains an estimate of the stochastic gradient variance from moving averages,. The underlying assumption is that the function does not change drastically horizon” of the moving average, such that the recent gradients can approx-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 682, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 505, + 693 + ], + "score": 1.0, + "content": "imately be considered to be iid draws from the stochastic gradient distribution. 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Hence,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "ADAM can be regarded as an approximate realization of this optimal variance adaptation scheme.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "We experimented with both variants and found them to have identical effects. 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(2013), where it appears together with diagonal curvature", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "estimates in element-wise step sizes for SGD. We refer to this method (without curvature estimates)", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 453, + 493, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 493, + 464 + ], + "score": 1.0, + "content": "as “Stochastic Variance-Adapted Gradient” (SVAG). 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Let", + "type": "text" + }, + { + "bbox": [ + 175, + 517, + 182, + 528 + ], + "score": 0.76, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 514, + 195, + 531 + ], + "score": 1.0, + "content": "be", + "type": "text" + }, + { + "bbox": [ + 195, + 519, + 202, + 528 + ], + "score": 0.78, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 514, + 286, + 531 + ], + "score": 1.0, + "content": "-strongly convex and", + "type": "text" + }, + { + "bbox": [ + 286, + 517, + 294, + 527 + ], + "score": 0.78, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 514, + 406, + 531 + ], + "score": 1.0, + "content": "-smooth. 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Then", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 514, + 506, + 554 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 252, + 559, + 359, + 587 + ], + "lines": [ + { + "bbox": [ + 252, + 559, + 359, + 587 + ], + "spans": [ + { + "bbox": [ + 252, + 559, + 359, + 587 + ], + "score": 0.95, + "content": "\\mathbf { E } [ f ( \\theta _ { t } ) - f _ { * } ] \\in \\mathcal { O } \\left( \\frac { 1 } { t } \\right) ,", + "type": "interline_equation", + "image_path": "c96fada4400f2865b94bc5250b3be005eba475fb5c198c224ff4708f21a28590.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 252, + 559, + 359, + 587 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 592, + 252, + 604 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 253, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 133, + 605 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 592, + 143, + 603 + ], + "score": 0.87, + "content": "f _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 590, + 243, + 605 + ], + "score": 1.0, + "content": "is the minimum value of", + "type": "text" + }, + { + "bbox": [ + 243, + 592, + 249, + 604 + ], + "score": 0.81, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 590, + 253, + 605 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 590, + 253, + 605 + ] + }, + { + "type": "text", + "bbox": [ + 443, + 592, + 504, + 604 + ], + "lines": [ + { + "bbox": [ + 442, + 591, + 503, + 605 + ], + "spans": [ + { + "bbox": [ + 442, + 591, + 481, + 605 + ], + "score": 1.0, + "content": "(Proof in", + "type": "text" + }, + { + "bbox": [ + 481, + 592, + 503, + 604 + ], + "score": 0.31, + "content": "\\ S B . 4 )", + "type": "inline_equation" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 442, + 591, + 503, + 605 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 616, + 282, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 283, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 283, + 630 + ], + "score": 1.0, + "content": "3.3 ESTIMATING GRADIENT VARIANCE", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 705 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "In practice, the relative variance is of course not known and must be estimated. As noted in the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 648, + 509, + 684 + ], + "spans": [ + { + "bbox": [ + 107, + 659, + 204, + 673 + ], + "score": 0.91, + "content": "\\sigma _ { t , i } ^ { 2 } \\approx \\widehat { s } _ { t , i } = v _ { t , i } - m _ { t , i } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 648, + 509, + 684 + ], + "score": 1.0, + "content": "ains an estimate of the stochastic gradient variance from moving averages,. The underlying assumption is that the function does not change drastically horizon” of the moving average, such that the recent gradients can approx-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 682, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 505, + 693 + ], + "score": 1.0, + "content": "imately be considered to be iid draws from the stochastic gradient distribution. An estimate of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 692, + 396, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 272, + 705 + ], + "score": 1.0, + "content": "relative variance can then be obtained by", + "type": "text" + }, + { + "bbox": [ + 272, + 692, + 340, + 705 + ], + "score": 0.91, + "content": "( v _ { t } - m _ { t } ^ { 2 } ) / ( m _ { t } ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 693, + 396, + 705 + ], + "score": 1.0, + "content": ", as in ADAM.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 638, + 509, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 386, + 722 + ], + "score": 1.0, + "content": "Unlike ADAM we do not use different moving average constants for", + "type": "text" + }, + { + "bbox": [ + 387, + 712, + 400, + 721 + ], + "score": 0.84, + "content": "m _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 709, + 419, + 722 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 419, + 712, + 429, + 721 + ], + "score": 0.83, + "content": "v _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 709, + 506, + 722 + ], + "score": 1.0, + "content": ". The constant for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "the moving average should define a time horizon over which the gradients can approximately be", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "considered to come from the same distribution. From this perspective, it is hardly justifiable to use", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "different horizons for the gradient and its square. 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From this perspective, it is hardly justifiable to use", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "different horizons for the gradient and its square. 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A more", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 319, + 145 + ], + "score": 1.0, + "content": "detailed discussion of both estimators can be found in", + "type": "text" + }, + { + "bbox": [ + 320, + 132, + 333, + 144 + ], + "score": 0.83, + "content": "\\mathrm { \\ S C }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "of the supplements. We have experimented", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "with both estimators and found them to work equally well for our purpose of variance adaptation.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "We thus stick to moving average-based estimates for the main paper. Appendix D provides details", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 312, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 312, + 177 + ], + "score": 1.0, + "content": "and experimental results for the mini-batch variant.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 108, + 201, + 259, + 213 + ], + "lines": [ + { + "bbox": [ + 106, + 201, + 261, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 261, + 213 + ], + "score": 1.0, + "content": "3.4 INCORPORATING MOMENTUM", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 262 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 373, + 239 + ], + "score": 1.0, + "content": "When we add momentum—i.e., we want to update in the direction", + "type": "text" + }, + { + "bbox": [ + 373, + 228, + 383, + 238 + ], + "score": 0.84, + "content": "r _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 225, + 394, + 239 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 395, + 226, + 429, + 239 + ], + "score": 0.87, + "content": "\\mathrm { s i g n } ( r _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 225, + 505, + 239 + ], + "score": 1.0, + "content": "with a momentum", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 238, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 238, + 127, + 255 + ], + "score": 1.0, + "content": "term", + "type": "text" + }, + { + "bbox": [ + 128, + 238, + 268, + 252 + ], + "score": 0.91, + "content": "\\begin{array} { r } { r _ { t } = \\mu r _ { t - 1 } + g _ { t } = \\sum _ { s = 0 } ^ { t } \\mu ^ { s } g _ { t - s } . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 238, + 506, + 255 + ], + "score": 1.0, + "content": "—the variance adaptation factors should be determined by", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 250, + 327, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 200, + 263 + ], + "score": 1.0, + "content": "the relative variance of", + "type": "text" + }, + { + "bbox": [ + 200, + 253, + 209, + 262 + ], + "score": 0.84, + "content": "r _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 250, + 327, + 263 + ], + "score": 1.0, + "content": ", according to Lemma 1. It is", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 277, + 459, + 311 + ], + "lines": [ + { + "bbox": [ + 152, + 277, + 459, + 311 + ], + "spans": [ + { + "bbox": [ + 152, + 277, + 459, + 311 + ], + "score": 0.94, + "content": "\\mathbf { E } [ r _ { t } ] = \\sum _ { s = 0 } ^ { t } \\mu ^ { s } \\nabla { \\mathcal { L } } _ { t - s } , \\quad \\mathbf { v a r } [ r _ { t , i } ] = \\sum _ { s = 0 } ^ { t } ( \\mu ^ { s } ) ^ { 2 } \\mathbf { v a r } [ g _ { t - s , i } ] = \\sum _ { s = 0 } ^ { t } \\mu ^ { 2 s } \\sigma _ { t - s , i } ^ { 2 } .", + "type": "interline_equation", + "image_path": "f171194e145aabca1c152fd192fb82d9b0ba5fcc7e511756aeaffe20a5f1a770.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 152, + 277, + 459, + 288.3333333333333 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 152, + 288.3333333333333, + 459, + 299.66666666666663 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 152, + 299.66666666666663, + 459, + 310.99999999999994 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 326, + 504, + 372 + ], + "lines": [ + { + "bbox": [ + 103, + 324, + 507, + 343 + ], + "spans": [ + { + "bbox": [ + 103, + 324, + 151, + 343 + ], + "score": 1.0, + "content": "Replacing", + "type": "text" + }, + { + "bbox": [ + 151, + 327, + 223, + 339 + ], + "score": 0.92, + "content": "\\mathbf { E } [ g _ { t - s } ] \\approx m _ { t - s }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 324, + 243, + 343 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 244, + 327, + 356, + 340 + ], + "score": 0.91, + "content": "\\mathbf { v a r } [ g _ { t - s } ] \\approx v _ { t - s } - m _ { t - s } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 324, + 507, + 343 + ], + "score": 1.0, + "content": "we could compute these quantities.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "However, this would require two additional moving averages and can thus be discarded as imprac-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 349, + 504, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 504, + 362 + ], + "score": 1.0, + "content": "tical. Fortunately, we can motivate an approximation that does not require any additional memory", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 360, + 199, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 179, + 372 + ], + "score": 1.0, + "content": "requirements (see", + "type": "text" + }, + { + "bbox": [ + 180, + 361, + 194, + 372 + ], + "score": 0.79, + "content": "\\mathrm { \\ S C ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 360, + 199, + 372 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 167, + 387, + 444, + 415 + ], + "lines": [ + { + "bbox": [ + 167, + 387, + 444, + 415 + ], + "spans": [ + { + "bbox": [ + 167, + 387, + 444, + 415 + ], + "score": 0.92, + "content": "\\frac { \\mathbf { v a r } [ r _ { t } ] } { \\mathbf { E } [ r _ { t } ] ^ { 2 } } \\approx \\kappa ( \\mu , t ) \\frac { v _ { t } - m _ { t } ^ { 2 } } { m _ { t } ^ { 2 } } ~ \\mathrm { w i t h } ~ \\kappa ( \\mu , t ) : = \\frac { ( 1 - \\mu ^ { 2 t } ) ( 1 - \\mu ) ^ { 2 } } { ( 1 - \\mu ^ { 2 } ) ( 1 - \\mu ^ { t } ) ^ { 2 } } .", + "type": "interline_equation", + "image_path": "6b47b32ad54de2f00e894c5a095152de344198a153811745f9609be234099aa3.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 167, + 387, + 444, + 396.3333333333333 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 167, + 396.3333333333333, + 444, + 405.66666666666663 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 167, + 405.66666666666663, + 444, + 414.99999999999994 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 429, + 505, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 234, + 443 + ], + "score": 1.0, + "content": "Note that the correction factor", + "type": "text" + }, + { + "bbox": [ + 235, + 430, + 263, + 442 + ], + "score": 0.93, + "content": "\\kappa ( \\mu , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "does not appear in ADAM, which updates in the direction", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 440, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 107, + 441, + 192, + 453 + ], + "score": 0.91, + "content": "\\mathrm { s i g n } ( m _ { t } ) = \\mathrm { s i g n } ( r _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 440, + 368, + 454 + ], + "score": 1.0, + "content": "but performs variance adaptation based on", + "type": "text" + }, + { + "bbox": [ + 368, + 441, + 429, + 453 + ], + "score": 0.93, + "content": "( v _ { t } - m _ { t } ^ { 2 } ) \\bar { / } m _ { t } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 440, + 505, + 454 + ], + "score": 1.0, + "content": ". The supplements", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 452, + 427, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 427, + 464 + ], + "score": 1.0, + "content": "contain experiments with a variant of ADAM that includes this correction factor.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 107, + 491, + 200, + 503 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 201, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 201, + 505 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "We compare momentum-SGD (M-SGD) and ADAM to two new methods: First, we consider M-SSD:", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "stochastic sign descent using a momentum term. The second method is M-SVAG, i.e., SGD with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 309, + 557 + ], + "score": 1.0, + "content": "momentum and variance adaptation of the form", + "type": "text" + }, + { + "bbox": [ + 309, + 544, + 357, + 557 + ], + "score": 0.93, + "content": "( 1 + \\eta ^ { 2 } ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 544, + 506, + 557 + ], + "score": 1.0, + "content": ", where the relative variance of the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "momentum term is estimated from moving averages according to Eq. (19). These four methods are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "the four possible recombinations of the sign aspect and the variance adaptation aspect of ADAM,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 578, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 590 + ], + "score": 1.0, + "content": "as laid out in Table 1. Algorithms 1 and 2 provide pseudo-code for M-SSD and M-SVAG. For", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 104, + 588, + 206, + 601 + ], + "score": 1.0, + "content": "all experiments, we use", + "type": "text" + }, + { + "bbox": [ + 206, + 589, + 243, + 599 + ], + "score": 0.89, + "content": "\\mu = 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 588, + 479, + 601 + ], + "score": 1.0, + "content": "for M-SGD, M-SSD and M-SVAG and default parameters", + "type": "text" + }, + { + "bbox": [ + 480, + 588, + 505, + 600 + ], + "score": 0.9, + "content": "\\beta _ { 1 } =", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 599, + 501, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 215, + 611 + ], + "score": 0.82, + "content": "0 . 9 , \\mathring { \\beta _ { 2 } } = 0 . 9 9 9 , \\varepsilon = 1 0 ^ { - 8 } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 599, + 403, + 612 + ], + "score": 1.0, + "content": "for ADAM. Note that M-SVAG does not use an", + "type": "text" + }, + { + "bbox": [ + 403, + 602, + 409, + 609 + ], + "score": 0.72, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 599, + 501, + 612 + ], + "score": 1.0, + "content": "-parameter, see Alg. 2.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29.5 + }, + { + "type": "title", + "bbox": [ + 107, + 634, + 365, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 365, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 365, + 648 + ], + "score": 1.0, + "content": "Algorithm 1 M-SSD (Stochastic Sign Descent with Momentum)", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "table", + "bbox": [ + 108, + 648, + 475, + 729 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 648, + 475, + 729 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 648, + 475, + 729 + ], + "spans": [ + { + "bbox": [ + 110, + 648, + 475, + 729 + ], + "score": 0.546, + "html": "
Require: initial value 0o,step size α, momentum parameter μ ∈ [0,1], number of steps T
1:Initialize m= O,v = 0
2: for t =1,...,T do
3: 4:Compute stochastic gradient g = g(0)
5:Update moving average m ← μm + g
6: end forUpdateθ ←θ-α sign(m)
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A more", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 319, + 145 + ], + "score": 1.0, + "content": "detailed discussion of both estimators can be found in", + "type": "text" + }, + { + "bbox": [ + 320, + 132, + 333, + 144 + ], + "score": 0.83, + "content": "\\mathrm { \\ S C }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "of the supplements. We have experimented", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "with both estimators and found them to work equally well for our purpose of variance adaptation.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "We thus stick to moving average-based estimates for the main paper. Appendix D provides details", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 312, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 312, + 177 + ], + "score": 1.0, + "content": "and experimental results for the mini-batch variant.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 121, + 505, + 177 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 201, + 259, + 213 + ], + "lines": [ + { + "bbox": [ + 106, + 201, + 261, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 261, + 213 + ], + "score": 1.0, + "content": "3.4 INCORPORATING MOMENTUM", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 262 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 373, + 239 + ], + "score": 1.0, + "content": "When we add momentum—i.e., we want to update in the direction", + "type": "text" + }, + { + "bbox": [ + 373, + 228, + 383, + 238 + ], + "score": 0.84, + "content": "r _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 225, + 394, + 239 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 395, + 226, + 429, + 239 + ], + "score": 0.87, + "content": "\\mathrm { s i g n } ( r _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 225, + 505, + 239 + ], + "score": 1.0, + "content": "with a momentum", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 238, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 238, + 127, + 255 + ], + "score": 1.0, + "content": "term", + "type": "text" + }, + { + "bbox": [ + 128, + 238, + 268, + 252 + ], + "score": 0.91, + "content": "\\begin{array} { r } { r _ { t } = \\mu r _ { t - 1 } + g _ { t } = \\sum _ { s = 0 } ^ { t } \\mu ^ { s } g _ { t - s } . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 238, + 506, + 255 + ], + "score": 1.0, + "content": "—the variance adaptation factors should be determined by", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 250, + 327, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 200, + 263 + ], + "score": 1.0, + "content": "the relative variance of", + "type": "text" + }, + { + "bbox": [ + 200, + 253, + 209, + 262 + ], + "score": 0.84, + "content": "r _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 250, + 327, + 263 + ], + "score": 1.0, + "content": ", according to Lemma 1. It is", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 104, + 225, + 506, + 263 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 277, + 459, + 311 + ], + "lines": [ + { + "bbox": [ + 152, + 277, + 459, + 311 + ], + "spans": [ + { + "bbox": [ + 152, + 277, + 459, + 311 + ], + "score": 0.94, + "content": "\\mathbf { E } [ r _ { t } ] = \\sum _ { s = 0 } ^ { t } \\mu ^ { s } \\nabla { \\mathcal { L } } _ { t - s } , \\quad \\mathbf { v a r } [ r _ { t , i } ] = \\sum _ { s = 0 } ^ { t } ( \\mu ^ { s } ) ^ { 2 } \\mathbf { v a r } [ g _ { t - s , i } ] = \\sum _ { s = 0 } ^ { t } \\mu ^ { 2 s } \\sigma _ { t - s , i } ^ { 2 } .", + "type": "interline_equation", + "image_path": "f171194e145aabca1c152fd192fb82d9b0ba5fcc7e511756aeaffe20a5f1a770.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 152, + 277, + 459, + 288.3333333333333 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 152, + 288.3333333333333, + 459, + 299.66666666666663 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 152, + 299.66666666666663, + 459, + 310.99999999999994 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 326, + 504, + 372 + ], + "lines": [ + { + "bbox": [ + 103, + 324, + 507, + 343 + ], + "spans": [ + { + "bbox": [ + 103, + 324, + 151, + 343 + ], + "score": 1.0, + "content": "Replacing", + "type": "text" + }, + { + "bbox": [ + 151, + 327, + 223, + 339 + ], + "score": 0.92, + "content": "\\mathbf { E } [ g _ { t - s } ] \\approx m _ { t - s }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 324, + 243, + 343 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 244, + 327, + 356, + 340 + ], + "score": 0.91, + "content": "\\mathbf { v a r } [ g _ { t - s } ] \\approx v _ { t - s } - m _ { t - s } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 324, + 507, + 343 + ], + "score": 1.0, + "content": "we could compute these quantities.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "However, this would require two additional moving averages and can thus be discarded as imprac-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 349, + 504, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 504, + 362 + ], + "score": 1.0, + "content": "tical. 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The supplements", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 452, + 427, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 427, + 464 + ], + "score": 1.0, + "content": "contain experiments with a variant of ADAM that includes this correction factor.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 429, + 505, + 464 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 491, + 200, + 503 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 201, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 201, + 505 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 505, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "We compare momentum-SGD (M-SGD) and ADAM to two new methods: First, we consider M-SSD:", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 545 + ], + "score": 1.0, + "content": "stochastic sign descent using a momentum term. The second method is M-SVAG, i.e., SGD with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 309, + 557 + ], + "score": 1.0, + "content": "momentum and variance adaptation of the form", + "type": "text" + }, + { + "bbox": [ + 309, + 544, + 357, + 557 + ], + "score": 0.93, + "content": "( 1 + \\eta ^ { 2 } ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 544, + 506, + 557 + ], + "score": 1.0, + "content": ", where the relative variance of the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "momentum term is estimated from moving averages according to Eq. (19). These four methods are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "the four possible recombinations of the sign aspect and the variance adaptation aspect of ADAM,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 578, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 590 + ], + "score": 1.0, + "content": "as laid out in Table 1. Algorithms 1 and 2 provide pseudo-code for M-SSD and M-SVAG. For", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 104, + 588, + 206, + 601 + ], + "score": 1.0, + "content": "all experiments, we use", + "type": "text" + }, + { + "bbox": [ + 206, + 589, + 243, + 599 + ], + "score": 0.89, + "content": "\\mu = 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 588, + 479, + 601 + ], + "score": 1.0, + "content": "for M-SGD, M-SSD and M-SVAG and default parameters", + "type": "text" + }, + { + "bbox": [ + 480, + 588, + 505, + 600 + ], + "score": 0.9, + "content": "\\beta _ { 1 } =", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 599, + 501, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 215, + 611 + ], + "score": 0.82, + "content": "0 . 9 , \\mathring { \\beta _ { 2 } } = 0 . 9 9 9 , \\varepsilon = 1 0 ^ { - 8 } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 599, + 403, + 612 + ], + "score": 1.0, + "content": "for ADAM. Note that M-SVAG does not use an", + "type": "text" + }, + { + "bbox": [ + 403, + 602, + 409, + 609 + ], + "score": 0.72, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 599, + 501, + 612 + ], + "score": 1.0, + "content": "-parameter, see Alg. 2.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 522, + 506, + 612 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 634, + 365, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 365, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 365, + 648 + ], + "score": 1.0, + "content": "Algorithm 1 M-SSD (Stochastic Sign Descent with Momentum)", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "table", + "bbox": [ + 108, + 648, + 475, + 729 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 648, + 475, + 729 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 648, + 475, + 729 + ], + "spans": [ + { + "bbox": [ + 110, + 648, + 475, + 729 + ], + "score": 0.546, + "html": "
Require: initial value 0o,step size α, momentum parameter μ ∈ [0,1], number of steps T
1:Initialize m= O,v = 0
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3: 4:Compute stochastic gradient g = g(0)
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6: end forUpdateθ ←θ-α sign(m)
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In the (rare) case that", + "type": "text" + }, + { + "bbox": [ + 383, + 215, + 418, + 226 + ], + "score": 0.91, + "content": "m _ { i } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 214, + 480, + 226 + ], + "score": 1.0, + "content": "for coordinate", + "type": "text" + }, + { + "bbox": [ + 481, + 215, + 485, + 224 + ], + "score": 0.69, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 214, + 505, + 226 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 225, + 463, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 463, + 238 + ], + "score": 1.0, + "content": "division by zero in line 6 is caught and the update magnitude will be set to zero in line 8.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 260, + 234, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 259, + 235, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 235, + 272 + ], + "score": 1.0, + "content": "4.1 EXPERIMENTAL SET-UP", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 106, + 282, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 293 + ], + "score": 1.0, + "content": "We tested all methods on three problems: a simple fully-connected neural network on the MNIST", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "score": 1.0, + "content": "data set (LeCun et al., 1998), as well as convolutional neural networks (CNNs) on the CIFAR-10", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "and CIFAR-100 data sets (Krizhevsky, 2009). On CIFAR-10, we used a simple CNN with three", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 315, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 326 + ], + "score": 1.0, + "content": "convolutional layers, interspersed with max-pooling, and three fully-connected layers. On CIFAR-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "score": 1.0, + "content": "100 we used the AllCNN architecture of Springenberg et al. (2014) with a total of nine convolutional", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 416, + 348 + ], + "score": 1.0, + "content": "layers. A complete description of all network architectures has been moved to", + "type": "text" + }, + { + "bbox": [ + 416, + 336, + 429, + 348 + ], + "score": 0.73, + "content": "\\ S \\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 337, + 505, + 348 + ], + "score": 1.0, + "content": ". While MNIST and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 324, + 360 + ], + "score": 1.0, + "content": "CIFAR-10 are trained with a constant global step size", + "type": "text" + }, + { + "bbox": [ + 324, + 348, + 338, + 358 + ], + "score": 0.64, + "content": "( \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 348, + 505, + 360 + ], + "score": 1.0, + "content": ", we used a fixed decreasing schedule for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 240, + 371 + ], + "score": 1.0, + "content": "CIFAR-100, dividing by 10 after", + "type": "text" + }, + { + "bbox": [ + 241, + 358, + 258, + 369 + ], + "score": 0.37, + "content": "4 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 358, + 276, + 371 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 276, + 358, + 293, + 369 + ], + "score": 0.29, + "content": "5 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "steps (adopted from Springenberg et al., 2014). We", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 369, + 392, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 392, + 382 + ], + "score": 1.0, + "content": "used a batch size of 128 on MNIST and 256 on the two CIFAR data sets.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 386, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "Step sizes (initial step sizes in the case of CIFAR-100) were tuned for each method individually by", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 397, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 408 + ], + "score": 1.0, + "content": "first finding the maximal stable step size by trial and error, then searching downwards over two orders", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 204, + 420 + ], + "score": 1.0, + "content": "of magnitude (details in", + "type": "text" + }, + { + "bbox": [ + 204, + 408, + 218, + 420 + ], + "score": 0.73, + "content": "\\ S \\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "). We selected the one that yielded maximal overall test accuracy within", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "score": 1.0, + "content": "the fixed number of training steps. Experiments with the best step size have been replicated ten", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "times with different random seeds and all performance indicators are reported as mean plus/minus", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 442, + 201, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 201, + 453 + ], + "score": 1.0, + "content": "one standard deviation.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 107, + 468, + 170, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 172, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 172, + 480 + ], + "score": 1.0, + "content": "4.2 RESULTS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 489, + 504, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "Results are shown in Figure 3. On MNIST, ADAM clearly outperforms M-SGD. Interestingly, there", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "score": 1.0, + "content": "is only a very small difference in performance between the two sign-based methods, M-SSD and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "ADAM. Apparently, the advantage of ADAM over M-SGD on this problem is primarily due to the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 521, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 535 + ], + "score": 1.0, + "content": "sign aspect. Going from M-SGD to M-SVAG, gives a considerable boost in performance, but M-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 533, + 343, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 343, + 545 + ], + "score": 1.0, + "content": "SVAG is still outperformed by the two sign-based methods.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 503, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "On CIFAR-10, the sign-based methods again have superior performance. Neither M-SSD nor M-SGD", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 561, + 338, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 338, + 573 + ], + "score": 1.0, + "content": "can benefit significantly from adding variance adaptation.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 590 + ], + "score": 1.0, + "content": "Finally, the situation is reversed on CIFAR-100, where M-SGD outperforms ADAM. It attains lower", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 589, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 505, + 600 + ], + "score": 1.0, + "content": "minimal loss values (both training and test) and converges faster. This is also reflected in the test", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "accuracies, where M-SGD beats ADAM by almost 10 percentage points. Furthermore, ADAM is much", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "less stable with significantly larger variance in performance. On this problem, variance adaptation", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "score": 1.0, + "content": "has a small but significant positive effect for the sign-based methods as well as for M-SGD. When", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "going from M-SGD to M-SVAG we gain some speed in the initial phase. The difference is later", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "evened out by the manual learning rate decrease (which was necessary, for all methods, to train this", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 266, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 266, + 667 + ], + "score": 1.0, + "content": "architecture to satisfying performance).", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5 + }, + { + "type": "title", + "bbox": [ + 108, + 683, + 287, + 696 + ], + "lines": [ + { + "bbox": [ + 104, + 681, + 289, + 699 + ], + "spans": [ + { + "bbox": [ + 104, + 681, + 289, + 699 + ], + "score": 1.0, + "content": "5 DISCUSSION AND CONCLUSION", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We have argued that ADAM combines two aspects: taking signs and variance adaptation. 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In the (rare) case that", + "type": "text" + }, + { + "bbox": [ + 383, + 215, + 418, + 226 + ], + "score": 0.91, + "content": "m _ { i } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 214, + 480, + 226 + ], + "score": 1.0, + "content": "for coordinate", + "type": "text" + }, + { + "bbox": [ + 481, + 215, + 485, + 224 + ], + "score": 0.69, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 214, + 505, + 226 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 225, + 463, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 463, + 238 + ], + "score": 1.0, + "content": "division by zero in line 6 is caught and the update magnitude will be set to zero in line 8.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 106, + 214, + 505, + 238 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 260, + 234, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 259, + 235, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 235, + 272 + ], + "score": 1.0, + "content": "4.1 EXPERIMENTAL SET-UP", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 106, + 282, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 293 + ], + "score": 1.0, + "content": "We tested all methods on three problems: a simple fully-connected neural network on the MNIST", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "score": 1.0, + "content": "data set (LeCun et al., 1998), as well as convolutional neural networks (CNNs) on the CIFAR-10", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "and CIFAR-100 data sets (Krizhevsky, 2009). On CIFAR-10, we used a simple CNN with three", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 315, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 326 + ], + "score": 1.0, + "content": "convolutional layers, interspersed with max-pooling, and three fully-connected layers. On CIFAR-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "score": 1.0, + "content": "100 we used the AllCNN architecture of Springenberg et al. (2014) with a total of nine convolutional", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 416, + 348 + ], + "score": 1.0, + "content": "layers. A complete description of all network architectures has been moved to", + "type": "text" + }, + { + "bbox": [ + 416, + 336, + 429, + 348 + ], + "score": 0.73, + "content": "\\ S \\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 337, + 505, + 348 + ], + "score": 1.0, + "content": ". While MNIST and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 324, + 360 + ], + "score": 1.0, + "content": "CIFAR-10 are trained with a constant global step size", + "type": "text" + }, + { + "bbox": [ + 324, + 348, + 338, + 358 + ], + "score": 0.64, + "content": "( \\alpha )", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 348, + 505, + 360 + ], + "score": 1.0, + "content": ", we used a fixed decreasing schedule for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 240, + 371 + ], + "score": 1.0, + "content": "CIFAR-100, dividing by 10 after", + "type": "text" + }, + { + "bbox": [ + 241, + 358, + 258, + 369 + ], + "score": 0.37, + "content": "4 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 358, + 276, + 371 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 276, + 358, + 293, + 369 + ], + "score": 0.29, + "content": "5 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "steps (adopted from Springenberg et al., 2014). We", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 369, + 392, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 392, + 382 + ], + "score": 1.0, + "content": "used a batch size of 128 on MNIST and 256 on the two CIFAR data sets.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 282, + 505, + 382 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 386, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "Step sizes (initial step sizes in the case of CIFAR-100) were tuned for each method individually by", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 397, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 408 + ], + "score": 1.0, + "content": "first finding the maximal stable step size by trial and error, then searching downwards over two orders", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 408, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 204, + 420 + ], + "score": 1.0, + "content": "of magnitude (details in", + "type": "text" + }, + { + "bbox": [ + 204, + 408, + 218, + 420 + ], + "score": 0.73, + "content": "\\ S \\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 408, + 505, + 420 + ], + "score": 1.0, + "content": "). We selected the one that yielded maximal overall test accuracy within", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 432 + ], + "score": 1.0, + "content": "the fixed number of training steps. Experiments with the best step size have been replicated ten", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "times with different random seeds and all performance indicators are reported as mean plus/minus", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 442, + 201, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 201, + 453 + ], + "score": 1.0, + "content": "one standard deviation.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 386, + 505, + 453 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 468, + 170, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 172, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 172, + 480 + ], + "score": 1.0, + "content": "4.2 RESULTS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 489, + 504, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "Results are shown in Figure 3. On MNIST, ADAM clearly outperforms M-SGD. Interestingly, there", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "score": 1.0, + "content": "is only a very small difference in performance between the two sign-based methods, M-SSD and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "ADAM. Apparently, the advantage of ADAM over M-SGD on this problem is primarily due to the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 521, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 535 + ], + "score": 1.0, + "content": "sign aspect. Going from M-SGD to M-SVAG, gives a considerable boost in performance, but M-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 533, + 343, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 343, + 545 + ], + "score": 1.0, + "content": "SVAG is still outperformed by the two sign-based methods.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 489, + 506, + 545 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 503, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "On CIFAR-10, the sign-based methods again have superior performance. Neither M-SSD nor M-SGD", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 561, + 338, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 338, + 573 + ], + "score": 1.0, + "content": "can benefit significantly from adding variance adaptation.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 549, + 505, + 573 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 590 + ], + "score": 1.0, + "content": "Finally, the situation is reversed on CIFAR-100, where M-SGD outperforms ADAM. It attains lower", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 589, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 505, + 600 + ], + "score": 1.0, + "content": "minimal loss values (both training and test) and converges faster. 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When", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "going from M-SGD to M-SVAG we gain some speed in the initial phase. The difference is later", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "evened out by the manual learning rate decrease (which was necessary, for all methods, to train this", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 266, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 266, + 667 + ], + "score": 1.0, + "content": "architecture to satisfying performance).", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 578, + 506, + 667 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 683, + 287, + 696 + ], + "lines": [ + { + "bbox": [ + 104, + 681, + 289, + 699 + ], + "spans": [ + { + "bbox": [ + 104, + 681, + 289, + 699 + ], + "score": 1.0, + "content": "5 DISCUSSION AND CONCLUSION", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We have argued that ADAM combines two aspects: taking signs and variance adaptation. 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We train a convolutional neural network (CNN) with three convolutional layers (64 filters", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 235, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 135, + 249 + ], + "score": 1.0, + "content": "of size", + "type": "text" + }, + { + "bbox": [ + 135, + 236, + 154, + 246 + ], + "score": 0.84, + "content": "5 \\times 5", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 235, + 225, + 249 + ], + "score": 1.0, + "content": ", 96 filters of size", + "type": "text" + }, + { + "bbox": [ + 225, + 236, + 244, + 246 + ], + "score": 0.89, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 235, + 336, + 249 + ], + "score": 1.0, + "content": ", and 128 filters of size", + "type": "text" + }, + { + "bbox": [ + 337, + 236, + 356, + 246 + ], + "score": 0.86, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 235, + 506, + 249 + ], + "score": 1.0, + "content": ") interspersed with max-pooling over", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 247, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 126, + 258 + ], + "score": 0.87, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "areas with stride 2. Two fully-connected layers with 512 and 256 units follow. We use ReLU", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "activation function for all layers. The output layer has 10 units for the 10 classes of CIFAR-10", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 268, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 407, + 281 + ], + "score": 1.0, + "content": "with softmax activation. 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We use", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 302, + 354, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 284, + 315 + ], + "score": 1.0, + "content": "a batch size of 256. The global learning rate", + "type": "text" + }, + { + "bbox": [ + 285, + 304, + 293, + 312 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 302, + 354, + 315 + ], + "score": 1.0, + "content": "stays constant.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 326, + 505, + 404 + ], + "lines": [ + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "CIFAR-100 We use the AllCNN architecture of Springenberg et al. (2014). It consists of seven", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "convolutional layers, some of them with stride, and no pooling layers. The fully-connected layers", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 235, + 361 + ], + "score": 1.0, + "content": "are replaced with two layers of", + "type": "text" + }, + { + "bbox": [ + 236, + 349, + 255, + 360 + ], + "score": 0.87, + "content": "1 \\times 1", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "convolutions with global spatial averaging in the end. 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The global learning", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 392, + 349, + 406 + ], + "spans": [ + { + "bbox": [ + 104, + 392, + 124, + 406 + ], + "score": 1.0, + "content": "rate", + "type": "text" + }, + { + "bbox": [ + 124, + 394, + 132, + 403 + ], + "score": 0.8, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 392, + 272, + 406 + ], + "score": 1.0, + "content": "is decreased by a factor of 10 after", + "type": "text" + }, + { + "bbox": [ + 273, + 393, + 289, + 403 + ], + "score": 0.54, + "content": "4 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 392, + 307, + 406 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 307, + 393, + 324, + 403 + ], + "score": 0.43, + "content": "5 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 392, + 349, + 406 + ], + "score": 1.0, + "content": "steps.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19 + }, + { + "type": "title", + "bbox": [ + 108, + 419, + 243, + 430 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 244, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 244, + 432 + ], + "score": 1.0, + "content": "A.2 LEARNING RATE TUNING", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 453 + ], + "score": 1.0, + "content": "Learning rates for each optimizer have been tuned by first finding the maximal stable learning rate", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 104, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "by trial and error and then searching downwards over two orders of magnitude with learning rates", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 137, + 473 + ], + "score": 0.73, + "content": "6 \\cdot 1 0 ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 461, + 141, + 475 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 141, + 462, + 172, + 473 + ], + "score": 0.74, + "content": "3 \\cdot 1 0 ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 461, + 193, + 475 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 193, + 462, + 224, + 473 + ], + "score": 0.89, + "content": "1 \\cdot 1 0 ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 461, + 318, + 475 + ], + "score": 1.0, + "content": "for order of magnitude", + "type": "text" + }, + { + "bbox": [ + 318, + 464, + 329, + 473 + ], + "score": 0.56, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 461, + 505, + 475 + ], + "score": 1.0, + "content": ". 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For SGD, we have", + "type": "text" + }, + { + "bbox": [ + 230, + 583, + 278, + 595 + ], + "score": 0.92, + "content": "\\mathbf { E } [ g ] = \\nabla \\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 580, + 297, + 596 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 297, + 582, + 458, + 595 + ], + "score": 0.92, + "content": "\\mathbf { E } [ g ^ { T } Q g ] = \\nabla \\mathcal { L } ^ { \\hat { T } } Q \\nabla \\mathcal { L } + \\mathrm { t r } ( Q \\mathbf { c o v } [ g ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 580, + 507, + 596 + ], + "score": 1.0, + "content": ", which is a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 592, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 104, + 592, + 506, + 606 + ], + "score": 1.0, + "content": "general fact for quadratic forms of random variables. For the stochastic QP the gradient covariance is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 603, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 107, + 604, + 174, + 616 + ], + "score": 0.92, + "content": "\\mathbf { \\bar { c o v } } [ g ] = \\nu ^ { 2 } Q \\dot { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 603, + 197, + 618 + ], + "score": 1.0, + "content": ", thus", + "type": "text" + }, + { + "bbox": [ + 198, + 604, + 365, + 617 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathrm { t r } ( Q \\mathbf { c o v } [ g ] ) = \\nu ^ { 2 } \\mathrm { t r } ( Q Q Q ) = \\nu ^ { 2 } \\sum _ { i } \\lambda _ { i } ^ { 3 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 603, + 506, + 618 + ], + "score": 1.0, + "content": ". Plugging everything into Eq. 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The global learning rate", + "type": "text" + }, + { + "bbox": [ + 300, + 191, + 308, + 199 + ], + "score": 0.74, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 189, + 370, + 201 + ], + "score": 1.0, + "content": "stays constant.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 156, + 505, + 201 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 213, + 505, + 313 + ], + "lines": [ + { + "bbox": [ + 106, + 213, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 300, + 226 + ], + "score": 1.0, + "content": "CIFAR-10 The CIFAR-10 data set consists of", + "type": "text" + }, + { + "bbox": [ + 300, + 214, + 339, + 225 + ], + "score": 0.84, + "content": "3 2 \\times 3 2 \\mathrm { p x }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 213, + 505, + 226 + ], + "score": 1.0, + "content": "RGB images with one of ten categorical", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "labels. 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For the stochastic QP the gradient covariance is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 603, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 107, + 604, + 174, + 616 + ], + "score": 0.92, + "content": "\\mathbf { \\bar { c o v } } [ g ] = \\nu ^ { 2 } Q \\dot { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 603, + 197, + 618 + ], + "score": 1.0, + "content": ", thus", + "type": "text" + }, + { + "bbox": [ + 198, + 604, + 365, + 617 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathrm { t r } ( Q \\mathbf { c o v } [ g ] ) = \\nu ^ { 2 } \\mathrm { t r } ( Q Q Q ) = \\nu ^ { 2 } \\sum _ { i } \\lambda _ { i } ^ { 3 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 603, + 506, + 618 + ], + "score": 1.0, + "content": ". Plugging everything into Eq. 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The shaded area under the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 246, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 506, + 259 + ], + "score": 1.0, + "content": "curve corresponds to the probability that a sample from the distribution has the opposite sign than its", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 257, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 477, + 272 + ], + "score": 1.0, + "content": "mean. For the Gaussian distribution, this probability is uniquely determined by the fraction", + "type": "text" + }, + { + "bbox": [ + 478, + 258, + 501, + 270 + ], + "score": 0.92, + "content": "\\sigma / | \\mu |", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 257, + 506, + 272 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 269, + 199, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 199, + 281 + ], + "score": 1.0, + "content": "as shown in Lemma 2.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 269, + 314 + ], + "lines": [ + { + "bbox": [ + 106, + 301, + 269, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 269, + 316 + ], + "score": 1.0, + "content": "Plugging everything into Eq. (11) yields", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 321, + 378, + 361 + ], + "lines": [ + { + "bbox": [ + 233, + 321, + 378, + 361 + ], + "spans": [ + { + "bbox": [ + 233, + 321, + 378, + 361 + ], + "score": 0.94, + "content": "\\mathcal { T } ( s ) \\geq \\frac { \\left( \\sum _ { i = 1 } ^ { d } ( 2 \\rho _ { i } - 1 ) | \\nabla \\mathcal { L } _ { i } | \\right) ^ { 2 } } { \\sum _ { i = 1 } ^ { d } | q _ { i j } | } .", + "type": "interline_equation", + "image_path": "2859f22318a0836e31650769e6c181a525c65c52d198d27de9df024b3a00b737.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 233, + 321, + 378, + 341.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 233, + 341.0, + 378, + 361.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 375, + 428, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 428, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 428, + 388 + ], + "score": 1.0, + "content": "B.2 SUCCESS PROBABILITIES OF THE SIGN OF A STOCHASTIC GRADIENT", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 379, + 411 + ], + "score": 1.0, + "content": "We have stated in the main text that the sign of a stochastic gradient,", + "type": "text" + }, + { + "bbox": [ + 379, + 397, + 453, + 409 + ], + "score": 0.92, + "content": "s ( \\theta ) = \\mathrm { s i g n } ( g ( \\theta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 395, + 505, + 411 + ], + "score": 1.0, + "content": ", has success", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 408, + 159, + 420 + ], + "spans": [ + { + "bbox": [ + 104, + 408, + 159, + 420 + ], + "score": 1.0, + "content": "probabilities", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 422, + 424, + 450 + ], + "lines": [ + { + "bbox": [ + 186, + 422, + 424, + 450 + ], + "spans": [ + { + "bbox": [ + 186, + 422, + 424, + 450 + ], + "score": 0.94, + "content": "\\rho _ { i } = \\mathbf { P } [ s ( \\theta ) _ { i } = \\operatorname { s i g n } ( \\nabla { \\mathcal { L } } ( \\theta ) _ { i } ) ] = { \\frac { 1 } { 2 } } + { \\frac { 1 } { 2 } } \\operatorname { e r f } \\left( { \\frac { | \\nabla { \\mathcal { L } } ( \\theta ) _ { i } | } { \\sqrt { 2 } \\sigma ( \\theta ) _ { i } } } \\right)", + "type": "interline_equation", + "image_path": "540546ab9a6fc7c98c6578f2fb7f2a54a1eb7ce6358cb465100ef7e57f55524f.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 422, + 424, + 436.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 186, + 436.0, + 424, + 450.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 480 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 214, + 470 + ], + "score": 1.0, + "content": "under the assumption that", + "type": "text" + }, + { + "bbox": [ + 214, + 457, + 280, + 469 + ], + "score": 0.93, + "content": "g \\sim \\mathcal { N } ( \\nabla \\mathcal { L } , \\Sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 457, + 505, + 470 + ], + "score": 1.0, + "content": ". The following Lemma formally proves this statement", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 469, + 286, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 286, + 480 + ], + "score": 1.0, + "content": "and Figure 4 provides a pictorial illustration.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 106, + 483, + 245, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 246, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 163, + 498 + ], + "score": 1.0, + "content": "Lemma 2. If", + "type": "text" + }, + { + "bbox": [ + 163, + 483, + 224, + 497 + ], + "score": 0.92, + "content": "X \\sim { \\mathcal { N } } ( \\mu , \\sigma ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 483, + 246, + 498 + ], + "score": 1.0, + "content": "then", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 504, + 416, + 532 + ], + "lines": [ + { + "bbox": [ + 194, + 504, + 416, + 532 + ], + "spans": [ + { + "bbox": [ + 194, + 504, + 416, + 532 + ], + "score": 0.92, + "content": "\\rho = \\mathbf { P } [ \\operatorname { s i g n } ( X ) = \\operatorname { s i g n } ( \\mu ) ] = { \\frac { 1 } { 2 } } \\left( 1 + \\operatorname { e r f } \\left( { \\frac { | \\mu | } { \\sqrt { 2 } \\sigma } } \\right) \\right) .", + "type": "interline_equation", + "image_path": "6e8d9691f416b6b49cde0cceceaa01a89f5cd95d1ef36688734e8a57ced030b7.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 194, + 504, + 416, + 518.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 194, + 518.0, + 416, + 532.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 505, + 578 + ], + "lines": [ + { + "bbox": [ + 106, + 552, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 304, + 567 + ], + "score": 1.0, + "content": "Proof. 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The shaded area under the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 246, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 506, + 259 + ], + "score": 1.0, + "content": "curve corresponds to the probability that a sample from the distribution has the opposite sign than its", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 257, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 477, + 272 + ], + "score": 1.0, + "content": "mean. For the Gaussian distribution, this probability is uniquely determined by the fraction", + "type": "text" + }, + { + "bbox": [ + 478, + 258, + 501, + 270 + ], + "score": 0.92, + "content": "\\sigma / | \\mu |", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 257, + 506, + 272 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 269, + 199, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 199, + 281 + ], + "score": 1.0, + "content": "as shown in Lemma 2.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 269, + 314 + ], + "lines": [ + { + "bbox": [ + 106, + 301, + 269, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 269, + 316 + ], + "score": 1.0, + "content": "Plugging everything into Eq. (11) yields", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 301, + 269, + 316 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 321, + 378, + 361 + ], + "lines": [ + { + "bbox": [ + 233, + 321, + 378, + 361 + ], + "spans": [ + { + "bbox": [ + 233, + 321, + 378, + 361 + ], + "score": 0.94, + "content": "\\mathcal { T } ( s ) \\geq \\frac { \\left( \\sum _ { i = 1 } ^ { d } ( 2 \\rho _ { i } - 1 ) | \\nabla \\mathcal { L } _ { i } | \\right) ^ { 2 } } { \\sum _ { i = 1 } ^ { d } | q _ { i j } | } .", + "type": "interline_equation", + "image_path": "2859f22318a0836e31650769e6c181a525c65c52d198d27de9df024b3a00b737.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 233, + 321, + 378, + 341.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 233, + 341.0, + 378, + 361.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 375, + 428, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 428, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 428, + 388 + ], + "score": 1.0, + "content": "B.2 SUCCESS PROBABILITIES OF THE SIGN OF A STOCHASTIC GRADIENT", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 106, + 375, + 428, + 388 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 105, + 395, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 379, + 411 + ], + "score": 1.0, + "content": "We have stated in the main text that the sign of a stochastic gradient,", + "type": "text" + }, + { + "bbox": [ + 379, + 397, + 453, + 409 + ], + "score": 0.92, + "content": "s ( \\theta ) = \\mathrm { s i g n } ( g ( \\theta ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 395, + 505, + 411 + ], + "score": 1.0, + "content": ", has success", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 408, + 159, + 420 + ], + "spans": [ + { + "bbox": [ + 104, + 408, + 159, + 420 + ], + "score": 1.0, + "content": "probabilities", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 395, + 505, + 420 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 422, + 424, + 450 + ], + "lines": [ + { + "bbox": [ + 186, + 422, + 424, + 450 + ], + "spans": [ + { + "bbox": [ + 186, + 422, + 424, + 450 + ], + "score": 0.94, + "content": "\\rho _ { i } = \\mathbf { P } [ s ( \\theta ) _ { i } = \\operatorname { s i g n } ( \\nabla { \\mathcal { L } } ( \\theta ) _ { i } ) ] = { \\frac { 1 } { 2 } } + { \\frac { 1 } { 2 } } \\operatorname { e r f } \\left( { \\frac { | \\nabla { \\mathcal { L } } ( \\theta ) _ { i } | } { \\sqrt { 2 } \\sigma ( \\theta ) _ { i } } } \\right)", + "type": "interline_equation", + "image_path": "540546ab9a6fc7c98c6578f2fb7f2a54a1eb7ce6358cb465100ef7e57f55524f.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 422, + 424, + 436.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 186, + 436.0, + 424, + 450.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 480 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 214, + 470 + ], + "score": 1.0, + "content": "under the assumption that", + "type": "text" + }, + { + "bbox": [ + 214, + 457, + 280, + 469 + ], + "score": 0.93, + "content": "g \\sim \\mathcal { N } ( \\nabla \\mathcal { L } , \\Sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 457, + 505, + 470 + ], + "score": 1.0, + "content": ". The following Lemma formally proves this statement", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 469, + 286, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 286, + 480 + ], + "score": 1.0, + "content": "and Figure 4 provides a pictorial illustration.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 106, + 457, + 505, + 480 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 483, + 245, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 483, + 246, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 163, + 498 + ], + "score": 1.0, + "content": "Lemma 2. If", + "type": "text" + }, + { + "bbox": [ + 163, + 483, + 224, + 497 + ], + "score": 0.92, + "content": "X \\sim { \\mathcal { N } } ( \\mu , \\sigma ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 483, + 246, + 498 + ], + "score": 1.0, + "content": "then", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 106, + 483, + 246, + 498 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 504, + 416, + 532 + ], + "lines": [ + { + "bbox": [ + 194, + 504, + 416, + 532 + ], + "spans": [ + { + "bbox": [ + 194, + 504, + 416, + 532 + ], + "score": 0.92, + "content": "\\rho = \\mathbf { P } [ \\operatorname { s i g n } ( X ) = \\operatorname { s i g n } ( \\mu ) ] = { \\frac { 1 } { 2 } } \\left( 1 + \\operatorname { e r f } \\left( { \\frac { | \\mu | } { \\sqrt { 2 } \\sigma } } \\right) \\right) .", + "type": "interline_equation", + "image_path": "6e8d9691f416b6b49cde0cceceaa01a89f5cd95d1ef36688734e8a57ced030b7.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 194, + 504, + 416, + 518.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 194, + 518.0, + 416, + 532.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 505, + 578 + ], + "lines": [ + { + "bbox": [ + 106, + 552, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 304, + 567 + ], + "score": 1.0, + "content": "Proof. 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It is closely approximated by", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 325, + 165, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 325, + 167, + 380, + 180 + ], + "score": 0.92, + "content": "( 1 + \\eta ^ { 2 } ) ^ { - 1 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 165, + 506, + 182 + ], + "score": 1.0, + "content": ", which is the factor implicitly", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 324, + 180, + 440, + 192 + ], + "spans": [ + { + "bbox": [ + 324, + 180, + 440, + 192 + ], + "score": 1.0, + "content": "employed by ADAM (Eq. 6).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11 + } + ], + "index": 7.0 + }, + { + "type": "text", + "bbox": [ + 106, + 213, + 335, + 225 + ], + "lines": [ + { + "bbox": [ + 105, + 212, + 335, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 335, + 226 + ], + "score": 1.0, + "content": "B.3 DETAILS ON VARIANCE ADAPTATION FACTORS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 232, + 375, + 247 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 377, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 173, + 248 + ], + "score": 1.0, + "content": "Proof of Lemma", + "type": "text" + }, + { + "bbox": [ + 174, + 235, + 179, + 244 + ], + "score": 0.43, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 232, + 212, + 248 + ], + "score": 1.0, + "content": ". 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p _ { i } ) ^ { 2 } ] = \\sum _ { i = 1 } ^ { d } \\gamma _ { i } ^ { 2 } { \\bf E } [ \\hat { p } _ { i } ^ { 2 } ] - 2 \\gamma _ { i } p _ { i } { \\bf E } [ \\hat { p } _ { i } ] + p _ { i } ^ { 2 } } } \\\\ { { \\displaystyle \\quad \\quad = \\sum _ { i = 1 } ^ { d } \\gamma _ { i } ^ { 2 } ( p _ { i } ^ { 2 } + \\sigma _ { i } ^ { 2 } ) - 2 \\gamma _ { i } p _ { i } ^ { 2 } + p _ { i } ^ { 2 } . } } \\end{array}", + "type": "interline_equation", + "image_path": "df331f4af5b71eb28daa2bf0b849cd5d16ae1aff63d184add01be14c14eeaa53.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 154, + 252, + 456, + 275.3333333333333 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 154, + 275.3333333333333, + 456, + 298.66666666666663 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 154, + 298.66666666666663, + 456, + 321.99999999999994 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 326, + 367, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 325, + 368, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 216, + 340 + ], + "score": 1.0, + "content": "Setting the derivative w.r.t.", + "type": "text" + }, + { + "bbox": [ + 216, + 329, + 226, + 338 + ], + "score": 0.85, + "content": "\\gamma _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 325, + 368, + 340 + ], + "score": 1.0, + "content": "to zero, we find the optimal choice", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 274, + 343, + 336, + 372 + ], + "lines": [ + { + "bbox": [ + 274, + 343, + 336, + 372 + ], + "spans": [ + { + "bbox": [ + 274, + 343, + 336, + 372 + ], + "score": 0.95, + "content": "\\gamma _ { i } = \\frac { p _ { i } ^ { 2 } } { p _ { i } ^ { 2 } + \\sigma _ { i } ^ { 2 } } .", + "type": "interline_equation", + "image_path": "175f5c793f07bfc4257ef76cedb3e4017c644e00bb6dad61d741abe955d2d60c.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 274, + 343, + 336, + 357.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 274, + 357.5, + 336, + 372.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 377, + 370, + 391 + ], + "lines": [ + { + "bbox": [ + 106, + 376, + 370, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 133, + 392 + ], + "score": 1.0, + "content": "Using", + "type": "text" + }, + { + "bbox": [ + 133, + 378, + 267, + 391 + ], + "score": 0.93, + "content": "\\mathbf { E } [ \\mathrm { s i g n } ( \\hat { p } _ { i } ) ] = ( 2 \\rho _ { i } - 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We", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 562, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 575 + ], + "score": 1.0, + "content": "have to clarify an aspect that we have glossed over in the main text. A stochastic optimizer gen-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 249, + 586 + ], + "score": 1.0, + "content": "erates a discrete stochastic process", + "type": "text" + }, + { + "bbox": [ + 249, + 573, + 285, + 585 + ], + "score": 0.93, + "content": "\\{ \\boldsymbol { \\theta } _ { t } \\} _ { t \\in { \\mathbb { N } } _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 573, + 348, + 586 + ], + "score": 1.0, + "content": ". 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Recall that", + "type": "text" + }, + { + "bbox": [ + 407, + 584, + 468, + 596 + ], + "score": 0.91, + "content": "\\mathbf { E } [ \\mathbf { E } _ { t } [ \\cdot ] ] = \\mathbf { E } [ \\cdot ]", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 583, + 474, + 597 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 104, + 551, + 505, + 597 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 609, + 505, + 635 + ], + "lines": [ + { + "bbox": [ + 106, + 608, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 179, + 623 + ], + "score": 1.0, + "content": "Proof of Theorem", + "type": "text" + }, + { + "bbox": [ + 179, + 610, + 185, + 619 + ], + "score": 0.31, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 608, + 330, + 623 + ], + "score": 1.0, + "content": ". Using the Lipschitz continuity of", + "type": "text" + }, + { + "bbox": [ + 330, + 610, + 345, + 621 + ], + "score": 0.89, + "content": "\\nabla f", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 608, + 411, + 623 + ], + "score": 1.0, + "content": ", we can bound", + "type": "text" + }, + { + "bbox": [ + 411, + 609, + 505, + 622 + ], + "score": 0.88, + "content": "f ( \\theta + \\Delta \\theta ) \\leq f ( \\theta ) +", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 619, + 237, + 636 + ], + "spans": [ + { + "bbox": [ + 107, + 621, + 202, + 635 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\nabla f ( \\theta ) ^ { T } \\Delta \\theta + \\frac { L } { 2 } \\| \\Delta \\theta \\| ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 619, + 237, + 636 + ], + "score": 1.0, + "content": ". Hence,", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 106, + 608, + 505, + 636 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 640, + 431, + 737 + ], + "lines": [ + { + "bbox": [ + 180, + 640, + 431, + 737 + ], + "spans": [ + { + "bbox": [ + 180, + 640, + 431, + 737 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\mathbf { E } _ { t } \\big [ f _ { t + 1 } \\big ] \\leq f _ { t } - \\alpha \\mathbf { E } _ { t } \\big [ \\nabla f _ { t } ^ { T } ( \\gamma _ { t } \\odot g _ { t } ) \\big ] + \\displaystyle \\frac { L \\alpha ^ { 2 } } { 2 } \\mathbf { E } _ { t } \\big [ \\| \\gamma _ { t } \\odot g _ { t } \\| ^ { 2 } \\big ] } \\\\ & { \\qquad = f _ { t } - \\displaystyle \\frac { 1 } { L } \\sum _ { i = 1 } ^ { d } \\gamma _ { t , i } \\nabla f _ { t , i } \\mathbf { E } [ g _ { t , i } ] + \\displaystyle \\frac { 1 } { 2 L } \\sum _ { i = 1 } ^ { d } \\gamma _ { t , i } ^ { 2 } \\mathbf { E } _ { t } [ g _ { t , i } ^ { 2 } ] } \\\\ & { \\qquad = f _ { t } - \\displaystyle \\frac { 1 } { L } \\sum _ { i = 1 } ^ { d } \\gamma _ { t , i } \\nabla f _ { t , i } ^ { 2 } + \\displaystyle \\frac { 1 } { 2 L } \\sum _ { i = 1 } ^ { d } \\gamma _ { t , i } ^ { 2 } ( \\nabla f _ { t , i } ^ { 2 } + \\sigma _ { t , i } ^ { 2 } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "7dc5cc13ef2ce527a5df0d6bb2c4db34101eff6dfa25acc3823b575eeb2dc38a.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 180, + 640, + 431, + 672.3333333333334 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 180, + 672.3333333333334, + 431, + 704.6666666666667 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 180, + 704.6666666666667, + 431, + 737.0000000000001 + ], + "spans": [], + "index": 41 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 210, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 211, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 211, + 95 + ], + "score": 1.0, + "content": "Plugging in the definition", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 265, + 95, + 346, + 125 + ], + "lines": [ + { + "bbox": [ + 265, + 95, + 346, + 125 + ], + "spans": [ + { + "bbox": [ + 265, + 95, + 346, + 125 + ], + "score": 0.94, + "content": "\\gamma _ { t , i } = \\frac { \\nabla f _ { t , i } ^ { 2 } } { \\nabla f _ { t , i } ^ { 2 } + \\sigma _ { t , i } ^ { 2 } }", + "type": "interline_equation", + "image_path": "c56a692e7aba0184ac5ff908977b5ee3d2910c876545df9eeac4d8d0b22e945a.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 265, + 95, + 346, + 125 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 129, + 203, + 141 + ], + "lines": [ + { + "bbox": [ + 106, + 128, + 203, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 203, + 144 + ], + "score": 1.0, + "content": "and simplifying, we get", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 148, + 395, + 182 + ], + "lines": [ + { + "bbox": [ + 216, + 148, + 395, + 182 + ], + "spans": [ + { + "bbox": [ + 216, + 148, + 395, + 182 + ], + "score": 0.95, + "content": "\\mathbf { E } _ { t } [ f _ { t + 1 } ] \\leq f _ { t } - \\frac { 1 } { 2 L } \\sum _ { i = 1 } ^ { d } \\frac { \\nabla f _ { t , i } ^ { 2 } } { \\nabla f _ { t , i } ^ { 2 } + \\sigma _ { t , i } ^ { 2 } } \\nabla f _ { t , i } ^ { 2 } .", + "type": "interline_equation", + "image_path": "6f11be44688f0d85f6a5bb8f9ada5475a3171c9cb6476cca92b47c6fcc22c2ff.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 216, + 148, + 395, + 165.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 216, + 165.0, + 395, + 182.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 189, + 214, + 201 + ], + "lines": [ + { + "bbox": [ + 106, + 189, + 214, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 214, + 203 + ], + "score": 1.0, + "content": "Using Jensen’s inequality4", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 209, + 439, + 315 + ], + "lines": [ + { + "bbox": [ + 172, + 209, + 439, + 315 + ], + "spans": [ + { + "bbox": [ + 172, + 209, + 439, + 315 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\displaystyle \\sum _ { i = 1 } ^ { d } \\frac { \\nabla f _ { t , i } ^ { 2 } } { \\nabla f _ { t , i } ^ { 2 } + \\sigma _ { t , i } ^ { 2 } } \\nabla f _ { t , i } ^ { 2 } = \\| \\nabla f _ { t } \\| ^ { 2 } \\sum _ { i = 1 } ^ { d } \\frac { \\nabla f _ { t , i } ^ { 2 } } { \\| \\nabla f _ { t } \\| ^ { 2 } } \\left( \\frac { \\nabla f _ { t , i } ^ { 2 } + \\sigma _ { t , i } ^ { 2 } } { \\nabla f _ { t , i } ^ { 2 } } \\right) ^ { - 1 } } \\\\ & { \\qquad \\geq \\| \\nabla f _ { t } \\| ^ { 2 } \\left( \\displaystyle \\sum _ { i = 1 } ^ { d } \\frac { \\nabla f _ { t , i } ^ { 2 } } { \\| \\nabla f _ { t } \\| ^ { 2 } } \\frac { \\nabla f _ { t , i } ^ { 2 } + \\sigma _ { t , i } ^ { 2 } } { \\nabla f _ { t , i } ^ { 2 } } \\right) ^ { - 1 } } \\\\ & { \\qquad = \\frac { \\| \\nabla f _ { t } \\| ^ { 4 } } { \\sum _ { i = 1 } ^ { d } ( \\nabla f _ { t , i } ^ { 2 } + \\sigma _ { t , i } ^ { 2 } ) } \\geq \\frac { \\| \\nabla f _ { t } \\| ^ { 4 } } { G ^ { 2 } } . } \\end{array}", + "type": "interline_equation", + "image_path": "385274eb888bdc4a638dacca3257c74106c0cc8623c83720f6f074b39fbfd9e5.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 172, + 209, + 439, + 244.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 172, + 244.33333333333334, + 439, + 279.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 172, + 279.6666666666667, + 439, + 315.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 319, + 427, + 333 + ], + "lines": [ + { + "bbox": [ + 106, + 319, + 427, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 241, + 334 + ], + "score": 1.0, + "content": "Due to strong convexity, we have", + "type": "text" + }, + { + "bbox": [ + 242, + 320, + 335, + 333 + ], + "score": 0.93, + "content": "\\| \\nabla f _ { t } \\| ^ { 2 } \\geq 2 \\mu ( f _ { t } - f _ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 319, + 427, + 334 + ], + "score": 1.0, + "content": "and can further bound", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 338, + 389, + 374 + ], + "lines": [ + { + "bbox": [ + 220, + 338, + 389, + 374 + ], + "spans": [ + { + "bbox": [ + 220, + 338, + 389, + 374 + ], + "score": 0.93, + "content": "\\sum _ { i = 1 } ^ { d } \\frac { \\nabla f _ { t , i } ^ { 2 } } { \\nabla f _ { t , i } ^ { 2 } + \\sigma _ { t , i } ^ { 2 } } \\nabla f _ { t , i } ^ { 2 } \\geq \\frac { 4 \\mu ^ { 2 } ( f _ { t } - f _ { * } ) ^ { 2 } } { G ^ { 2 } } .", + "type": "interline_equation", + "image_path": "2eceef0bbfb11f7ce6d28aa07a8fb65464d0b4b46517b5bafcaffb113d254f74.jpg" + } + ] + } + ], + "index": 10.5, + "virtual_lines": [ + { + "bbox": [ + 220, + 338, + 389, + 356.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 220, + 356.0, + 389, + 374.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 297, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 377, + 298, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 254, + 394 + ], + "score": 1.0, + "content": "Inserting this in (33) and subtracting", + "type": "text" + }, + { + "bbox": [ + 254, + 380, + 264, + 391 + ], + "score": 0.89, + "content": "f _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 377, + 298, + 394 + ], + "score": 1.0, + "content": ", we get", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 398, + 396, + 424 + ], + "lines": [ + { + "bbox": [ + 214, + 398, + 396, + 424 + ], + "spans": [ + { + "bbox": [ + 214, + 398, + 396, + 424 + ], + "score": 0.93, + "content": "\\mathbf { E } _ { t } [ f _ { t + 1 } ] - f _ { * } \\leq f _ { t } - f _ { * } - \\frac { 2 \\mu ^ { 2 } } { L G ^ { 2 } } ( f _ { t } - f _ { * } ) ^ { 2 } ,", + "type": "interline_equation", + "image_path": "f1628a5dc4e05d475082946d5960a8d208b8c3ebe0524f1f49c6fd2b55f0f7f7.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 214, + 398, + 396, + 424 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 430, + 264, + 442 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 264, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 264, + 444 + ], + "score": 1.0, + "content": "and, consequently, by total expectation", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 448, + 449, + 501 + ], + "lines": [ + { + "bbox": [ + 163, + 448, + 449, + 501 + ], + "spans": [ + { + "bbox": [ + 163, + 448, + 449, + 501 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { \\displaystyle { \\bf E } [ f _ { t + 1 } - f _ { * } ] = { \\bf E } \\left[ { \\bf E } _ { t } [ f _ { t + 1 } ] - f _ { * } \\right] \\leq { \\bf E } [ f _ { t } - f _ { * } ] - \\frac { 2 \\mu ^ { 2 } } { L G ^ { 2 } } { \\bf E } [ ( f _ { t } - f _ { * } ) ^ { 2 } ] } \\ ~ } \\\\ { { \\displaystyle \\phantom { \\frac { 1 } { 1 } } \\leq { \\bf E } [ f _ { t } - f _ { * } ] - \\frac { 2 \\mu ^ { 2 } } { L G ^ { 2 } } { \\bf E } [ f _ { t } - f _ { * } ] ^ { 2 } } , } \\end{array}", + "type": "interline_equation", + "image_path": "2573526d54b78495bc64b7a8a748a4c4ed63f83a4ddbc51a4a3eb01b411e1aad.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 163, + 448, + 449, + 465.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 163, + 465.6666666666667, + 449, + 483.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 163, + 483.33333333333337, + 449, + 501.00000000000006 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 505, + 344, + 518 + ], + "lines": [ + { + "bbox": [ + 106, + 504, + 344, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 261, + 520 + ], + "score": 1.0, + "content": "which we rewrite, using the shorthand", + "type": "text" + }, + { + "bbox": [ + 261, + 506, + 329, + 518 + ], + "score": 0.93, + "content": "e _ { t } : = \\mathbf { E } [ f _ { t } - f _ { * } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 504, + 344, + 520 + ], + "score": 1.0, + "content": ", as", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 524, + 383, + 550 + ], + "lines": [ + { + "bbox": [ + 228, + 524, + 383, + 550 + ], + "spans": [ + { + "bbox": [ + 228, + 524, + 383, + 550 + ], + "score": 0.93, + "content": "0 \\leq e _ { t + 1 } \\leq e _ { t } ( 1 - c e _ { t } ) , \\quad c = \\frac { 2 \\mu ^ { 2 } } { L G ^ { 2 } } .", + "type": "interline_equation", + "image_path": "07ff0b5c2b9b1af4d16c0101f7e1f502479e4135e97344bd949d7a15621587c1.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 228, + 524, + 383, + 550 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 556, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 105, + 555, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 327, + 572 + ], + "score": 1.0, + "content": "To conclude the proof, we will show that this implies", + "type": "text" + }, + { + "bbox": [ + 328, + 556, + 374, + 570 + ], + "score": 0.93, + "content": "\\textstyle e _ { t } \\in { \\mathcal { O } } { \\bigl ( } { \\frac { 1 } { t } } { \\bigr ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 555, + 505, + 572 + ], + "score": 1.0, + "content": ". Without loss of generality, we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 568, + 210, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 138, + 581 + ], + "score": 1.0, + "content": "assume", + "type": "text" + }, + { + "bbox": [ + 138, + 569, + 176, + 580 + ], + "score": 0.92, + "content": "e _ { t + 1 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 568, + 210, + 581 + ], + "score": 1.0, + "content": "and get", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 587, + 411, + 603 + ], + "lines": [ + { + "bbox": [ + 200, + 587, + 411, + 603 + ], + "spans": [ + { + "bbox": [ + 200, + 587, + 411, + 603 + ], + "score": 0.89, + "content": "e _ { t + 1 } ^ { - 1 } \\geq e _ { t } ^ { - 1 } ( 1 - c e _ { t } ) ^ { - 1 } \\geq e _ { t } ^ { - 1 } ( 1 + c e _ { t } ) = e _ { t } ^ { - 1 } + c ,", + "type": "interline_equation", + "image_path": "34f8fbbf0779998854496aca0bc96e14b66cb858553ac79d27e8e3d06b127c97.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 200, + 587, + 411, + 603 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 505, + 635 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 308, + 625 + ], + "score": 1.0, + "content": "where the second step is due to the simple fact that", + "type": "text" + }, + { + "bbox": [ + 308, + 609, + 388, + 622 + ], + "score": 0.91, + "content": "( 1 - x ) ^ { - 1 } \\geq ( 1 + x )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 607, + 419, + 625 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 420, + 610, + 459, + 623 + ], + "score": 0.92, + "content": "x \\in [ 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 607, + 505, + 625 + ], + "score": 1.0, + "content": ". 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\\frac { 1 } { 2 L } \\sum _ { i = 1 } ^ { d } \\frac { \\nabla f _ { t , i } ^ { 2 } } { \\nabla f _ { t , i } ^ { 2 } + \\sigma _ { t , i } ^ { 2 } } \\nabla f _ { t , i } ^ { 2 } .", + "type": "interline_equation", + "image_path": "6f11be44688f0d85f6a5bb8f9ada5475a3171c9cb6476cca92b47c6fcc22c2ff.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 216, + 148, + 395, + 165.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 216, + 165.0, + 395, + 182.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 189, + 214, + 201 + ], + "lines": [ + { + "bbox": [ + 106, + 189, + 214, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 214, + 203 + ], + "score": 1.0, + "content": "Using Jensen’s inequality4", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 106, + 189, + 214, + 203 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 209, + 439, + 315 + ], + "lines": [ + { + "bbox": [ + 172, + 209, + 439, + 315 + ], + "spans": [ + { + "bbox": [ + 172, + 209, + 439, + 315 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\displaystyle \\sum _ { i = 1 } ^ { d } \\frac { \\nabla f _ { t , i } ^ { 2 } } { \\nabla f _ { t , i } ^ { 2 } + \\sigma _ { t , i } ^ { 2 } } \\nabla f _ { t , i } ^ { 2 } = \\| \\nabla f _ { t } \\| ^ { 2 } \\sum _ { i = 1 } ^ { d } \\frac { \\nabla f _ { t , i } ^ { 2 } } { \\| \\nabla f _ { t } \\| ^ { 2 } } \\left( \\frac { \\nabla f _ { t , i } ^ { 2 } + \\sigma _ { t , i } ^ { 2 } } { \\nabla f _ { t , i } ^ { 2 } } \\right) ^ { - 1 } } \\\\ & { \\qquad \\geq \\| \\nabla f _ { t } \\| ^ { 2 } \\left( \\displaystyle \\sum _ { i = 1 } ^ { d } \\frac { \\nabla f _ { t , i } ^ { 2 } } { \\| \\nabla f _ { t } \\| ^ { 2 } } \\frac { \\nabla f _ { t , i } ^ { 2 } + \\sigma _ { t , i } ^ { 2 } } { \\nabla f _ { t , i } ^ { 2 } } \\right) ^ { - 1 } } \\\\ & { \\qquad = \\frac { \\| \\nabla f _ { t } \\| ^ { 4 } } { \\sum _ { i = 1 } ^ { d } ( \\nabla f _ { t , i } ^ { 2 } + \\sigma _ { t , i } ^ { 2 } ) } \\geq \\frac { \\| \\nabla f _ { t } \\| ^ { 4 } } { G ^ { 2 } } . } \\end{array}", + "type": "interline_equation", + "image_path": "385274eb888bdc4a638dacca3257c74106c0cc8623c83720f6f074b39fbfd9e5.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 172, + 209, + 439, + 244.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 172, + 244.33333333333334, + 439, + 279.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 172, + 279.6666666666667, + 439, + 315.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 319, + 427, + 333 + ], + "lines": [ + { + "bbox": [ + 106, + 319, + 427, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 241, + 334 + ], + "score": 1.0, + "content": "Due to strong convexity, we have", + "type": "text" + }, + { + "bbox": [ + 242, + 320, + 335, + 333 + ], + "score": 0.93, + "content": "\\| \\nabla f _ { t } \\| ^ { 2 } \\geq 2 \\mu ( f _ { t } - f _ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 319, + 427, + 334 + ], + "score": 1.0, + "content": "and can further bound", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 106, + 319, + 427, + 334 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 338, + 389, + 374 + ], + "lines": [ + { + "bbox": [ + 220, + 338, + 389, + 374 + ], + "spans": [ + { + "bbox": [ + 220, + 338, + 389, + 374 + ], + "score": 0.93, + "content": "\\sum _ { i = 1 } ^ { d } \\frac { \\nabla f _ { t , i } ^ { 2 } } { \\nabla f _ { t , i } ^ { 2 } + \\sigma _ { t , i } ^ { 2 } } \\nabla f _ { t , i } ^ { 2 } \\geq \\frac { 4 \\mu ^ { 2 } ( f _ { t } - f _ { * } ) ^ { 2 } } { G ^ { 2 } } .", + "type": "interline_equation", + "image_path": "2eceef0bbfb11f7ce6d28aa07a8fb65464d0b4b46517b5bafcaffb113d254f74.jpg" + } + ] + } + ], + "index": 10.5, + "virtual_lines": [ + { + "bbox": [ + 220, + 338, + 389, + 356.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 220, + 356.0, + 389, + 374.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 297, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 377, + 298, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 254, + 394 + ], + "score": 1.0, + "content": "Inserting this in (33) and subtracting", + "type": "text" + }, + { + "bbox": [ + 254, + 380, + 264, + 391 + ], + "score": 0.89, + "content": "f _ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 377, + 298, + 394 + ], + "score": 1.0, + "content": ", we get", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 377, + 298, + 394 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 398, + 396, + 424 + ], + "lines": [ + { + "bbox": [ + 214, + 398, + 396, + 424 + ], + "spans": [ + { + "bbox": [ + 214, + 398, + 396, + 424 + ], + "score": 0.93, + "content": "\\mathbf { E } _ { t } [ f _ { t + 1 } ] - f _ { * } \\leq f _ { t } - f _ { * } - \\frac { 2 \\mu ^ { 2 } } { L G ^ { 2 } } ( f _ { t } - f _ { * } ) ^ { 2 } ,", + "type": "interline_equation", + "image_path": "f1628a5dc4e05d475082946d5960a8d208b8c3ebe0524f1f49c6fd2b55f0f7f7.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 214, + 398, + 396, + 424 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 430, + 264, + 442 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 264, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 264, + 444 + ], + "score": 1.0, + "content": "and, consequently, by total expectation", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 429, + 264, + 444 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 448, + 449, + 501 + ], + "lines": [ + { + "bbox": [ + 163, + 448, + 449, + 501 + ], + "spans": [ + { + "bbox": [ + 163, + 448, + 449, + 501 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { \\displaystyle { \\bf E } [ f _ { t + 1 } - f _ { * } ] = { \\bf E } \\left[ { \\bf E } _ { t } [ f _ { t + 1 } ] - f _ { * } \\right] \\leq { \\bf E } [ f _ { t } - f _ { * } ] - \\frac { 2 \\mu ^ { 2 } } { L G ^ { 2 } } { \\bf E } [ ( f _ { t } - f _ { * } ) ^ { 2 } ] } \\ ~ } \\\\ { { \\displaystyle \\phantom { \\frac { 1 } { 1 } } \\leq { \\bf E } [ f _ { t } - f _ { * } ] - \\frac { 2 \\mu ^ { 2 } } { L G ^ { 2 } } { \\bf E } [ f _ { t } - f _ { * } ] ^ { 2 } } , } \\end{array}", + "type": "interline_equation", + "image_path": "2573526d54b78495bc64b7a8a748a4c4ed63f83a4ddbc51a4a3eb01b411e1aad.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 163, + 448, + 449, + 465.6666666666667 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 163, + 465.6666666666667, + 449, + 483.33333333333337 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 163, + 483.33333333333337, + 449, + 501.00000000000006 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 505, + 344, + 518 + ], + "lines": [ + { + "bbox": [ + 106, + 504, + 344, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 261, + 520 + ], + "score": 1.0, + "content": "which we rewrite, using the shorthand", + "type": "text" + }, + { + "bbox": [ + 261, + 506, + 329, + 518 + ], + "score": 0.93, + "content": "e _ { t } : = \\mathbf { E } [ f _ { t } - f _ { * } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 504, + 344, + 520 + ], + "score": 1.0, + "content": ", as", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 106, + 504, + 344, + 520 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 524, + 383, + 550 + ], + "lines": [ + { + "bbox": [ + 228, + 524, + 383, + 550 + ], + "spans": [ + { + "bbox": [ + 228, + 524, + 383, + 550 + ], + "score": 0.93, + "content": "0 \\leq e _ { t + 1 } \\leq e _ { t } ( 1 - c e _ { t } ) , \\quad c = \\frac { 2 \\mu ^ { 2 } } { L G ^ { 2 } } .", + "type": "interline_equation", + "image_path": "07ff0b5c2b9b1af4d16c0101f7e1f502479e4135e97344bd949d7a15621587c1.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 228, + 524, + 383, + 550 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 556, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 105, + 555, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 327, + 572 + ], + "score": 1.0, + "content": "To conclude the proof, we will show that this implies", + "type": "text" + }, + { + "bbox": [ + 328, + 556, + 374, + 570 + ], + "score": 0.93, + "content": "\\textstyle e _ { t } \\in { \\mathcal { O } } { \\bigl ( } { \\frac { 1 } { t } } { \\bigr ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 555, + 505, + 572 + ], + "score": 1.0, + "content": ". Without loss of generality, we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 568, + 210, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 138, + 581 + ], + "score": 1.0, + "content": "assume", + "type": "text" + }, + { + "bbox": [ + 138, + 569, + 176, + 580 + ], + "score": 0.92, + "content": "e _ { t + 1 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 568, + 210, + 581 + ], + "score": 1.0, + "content": "and get", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 555, + 505, + 581 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 587, + 411, + 603 + ], + "lines": [ + { + "bbox": [ + 200, + 587, + 411, + 603 + ], + "spans": [ + { + "bbox": [ + 200, + 587, + 411, + 603 + ], + "score": 0.89, + "content": "e _ { t + 1 } ^ { - 1 } \\geq e _ { t } ^ { - 1 } ( 1 - c e _ { t } ) ^ { - 1 } \\geq e _ { t } ^ { - 1 } ( 1 + c e _ { t } ) = e _ { t } ^ { - 1 } + c ,", + "type": "interline_equation", + "image_path": "34f8fbbf0779998854496aca0bc96e14b66cb858553ac79d27e8e3d06b127c97.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 200, + 587, + 411, + 603 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 505, + 635 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 308, + 625 + ], + "score": 1.0, + "content": "where the second step is due to the simple fact that", + "type": "text" + }, + { + "bbox": [ + 308, + 609, + 388, + 622 + ], + "score": 0.91, + "content": "( 1 - x ) ^ { - 1 } \\geq ( 1 + x )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 607, + 419, + 625 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 420, + 610, + 459, + 623 + ], + "score": 0.92, + "content": "x \\in [ 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 607, + 505, + 625 + ], + "score": 1.0, + "content": ". Summing", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 621, + 395, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 185, + 637 + ], + "score": 1.0, + "content": "this inequality over", + "type": "text" + }, + { + "bbox": [ + 186, + 623, + 255, + 634 + ], + "score": 0.93, + "content": "t = 0 , \\ldots , T - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 621, + 283, + 637 + ], + "score": 1.0, + "content": "yields", + "type": "text" + }, + { + "bbox": [ + 283, + 622, + 352, + 635 + ], + "score": 0.92, + "content": "e _ { T } ^ { - 1 } \\geq e _ { 0 } ^ { - 1 } + T c", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 621, + 395, + 637 + ], + "score": 1.0, + "content": "and, thus,", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 607, + 505, + 637 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 227, + 641, + 383, + 671 + ], + "lines": [ + { + "bbox": [ + 227, + 641, + 383, + 671 + ], + "spans": [ + { + "bbox": [ + 227, + 641, + 383, + 671 + ], + "score": 0.95, + "content": "T e _ { T } \\le \\left( \\frac { 1 } { T e _ { 0 } } + c \\right) ^ { - 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This will of course be utterly wrong for gradient observations", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 247, + 277 + ], + "score": 1.0, + "content": "that are far in the past, but since", + "type": "text" + }, + { + "bbox": [ + 247, + 264, + 283, + 276 + ], + "score": 0.93, + "content": "c ( \\mu , t , s )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 263, + 381, + 277 + ], + "score": 1.0, + "content": "is very small for large", + "type": "text" + }, + { + "bbox": [ + 382, + 264, + 406, + 275 + ], + "score": 0.86, + "content": "t - s", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 263, + 505, + 277 + ], + "score": 1.0, + "content": ", these won’t contribute", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "significantly to the moving average. 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The error arising from this bias", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 427, + 423, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 423, + 441 + ], + "score": 1.0, + "content": "should generally be dominated by other error sources and will thus be ignored.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 107, + 452, + 240, + 463 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 241, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 241, + 465 + ], + "score": 1.0, + "content": "C.2 MINI-BATCH ESTIMATES", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 472, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 484 + ], + "score": 1.0, + "content": "An alternative gradient variance estimate can be obtained locally, within a single mini-batch. The", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 484, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 189, + 496 + ], + "score": 1.0, + "content": "individual gradients", + "type": "text" + }, + { + "bbox": [ + 190, + 484, + 230, + 495 + ], + "score": 0.89, + "content": "\\nabla \\ell ( \\theta , x _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 484, + 505, + 496 + ], + "score": 1.0, + "content": "in a mini-batch are iid random variables and, as noted in the intro-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 493, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 142, + 508 + ], + "score": 1.0, + "content": "duction,", + "type": "text" + }, + { + "bbox": [ + 142, + 495, + 282, + 506 + ], + "score": 0.91, + "content": "\\mathbf { v a r } [ g ( \\theta ) ] = | \\vartheta | ^ { - 1 } \\mathbf { v a r } [ \\nabla \\ell ( \\theta , x _ { k } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 493, + 378, + 508 + ], + "score": 1.0, + "content": ". We can thus estimate", + "type": "text" + }, + { + "bbox": [ + 379, + 495, + 397, + 507 + ], + "score": 0.91, + "content": "g ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 493, + 505, + 508 + ], + "score": 1.0, + "content": "’s variances by computing", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 504, + 375, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 214, + 519 + ], + "score": 1.0, + "content": "the sample variance of the", + "type": "text" + }, + { + "bbox": [ + 214, + 506, + 280, + 518 + ], + "score": 0.9, + "content": "\\{ \\nabla \\ell ( \\theta , x _ { k } ) \\} _ { k \\in B }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 504, + 347, + 519 + ], + "score": 1.0, + "content": ", then scaling by", + "type": "text" + }, + { + "bbox": [ + 347, + 505, + 371, + 518 + ], + "score": 0.92, + "content": "| B | ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 504, + 375, + 519 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 523, + 405, + 557 + ], + "lines": [ + { + "bbox": [ + 205, + 523, + 405, + 557 + ], + "spans": [ + { + "bbox": [ + 205, + 523, + 405, + 557 + ], + "score": 0.93, + "content": "\\hat { s } ( \\theta ) = \\frac { 1 } { | \\mathcal { B } | } \\left( \\frac { 1 } { | \\mathcal { B } | - 1 } \\sum _ { k \\in \\mathcal { B } } \\nabla \\ell ( \\theta , x _ { k } ) ^ { 2 } - g ( \\theta ) ^ { 2 } \\right) .", + "type": "interline_equation", + "image_path": "04e79c267a2db23585509438da63beed31209809aa239193fa93da6852b2522f.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 205, + 523, + 405, + 540.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 205, + 540.0, + 405, + 557.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "score": 1.0, + "content": "Several recent papers (Mahsereci & Hennig, 2015; Balles et al., 2017b; Mahsereci et al., 2017) have", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 572, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 433, + 586 + ], + "score": 1.0, + "content": "used this variance estimate for other aspects of stochastic optimizers. In contrast to", + "type": "text" + }, + { + "bbox": [ + 434, + 574, + 464, + 585 + ], + "score": 0.92, + "content": "v _ { t } - m _ { t } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 572, + 505, + 586 + ], + "score": 1.0, + "content": ", this is an", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "unbiased estimate of the local gradient variance. The (non-trivial) implementation of this estimator", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 594, + 332, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 332, + 607 + ], + "score": 1.0, + "content": "for neural networks is described in Balles et al. 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The error arising from this bias", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 427, + 423, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 423, + 441 + ], + "score": 1.0, + "content": "should generally be dominated by other error sources and will thus be ignored.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 404, + 506, + 441 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 452, + 240, + 463 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 241, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 241, + 465 + ], + "score": 1.0, + "content": "C.2 MINI-BATCH ESTIMATES", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 472, + 505, + 518 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 505, + 484 + ], + "score": 1.0, + "content": "An alternative gradient variance estimate can be obtained locally, within a single mini-batch. 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We can thus estimate", + "type": "text" + }, + { + "bbox": [ + 379, + 495, + 397, + 507 + ], + "score": 0.91, + "content": "g ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 493, + 505, + 508 + ], + "score": 1.0, + "content": "’s variances by computing", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 504, + 375, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 214, + 519 + ], + "score": 1.0, + "content": "the sample variance of the", + "type": "text" + }, + { + "bbox": [ + 214, + 506, + 280, + 518 + ], + "score": 0.9, + "content": "\\{ \\nabla \\ell ( \\theta , x _ { k } ) \\} _ { k \\in B }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 504, + 347, + 519 + ], + "score": 1.0, + "content": ", then scaling by", + "type": "text" + }, + { + "bbox": [ + 347, + 505, + 371, + 518 + ], + "score": 0.92, + "content": "| B | ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 504, + 375, + 519 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 473, + 505, + 519 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 523, + 405, + 557 + ], + "lines": [ + { + "bbox": [ + 205, + 523, + 405, + 557 + ], + "spans": [ + { + "bbox": [ + 205, + 523, + 405, + 557 + ], + "score": 0.93, + "content": "\\hat { s } ( \\theta ) = \\frac { 1 } { | \\mathcal { B } | } \\left( \\frac { 1 } { | \\mathcal { B } | - 1 } \\sum _ { k \\in \\mathcal { B } } \\nabla \\ell ( \\theta , x _ { k } ) ^ { 2 } - g ( \\theta ) ^ { 2 } \\right) .", + "type": "interline_equation", + "image_path": "04e79c267a2db23585509438da63beed31209809aa239193fa93da6852b2522f.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 205, + 523, + 405, + 540.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 205, + 540.0, + 405, + 557.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "score": 1.0, + "content": "Several recent papers (Mahsereci & Hennig, 2015; Balles et al., 2017b; Mahsereci et al., 2017) have", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 572, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 433, + 586 + ], + "score": 1.0, + "content": "used this variance estimate for other aspects of stochastic optimizers. In contrast to", + "type": "text" + }, + { + "bbox": [ + 434, + 574, + 464, + 585 + ], + "score": 0.92, + "content": "v _ { t } - m _ { t } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 572, + 505, + 586 + ], + "score": 1.0, + "content": ", this is an", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "unbiased estimate of the local gradient variance. The (non-trivial) implementation of this estimator", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 594, + 332, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 332, + 607 + ], + "score": 1.0, + "content": "for neural networks is described in Balles et al. 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