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An enormous body of research has appeared on more efficient variants of attention to overcome", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 586, 506, 599 ], "spans": [ { "bbox": [ 105, 586, 506, 599 ], "score": 1.0, "content": "these drawbacks (Tay et al., 2022), but often at the expense of the very properties that makes it effective.", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 597, 504, 609 ], "spans": [ { "bbox": [ 106, 597, 504, 609 ], "score": 1.0, "content": "As of yet, none of these variants have been shown to be empirically effective at scale across domains.", "type": "text" } ], "index": 39 } ], "index": 33, "bbox_fs": [ 105, 469, 506, 609 ] }, { "type": "text", "bbox": [ 107, 613, 505, 732 ], "lines": [ { "bbox": [ 105, 613, 505, 625 ], "spans": [ { "bbox": [ 105, 613, 505, 625 ], "score": 1.0, "content": "Recently, structured state space sequence models (SSMs) (Gu et al., 2021; 2022a) have emerged", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 624, 506, 636 ], "spans": [ { "bbox": [ 105, 624, 506, 636 ], "score": 1.0, "content": "as a promising class of architectures for sequence modeling. These models can be interpreted as a", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 635, 504, 646 ], "spans": [ { "bbox": [ 106, 635, 504, 646 ], "score": 1.0, "content": "combination of recurrent neural networks (RNNs) and convolutional neural networks (CNNs), with", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 645, 505, 657 ], "spans": [ { "bbox": [ 105, 645, 505, 657 ], "score": 1.0, "content": "inspiration from classical state space models (Kalman, 1960). This class of models can be computed", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 656, 505, 668 ], "spans": [ { "bbox": [ 105, 656, 505, 668 ], "score": 1.0, "content": "very efficiently as either a recurrence or convolution, with linear or near-linear scaling in sequence", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 667, 505, 679 ], "spans": [ { "bbox": [ 105, 667, 505, 679 ], "score": 1.0, "content": "length. Additionally, they have principled mechanisms for modeling long-range dependencies (Gu", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 676, 505, 690 ], "spans": [ { "bbox": [ 105, 676, 505, 690 ], "score": 1.0, "content": "et al., 2020a) in certain data modalities, and have dominated benchmarks such as the Long Range", "type": "text" } ], "index": 46 }, { "bbox": [ 106, 689, 505, 700 ], "spans": [ { "bbox": [ 106, 689, 505, 700 ], "score": 1.0, "content": "Arena (Tay et al., 2021). Many flavors of SSMs (Gu et al., 2022a; Gupta, 2022; Gu et al., 2022b; Li et al.,", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 698, 505, 711 ], "spans": [ { "bbox": [ 105, 698, 505, 711 ], "score": 1.0, "content": "2023; Ma et al., 2023; Smith et al., 2023; Orvieto et al., 2023) have been successful in domains involving", "type": "text" } ], "index": 48 }, { "bbox": [ 105, 710, 506, 722 ], "spans": [ { "bbox": [ 105, 710, 506, 722 ], "score": 1.0, "content": "continuous signal data such as audio and vision (Goel et al., 2022; Saon et al., 2023; Nguyen et al., 2022).", "type": "text" } ], "index": 49 }, { "bbox": [ 106, 720, 501, 731 ], "spans": [ { "bbox": [ 106, 720, 501, 731 ], "score": 1.0, "content": "However, they have been less effective at modeling discrete and information-dense data such as text.", "type": "text" } ], "index": 50 } ], "index": 45, "bbox_fs": [ 105, 613, 506, 731 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 82, 503, 105 ], "lines": [ { "bbox": [ 106, 82, 505, 95 ], "spans": [ { "bbox": [ 106, 82, 505, 95 ], "score": 1.0, "content": "We propose a new class of selective state space models, that improves on prior work on several axes", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 93, 459, 106 ], "spans": [ { "bbox": [ 106, 93, 459, 106 ], "score": 1.0, "content": "to achieve the modeling power of Transformers while scaling linearly in sequence length.", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "text", "bbox": [ 107, 109, 504, 163 ], "lines": [ { "bbox": [ 105, 108, 506, 122 ], "spans": [ { "bbox": [ 105, 108, 506, 122 ], "score": 1.0, "content": "Selection Mechanism. First, we identify a key limitation of prior models: the ability to efficiently", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 119, 506, 133 ], "spans": [ { "bbox": [ 105, 119, 506, 133 ], "score": 1.0, "content": "select data in an input-dependent manner (i.e. focus on or ignore particular inputs). Building on", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 130, 506, 144 ], "spans": [ { "bbox": [ 105, 130, 506, 144 ], "score": 1.0, "content": "intuition based on important synthetic tasks such as selective copy and induction heads, we design", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 140, 506, 154 ], "spans": [ { "bbox": [ 105, 140, 506, 154 ], "score": 1.0, "content": "a simple selection mechanism by parameterizing the SSM parameters based on the input. This allows", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 151, 472, 164 ], "spans": [ { "bbox": [ 105, 151, 472, 164 ], "score": 1.0, "content": "the model to filter out irrelevant information and remember relevant information indefinitely.", "type": "text" } ], "index": 6 } ], "index": 4 }, { "type": "text", "bbox": [ 107, 168, 505, 244 ], "lines": [ { "bbox": [ 105, 167, 506, 181 ], "spans": [ { "bbox": [ 105, 167, 506, 181 ], "score": 1.0, "content": "Hardware-aware Algorithm. This simple change poses a technical challenge for the computation of", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 178, 506, 191 ], "spans": [ { "bbox": [ 105, 178, 506, 191 ], "score": 1.0, "content": "the model; in fact, all prior SSMs models must be time- and input-invariant in order to be computation-", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 188, 505, 202 ], "spans": [ { "bbox": [ 105, 188, 505, 202 ], "score": 1.0, "content": "ally efficient. We overcome this with a hardware-aware algorithm that computes the model recurrently", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 200, 504, 211 ], "spans": [ { "bbox": [ 106, 200, 504, 211 ], "score": 1.0, "content": "with a scan instead of convolution, but does not materialize the expanded state in order to avoid IO", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 210, 505, 223 ], "spans": [ { "bbox": [ 105, 210, 505, 223 ], "score": 1.0, "content": "access between different levels of the GPU memory hierarchy. The resulting implementation is faster", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 222, 506, 234 ], "spans": [ { "bbox": [ 105, 222, 506, 234 ], "score": 1.0, "content": "than previous methods both in theory (scaling linearly in sequence length, compared to pseudo-linear", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 231, 468, 245 ], "spans": [ { "bbox": [ 105, 231, 361, 245 ], "score": 1.0, "content": "for all convolution-based SSMs) and on modern hardware (up to", "type": "text" }, { "bbox": [ 361, 233, 375, 243 ], "score": 0.87, "content": "3 \\times", "type": "inline_equation" }, { "bbox": [ 375, 231, 468, 245 ], "score": 1.0, "content": "faster on A100 GPUs).", "type": "text" } ], "index": 13 } ], "index": 10 }, { "type": "text", "bbox": [ 107, 248, 505, 281 ], "lines": [ { "bbox": [ 105, 247, 506, 261 ], "spans": [ { "bbox": [ 105, 247, 506, 261 ], "score": 1.0, "content": "Architecture. We simplify prior deep sequence model architectures by combining the design of prior", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 256, 505, 273 ], "spans": [ { "bbox": [ 105, 256, 505, 273 ], "score": 1.0, "content": "SSM architectures (Dao et al., 2023) with the MLP block of Transformers into a single block, leading", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 269, 482, 283 ], "spans": [ { "bbox": [ 105, 269, 482, 283 ], "score": 1.0, "content": "to a simple and homogenous architecture design (Mamba) incorporating selective state spaces.", "type": "text" } ], "index": 16 } ], "index": 15 }, { "type": "text", "bbox": [ 107, 285, 505, 361 ], "lines": [ { "bbox": [ 105, 284, 505, 299 ], "spans": [ { "bbox": [ 105, 284, 505, 299 ], "score": 1.0, "content": "Selective SSMs, and by extension the Mamba architecture, are fully recurrent models with key", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 295, 505, 309 ], "spans": [ { "bbox": [ 105, 295, 505, 309 ], "score": 1.0, "content": "properties that make them suitable as the backbone of general foundation models operating on", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 307, 505, 319 ], "spans": [ { "bbox": [ 105, 307, 505, 319 ], "score": 1.0, "content": "sequences. (i) High quality: selectivity brings strong performance on dense modalities such as", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 317, 505, 330 ], "spans": [ { "bbox": [ 105, 317, 505, 330 ], "score": 1.0, "content": "language and genomics. (ii) Fast training and inference: computation and memory scales linearly", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 329, 505, 341 ], "spans": [ { "bbox": [ 105, 329, 505, 341 ], "score": 1.0, "content": "in sequence length during training, and unrolling the model autoregressively during inference requires", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 338, 505, 352 ], "spans": [ { "bbox": [ 105, 338, 505, 352 ], "score": 1.0, "content": "only constant time per step since it does not require a cache of previous elements. (iii) Long context: the", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 349, 507, 363 ], "spans": [ { "bbox": [ 105, 349, 507, 363 ], "score": 1.0, "content": "quality and efficiency together yield performance improvements on real data up to sequence length 1M.", "type": "text" } ], "index": 23 } ], "index": 20 }, { "type": "text", "bbox": [ 108, 365, 503, 387 ], "lines": [ { "bbox": [ 105, 364, 505, 379 ], "spans": [ { "bbox": [ 105, 364, 505, 379 ], "score": 1.0, "content": "We empirically validate Mamba’s potential as a general sequence FM backbone, in both pretraining", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 376, 462, 389 ], "spans": [ { "bbox": [ 105, 376, 462, 389 ], "score": 1.0, "content": "quality and domain-specific task performance, on several types of modalities and settings:", "type": "text" } ], "index": 25 } ], "index": 24.5 }, { "type": "text", "bbox": [ 106, 392, 506, 554 ], "lines": [ { "bbox": [ 105, 390, 506, 405 ], "spans": [ { "bbox": [ 105, 390, 506, 405 ], "score": 1.0, "content": "• Synthetics. On important synthetic tasks such as copying and induction heads that have been", "type": "text" } ], "index": 26 }, { "bbox": [ 113, 402, 505, 415 ], "spans": [ { "bbox": [ 113, 402, 505, 415 ], "score": 1.0, "content": "proposed as being key to large language models, Mamba not only solves them easily but can", "type": "text" } ], "index": 27 }, { "bbox": [ 114, 414, 327, 425 ], "spans": [ { "bbox": [ 114, 414, 269, 425 ], "score": 1.0, "content": "extrapolate solutions indefinitely long", "type": "text" }, { "bbox": [ 269, 414, 293, 424 ], "score": 0.81, "content": "{ \\bf \\Phi } > 1 { \\bf M }", "type": "inline_equation" }, { "bbox": [ 293, 414, 327, 425 ], "score": 1.0, "content": "tokens).", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 429, 506, 442 ], "spans": [ { "bbox": [ 106, 429, 506, 442 ], "score": 1.0, "content": "• Audio and Genomics. Mamba out-performs prior state-of-the-art models such as SaShiMi, Hyena,", "type": "text" } ], "index": 29 }, { "bbox": [ 113, 440, 505, 453 ], "spans": [ { "bbox": [ 113, 440, 505, 453 ], "score": 1.0, "content": "and Transformers on modeling audio waveforms and DNA sequences, both in pretraining quality", "type": "text" } ], "index": 30 }, { "bbox": [ 114, 451, 505, 464 ], "spans": [ { "bbox": [ 114, 451, 505, 464 ], "score": 1.0, "content": "and downstream metrics (e.g. reducing FID on a challenging speech generation dataset by more than", "type": "text" } ], "index": 31 }, { "bbox": [ 113, 461, 506, 475 ], "spans": [ { "bbox": [ 113, 461, 506, 475 ], "score": 1.0, "content": "half). In both settings, its performance improves with longer context up to million-length sequences.", "type": "text" } ], "index": 32 }, { "bbox": [ 107, 478, 506, 491 ], "spans": [ { "bbox": [ 107, 478, 506, 491 ], "score": 1.0, "content": "• Language Modeling. Mamba is the first linear-time sequence model that truly achieves", "type": "text" } ], "index": 33 }, { "bbox": [ 114, 489, 506, 502 ], "spans": [ { "bbox": [ 114, 489, 506, 502 ], "score": 1.0, "content": "Transformer-quality performance, both in pretraining perplexity and downstream evaluations. With", "type": "text" } ], "index": 34 }, { "bbox": [ 113, 498, 506, 514 ], "spans": [ { "bbox": [ 113, 498, 506, 514 ], "score": 1.0, "content": "scaling laws up to 1B parameters, we show that Mamba exceeds the performance of a large range", "type": "text" } ], "index": 35 }, { "bbox": [ 114, 511, 506, 523 ], "spans": [ { "bbox": [ 114, 511, 506, 523 ], "score": 1.0, "content": "of baselines, including very strong modern Transformer training recipes based on LLaMa (Touvron", "type": "text" } ], "index": 36 }, { "bbox": [ 113, 521, 506, 534 ], "spans": [ { "bbox": [ 113, 521, 343, 534 ], "score": 1.0, "content": "et al., 2023). Our 1.4B Mamba language model has", "type": "text" }, { "bbox": [ 343, 522, 358, 532 ], "score": 0.87, "content": "5 \\times", "type": "inline_equation" }, { "bbox": [ 358, 521, 506, 534 ], "score": 1.0, "content": "inference throughput compared to", "type": "text" } ], "index": 37 }, { "bbox": [ 113, 531, 506, 546 ], "spans": [ { "bbox": [ 113, 531, 506, 546 ], "score": 1.0, "content": "Transformers of similar size, and its quality matches that of Transformers twice its size (e.g. 5 points", "type": "text" } ], "index": 38 }, { "bbox": [ 113, 542, 487, 556 ], "spans": [ { "bbox": [ 113, 542, 487, 556 ], "score": 1.0, "content": "higher avg. on common sense reasoning compared to Pythia-1.4B and matching Pythia-2.8B).", "type": "text" } ], "index": 39 } ], "index": 32.5 }, { "type": "title", "bbox": [ 107, 569, 241, 582 ], "lines": [ { "bbox": [ 105, 568, 242, 585 ], "spans": [ { "bbox": [ 105, 568, 242, 585 ], "score": 1.0, "content": "2 STATE SPACE MODELS", "type": "text" } ], "index": 40 } ], "index": 40 }, { "type": "text", "bbox": [ 107, 587, 504, 643 ], "lines": [ { "bbox": [ 105, 587, 506, 601 ], "spans": [ { "bbox": [ 105, 587, 506, 601 ], "score": 1.0, "content": "Structured state space sequence models (S4) are a recent class of sequence models for deep learning", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 597, 505, 612 ], "spans": [ { "bbox": [ 105, 597, 505, 612 ], "score": 1.0, "content": "that are broadly related to RNNs, and CNNs, and classical state space models. They are inspired by", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 609, 504, 622 ], "spans": [ { "bbox": [ 105, 609, 425, 622 ], "score": 1.0, "content": "a particular continuous system (1) that maps a 1-dimensional function or sequence", "type": "text" }, { "bbox": [ 425, 609, 504, 621 ], "score": 0.91, "content": "x ( t ) \\in \\mathbb { R } \\mapsto y ( t ) \\in \\mathbb { R }", "type": "inline_equation" } ], "index": 43 }, { "bbox": [ 105, 619, 506, 635 ], "spans": [ { "bbox": [ 105, 619, 231, 635 ], "score": 1.0, "content": "through an implicit latent state", "type": "text" }, { "bbox": [ 231, 621, 272, 633 ], "score": 0.92, "content": "h ( t ) \\in \\mathbb { R } ^ { N }", "type": "inline_equation" }, { "bbox": [ 273, 619, 506, 635 ], "score": 1.0, "content": ". Concretely, S4 models are defined with four parameters", "type": "text" } ], "index": 44 }, { "bbox": [ 106, 631, 426, 645 ], "spans": [ { "bbox": [ 106, 632, 157, 644 ], "score": 0.85, "content": "( \\Delta , A , B , C )", "type": "inline_equation" }, { "bbox": [ 157, 631, 426, 645 ], "score": 1.0, "content": ", which define a sequence-to-sequence transformation in two stages.", "type": "text" } ], "index": 45 } ], "index": 43 }, { "type": "interline_equation", "bbox": [ 113, 649, 487, 682 ], "lines": [ { "bbox": [ 113, 649, 487, 682 ], "spans": [ { "bbox": [ 113, 649, 487, 682 ], "score": 0.92, "content": "\\begin{array} { r l r l r l } & { h ^ { \\prime } ( t ) = A h ( t ) + B x ( t ) } & { } & { ( 1 ) } & { \\quad h _ { k } = \\overline { A } h _ { k - 1 } + \\overline { B } x _ { k } } & { \\quad ( 2 \\mathbf { a } ) } & { \\overline { K } = ( C \\overline { B } , C \\overline { A } B , . . . , C \\overline { A } ^ { k } \\overline { B } , . . . ) } \\\\ & { y ( t ) = C h ( t ) } & { } & { y _ { k } = C h _ { k } } & { \\quad ( 2 \\mathbf { b } ) } & { y = x * \\overline { K } } \\end{array}", "type": "interline_equation", "image_path": "8e5eafa5d2465f7371dad1c7d8f03f69711bf43a103db356ab0d729b6c1171af.jpg" } ] } ], "index": 47, "virtual_lines": [ { "bbox": [ 113, 649, 487, 660.0 ], "spans": [], "index": 46 }, { "bbox": [ 113, 660.0, 487, 671.0 ], "spans": [], "index": 47 }, { "bbox": [ 113, 671.0, 487, 682.0 ], "spans": [], "index": 48 } ] }, { "type": "text", "bbox": [ 106, 685, 505, 732 ], "lines": [ { "bbox": [ 105, 684, 506, 698 ], "spans": [ { "bbox": [ 105, 684, 406, 698 ], "score": 1.0, "content": "Discretization The first stage transforms the “continuous parameters”", "type": "text" }, { "bbox": [ 407, 685, 451, 697 ], "score": 0.9, "content": "( \\Delta , A , B )", "type": "inline_equation" }, { "bbox": [ 452, 684, 506, 698 ], "score": 1.0, "content": "to “discrete", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 696, 506, 710 ], "spans": [ { "bbox": [ 105, 696, 158, 710 ], "score": 1.0, "content": "parameters”", "type": "text" }, { "bbox": [ 159, 696, 188, 709 ], "score": 0.91, "content": "( \\overline { { A } } , \\overline { { B } } )", "type": "inline_equation" }, { "bbox": [ 189, 696, 286, 710 ], "score": 1.0, "content": "through fixed formulas", "type": "text" }, { "bbox": [ 286, 696, 347, 709 ], "score": 0.93, "content": "\\overline { { { \\cal A } } } = f _ { A } ( \\Delta , A )", "type": "inline_equation" }, { "bbox": [ 347, 696, 366, 710 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 366, 696, 440, 709 ], "score": 0.92, "content": "\\overline { { B } } = f _ { B } ( \\Delta , A , B )", "type": "inline_equation" }, { "bbox": [ 441, 696, 506, 710 ], "score": 1.0, "content": ", where the pair", "type": "text" } ], "index": 50 }, { "bbox": [ 106, 708, 505, 721 ], "spans": [ { "bbox": [ 106, 708, 141, 720 ], "score": 0.9, "content": "\\bar { ( } f _ { A } , f _ { B } )", "type": "inline_equation" }, { "bbox": [ 141, 708, 505, 721 ], "score": 1.0, "content": "is called a discretization rule. The most common is zero-order hold (ZOH) defined by", "type": "text" } ], "index": 51 }, { "bbox": [ 107, 720, 327, 733 ], "spans": [ { "bbox": [ 107, 720, 166, 732 ], "score": 0.9, "content": "\\overline { { { \\cal A } } } = \\exp ( \\Delta { \\cal A } )", "type": "inline_equation" }, { "bbox": [ 166, 720, 184, 733 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 184, 720, 323, 733 ], "score": 0.9, "content": "\\overline { { B } } = ( \\Delta A ) ^ { - 1 } ( \\exp ( \\Delta A ) - I ) \\cdot \\Delta B", "type": "inline_equation" }, { "bbox": [ 323, 720, 327, 733 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 52 } ], "index": 50.5 } ], "page_idx": 1, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 304, 37 ], "lines": [ { "bbox": [ 106, 26, 304, 38 ], "spans": [ { "bbox": [ 106, 26, 304, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 309, 760 ], "lines": [ { "bbox": [ 301, 750, 310, 763 ], "spans": [ { "bbox": [ 301, 750, 310, 763 ], "score": 1.0, "content": "2", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 82, 503, 105 ], "lines": [ { "bbox": [ 106, 82, 505, 95 ], "spans": [ { "bbox": [ 106, 82, 505, 95 ], "score": 1.0, "content": "We propose a new class of selective state space models, that improves on prior work on several axes", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 93, 459, 106 ], "spans": [ { "bbox": [ 106, 93, 459, 106 ], "score": 1.0, "content": "to achieve the modeling power of Transformers while scaling linearly in sequence length.", "type": "text" } ], "index": 1 } ], "index": 0.5, "bbox_fs": [ 106, 82, 505, 106 ] }, { "type": "text", "bbox": [ 107, 109, 504, 163 ], "lines": [ { "bbox": [ 105, 108, 506, 122 ], "spans": [ { "bbox": [ 105, 108, 506, 122 ], "score": 1.0, "content": "Selection Mechanism. First, we identify a key limitation of prior models: the ability to efficiently", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 119, 506, 133 ], "spans": [ { "bbox": [ 105, 119, 506, 133 ], "score": 1.0, "content": "select data in an input-dependent manner (i.e. focus on or ignore particular inputs). Building on", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 130, 506, 144 ], "spans": [ { "bbox": [ 105, 130, 506, 144 ], "score": 1.0, "content": "intuition based on important synthetic tasks such as selective copy and induction heads, we design", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 140, 506, 154 ], "spans": [ { "bbox": [ 105, 140, 506, 154 ], "score": 1.0, "content": "a simple selection mechanism by parameterizing the SSM parameters based on the input. This allows", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 151, 472, 164 ], "spans": [ { "bbox": [ 105, 151, 472, 164 ], "score": 1.0, "content": "the model to filter out irrelevant information and remember relevant information indefinitely.", "type": "text" } ], "index": 6 } ], "index": 4, "bbox_fs": [ 105, 108, 506, 164 ] }, { "type": "text", "bbox": [ 107, 168, 505, 244 ], "lines": [ { "bbox": [ 105, 167, 506, 181 ], "spans": [ { "bbox": [ 105, 167, 506, 181 ], "score": 1.0, "content": "Hardware-aware Algorithm. This simple change poses a technical challenge for the computation of", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 178, 506, 191 ], "spans": [ { "bbox": [ 105, 178, 506, 191 ], "score": 1.0, "content": "the model; in fact, all prior SSMs models must be time- and input-invariant in order to be computation-", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 188, 505, 202 ], "spans": [ { "bbox": [ 105, 188, 505, 202 ], "score": 1.0, "content": "ally efficient. We overcome this with a hardware-aware algorithm that computes the model recurrently", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 200, 504, 211 ], "spans": [ { "bbox": [ 106, 200, 504, 211 ], "score": 1.0, "content": "with a scan instead of convolution, but does not materialize the expanded state in order to avoid IO", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 210, 505, 223 ], "spans": [ { "bbox": [ 105, 210, 505, 223 ], "score": 1.0, "content": "access between different levels of the GPU memory hierarchy. The resulting implementation is faster", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 222, 506, 234 ], "spans": [ { "bbox": [ 105, 222, 506, 234 ], "score": 1.0, "content": "than previous methods both in theory (scaling linearly in sequence length, compared to pseudo-linear", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 231, 468, 245 ], "spans": [ { "bbox": [ 105, 231, 361, 245 ], "score": 1.0, "content": "for all convolution-based SSMs) and on modern hardware (up to", "type": "text" }, { "bbox": [ 361, 233, 375, 243 ], "score": 0.87, "content": "3 \\times", "type": "inline_equation" }, { "bbox": [ 375, 231, 468, 245 ], "score": 1.0, "content": "faster on A100 GPUs).", "type": "text" } ], "index": 13 } ], "index": 10, "bbox_fs": [ 105, 167, 506, 245 ] }, { "type": "text", "bbox": [ 107, 248, 505, 281 ], "lines": [ { "bbox": [ 105, 247, 506, 261 ], "spans": [ { "bbox": [ 105, 247, 506, 261 ], "score": 1.0, "content": "Architecture. We simplify prior deep sequence model architectures by combining the design of prior", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 256, 505, 273 ], "spans": [ { "bbox": [ 105, 256, 505, 273 ], "score": 1.0, "content": "SSM architectures (Dao et al., 2023) with the MLP block of Transformers into a single block, leading", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 269, 482, 283 ], "spans": [ { "bbox": [ 105, 269, 482, 283 ], "score": 1.0, "content": "to a simple and homogenous architecture design (Mamba) incorporating selective state spaces.", "type": "text" } ], "index": 16 } ], "index": 15, "bbox_fs": [ 105, 247, 506, 283 ] }, { "type": "text", "bbox": [ 107, 285, 505, 361 ], "lines": [ { "bbox": [ 105, 284, 505, 299 ], "spans": [ { "bbox": [ 105, 284, 505, 299 ], "score": 1.0, "content": "Selective SSMs, and by extension the Mamba architecture, are fully recurrent models with key", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 295, 505, 309 ], "spans": [ { "bbox": [ 105, 295, 505, 309 ], "score": 1.0, "content": "properties that make them suitable as the backbone of general foundation models operating on", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 307, 505, 319 ], "spans": [ { "bbox": [ 105, 307, 505, 319 ], "score": 1.0, "content": "sequences. (i) High quality: selectivity brings strong performance on dense modalities such as", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 317, 505, 330 ], "spans": [ { "bbox": [ 105, 317, 505, 330 ], "score": 1.0, "content": "language and genomics. (ii) Fast training and inference: computation and memory scales linearly", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 329, 505, 341 ], "spans": [ { "bbox": [ 105, 329, 505, 341 ], "score": 1.0, "content": "in sequence length during training, and unrolling the model autoregressively during inference requires", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 338, 505, 352 ], "spans": [ { "bbox": [ 105, 338, 505, 352 ], "score": 1.0, "content": "only constant time per step since it does not require a cache of previous elements. (iii) Long context: the", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 349, 507, 363 ], "spans": [ { "bbox": [ 105, 349, 507, 363 ], "score": 1.0, "content": "quality and efficiency together yield performance improvements on real data up to sequence length 1M.", "type": "text" } ], "index": 23 } ], "index": 20, "bbox_fs": [ 105, 284, 507, 363 ] }, { "type": "text", "bbox": [ 108, 365, 503, 387 ], "lines": [ { "bbox": [ 105, 364, 505, 379 ], "spans": [ { "bbox": [ 105, 364, 505, 379 ], "score": 1.0, "content": "We empirically validate Mamba’s potential as a general sequence FM backbone, in both pretraining", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 376, 462, 389 ], "spans": [ { "bbox": [ 105, 376, 462, 389 ], "score": 1.0, "content": "quality and domain-specific task performance, on several types of modalities and settings:", "type": "text" } ], "index": 25 } ], "index": 24.5, "bbox_fs": [ 105, 364, 505, 389 ] }, { "type": "text", "bbox": [ 106, 392, 506, 554 ], "lines": [ { "bbox": [ 105, 390, 506, 405 ], "spans": [ { "bbox": [ 105, 390, 506, 405 ], "score": 1.0, "content": "• Synthetics. On important synthetic tasks such as copying and induction heads that have been", "type": "text" } ], "index": 26 }, { "bbox": [ 113, 402, 505, 415 ], "spans": [ { "bbox": [ 113, 402, 505, 415 ], "score": 1.0, "content": "proposed as being key to large language models, Mamba not only solves them easily but can", "type": "text" } ], "index": 27 }, { "bbox": [ 114, 414, 327, 425 ], "spans": [ { "bbox": [ 114, 414, 269, 425 ], "score": 1.0, "content": "extrapolate solutions indefinitely long", "type": "text" }, { "bbox": [ 269, 414, 293, 424 ], "score": 0.81, "content": "{ \\bf \\Phi } > 1 { \\bf M }", "type": "inline_equation" }, { "bbox": [ 293, 414, 327, 425 ], "score": 1.0, "content": "tokens).", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 429, 506, 442 ], "spans": [ { "bbox": [ 106, 429, 506, 442 ], "score": 1.0, "content": "• Audio and Genomics. Mamba out-performs prior state-of-the-art models such as SaShiMi, Hyena,", "type": "text" } ], "index": 29 }, { "bbox": [ 113, 440, 505, 453 ], "spans": [ { "bbox": [ 113, 440, 505, 453 ], "score": 1.0, "content": "and Transformers on modeling audio waveforms and DNA sequences, both in pretraining quality", "type": "text" } ], "index": 30 }, { "bbox": [ 114, 451, 505, 464 ], "spans": [ { "bbox": [ 114, 451, 505, 464 ], "score": 1.0, "content": "and downstream metrics (e.g. reducing FID on a challenging speech generation dataset by more than", "type": "text" } ], "index": 31 }, { "bbox": [ 113, 461, 506, 475 ], "spans": [ { "bbox": [ 113, 461, 506, 475 ], "score": 1.0, "content": "half). In both settings, its performance improves with longer context up to million-length sequences.", "type": "text" } ], "index": 32 }, { "bbox": [ 107, 478, 506, 491 ], "spans": [ { "bbox": [ 107, 478, 506, 491 ], "score": 1.0, "content": "• Language Modeling. Mamba is the first linear-time sequence model that truly achieves", "type": "text" } ], "index": 33 }, { "bbox": [ 114, 489, 506, 502 ], "spans": [ { "bbox": [ 114, 489, 506, 502 ], "score": 1.0, "content": "Transformer-quality performance, both in pretraining perplexity and downstream evaluations. With", "type": "text" } ], "index": 34 }, { "bbox": [ 113, 498, 506, 514 ], "spans": [ { "bbox": [ 113, 498, 506, 514 ], "score": 1.0, "content": "scaling laws up to 1B parameters, we show that Mamba exceeds the performance of a large range", "type": "text" } ], "index": 35 }, { "bbox": [ 114, 511, 506, 523 ], "spans": [ { "bbox": [ 114, 511, 506, 523 ], "score": 1.0, "content": "of baselines, including very strong modern Transformer training recipes based on LLaMa (Touvron", "type": "text" } ], "index": 36 }, { "bbox": [ 113, 521, 506, 534 ], "spans": [ { "bbox": [ 113, 521, 343, 534 ], "score": 1.0, "content": "et al., 2023). Our 1.4B Mamba language model has", "type": "text" }, { "bbox": [ 343, 522, 358, 532 ], "score": 0.87, "content": "5 \\times", "type": "inline_equation" }, { "bbox": [ 358, 521, 506, 534 ], "score": 1.0, "content": "inference throughput compared to", "type": "text" } ], "index": 37 }, { "bbox": [ 113, 531, 506, 546 ], "spans": [ { "bbox": [ 113, 531, 506, 546 ], "score": 1.0, "content": "Transformers of similar size, and its quality matches that of Transformers twice its size (e.g. 5 points", "type": "text" } ], "index": 38 }, { "bbox": [ 113, 542, 487, 556 ], "spans": [ { "bbox": [ 113, 542, 487, 556 ], "score": 1.0, "content": "higher avg. on common sense reasoning compared to Pythia-1.4B and matching Pythia-2.8B).", "type": "text" } ], "index": 39 } ], "index": 32.5, "bbox_fs": [ 105, 390, 506, 556 ] }, { "type": "title", "bbox": [ 107, 569, 241, 582 ], "lines": [ { "bbox": [ 105, 568, 242, 585 ], "spans": [ { "bbox": [ 105, 568, 242, 585 ], "score": 1.0, "content": "2 STATE SPACE MODELS", "type": "text" } ], "index": 40 } ], "index": 40 }, { "type": "text", "bbox": [ 107, 587, 504, 643 ], "lines": [ { "bbox": [ 105, 587, 506, 601 ], "spans": [ { "bbox": [ 105, 587, 506, 601 ], "score": 1.0, "content": "Structured state space sequence models (S4) are a recent class of sequence models for deep learning", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 597, 505, 612 ], "spans": [ { "bbox": [ 105, 597, 505, 612 ], "score": 1.0, "content": "that are broadly related to RNNs, and CNNs, and classical state space models. They are inspired by", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 609, 504, 622 ], "spans": [ { "bbox": [ 105, 609, 425, 622 ], "score": 1.0, "content": "a particular continuous system (1) that maps a 1-dimensional function or sequence", "type": "text" }, { "bbox": [ 425, 609, 504, 621 ], "score": 0.91, "content": "x ( t ) \\in \\mathbb { R } \\mapsto y ( t ) \\in \\mathbb { R }", "type": "inline_equation" } ], "index": 43 }, { "bbox": [ 105, 619, 506, 635 ], "spans": [ { "bbox": [ 105, 619, 231, 635 ], "score": 1.0, "content": "through an implicit latent state", "type": "text" }, { "bbox": [ 231, 621, 272, 633 ], "score": 0.92, "content": "h ( t ) \\in \\mathbb { R } ^ { N }", "type": "inline_equation" }, { "bbox": [ 273, 619, 506, 635 ], "score": 1.0, "content": ". Concretely, S4 models are defined with four parameters", "type": "text" } ], "index": 44 }, { "bbox": [ 106, 631, 426, 645 ], "spans": [ { "bbox": [ 106, 632, 157, 644 ], "score": 0.85, "content": "( \\Delta , A , B , C )", "type": "inline_equation" }, { "bbox": [ 157, 631, 426, 645 ], "score": 1.0, "content": ", which define a sequence-to-sequence transformation in two stages.", "type": "text" } ], "index": 45 } ], "index": 43, "bbox_fs": [ 105, 587, 506, 645 ] }, { "type": "interline_equation", "bbox": [ 113, 649, 487, 682 ], "lines": [ { "bbox": [ 113, 649, 487, 682 ], "spans": [ { "bbox": [ 113, 649, 487, 682 ], "score": 0.92, "content": "\\begin{array} { r l r l r l } & { h ^ { \\prime } ( t ) = A h ( t ) + B x ( t ) } & { } & { ( 1 ) } & { \\quad h _ { k } = \\overline { A } h _ { k - 1 } + \\overline { B } x _ { k } } & { \\quad ( 2 \\mathbf { a } ) } & { \\overline { K } = ( C \\overline { B } , C \\overline { A } B , . . . , C \\overline { A } ^ { k } \\overline { B } , . . . ) } \\\\ & { y ( t ) = C h ( t ) } & { } & { y _ { k } = C h _ { k } } & { \\quad ( 2 \\mathbf { b } ) } & { y = x * \\overline { K } } \\end{array}", "type": "interline_equation", "image_path": "8e5eafa5d2465f7371dad1c7d8f03f69711bf43a103db356ab0d729b6c1171af.jpg" } ] } ], "index": 47, "virtual_lines": [ { "bbox": [ 113, 649, 487, 660.0 ], "spans": [], "index": 46 }, { "bbox": [ 113, 660.0, 487, 671.0 ], "spans": [], "index": 47 }, { "bbox": [ 113, 671.0, 487, 682.0 ], "spans": [], "index": 48 } ] }, { "type": "text", "bbox": [ 106, 685, 505, 732 ], "lines": [ { "bbox": [ 105, 684, 506, 698 ], "spans": [ { "bbox": [ 105, 684, 406, 698 ], "score": 1.0, "content": "Discretization The first stage transforms the “continuous parameters”", "type": "text" }, { "bbox": [ 407, 685, 451, 697 ], "score": 0.9, "content": "( \\Delta , A , B )", "type": "inline_equation" }, { "bbox": [ 452, 684, 506, 698 ], "score": 1.0, "content": "to “discrete", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 696, 506, 710 ], "spans": [ { "bbox": [ 105, 696, 158, 710 ], "score": 1.0, "content": "parameters”", "type": "text" }, { "bbox": [ 159, 696, 188, 709 ], "score": 0.91, "content": "( \\overline { { A } } , \\overline { { B } } )", "type": "inline_equation" }, { "bbox": [ 189, 696, 286, 710 ], "score": 1.0, "content": "through fixed formulas", "type": "text" }, { "bbox": [ 286, 696, 347, 709 ], "score": 0.93, "content": "\\overline { { { \\cal A } } } = f _ { A } ( \\Delta , A )", "type": "inline_equation" }, { "bbox": [ 347, 696, 366, 710 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 366, 696, 440, 709 ], "score": 0.92, "content": "\\overline { { B } } = f _ { B } ( \\Delta , A , B )", "type": "inline_equation" }, { "bbox": [ 441, 696, 506, 710 ], "score": 1.0, "content": ", where the pair", "type": "text" } ], "index": 50 }, { "bbox": [ 106, 708, 505, 721 ], "spans": [ { "bbox": [ 106, 708, 141, 720 ], "score": 0.9, "content": "\\bar { ( } f _ { A } , f _ { B } )", "type": "inline_equation" }, { "bbox": [ 141, 708, 505, 721 ], "score": 1.0, "content": "is called a discretization rule. The most common is zero-order hold (ZOH) defined by", "type": "text" } ], "index": 51 }, { "bbox": [ 107, 720, 327, 733 ], "spans": [ { "bbox": [ 107, 720, 166, 732 ], "score": 0.9, "content": "\\overline { { { \\cal A } } } = \\exp ( \\Delta { \\cal A } )", "type": "inline_equation" }, { "bbox": [ 166, 720, 184, 733 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 184, 720, 323, 733 ], "score": 0.9, "content": "\\overline { { B } } = ( \\Delta A ) ^ { - 1 } ( \\exp ( \\Delta A ) - I ) \\cdot \\Delta B", "type": "inline_equation" }, { "bbox": [ 323, 720, 327, 733 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 52 } ], "index": 50.5, "bbox_fs": [ 105, 684, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 147 ], "lines": [ { "bbox": [ 105, 82, 505, 95 ], "spans": [ { "bbox": [ 105, 82, 505, 95 ], "score": 1.0, "content": "Discretization has deep connections to continuous-time systems which can endow them with additional", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 505, 105 ], "spans": [ { "bbox": [ 105, 93, 505, 105 ], "score": 1.0, "content": "properties such as resolution invariance (Nguyen et al., 2022) and automatically ensuring that the", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 103, 505, 117 ], "spans": [ { "bbox": [ 105, 103, 505, 117 ], "score": 1.0, "content": "model is properly normalized (Gu et al., 2023; Orvieto et al., 2023). It also has connections to gating", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 115, 506, 126 ], "spans": [ { "bbox": [ 106, 115, 506, 126 ], "score": 1.0, "content": "mechanisms of RNNs (Tallec & Ollivier, 2018; Gu et al., 2020b) which we will revisit in Section 3.5.", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 125, 506, 137 ], "spans": [ { "bbox": [ 105, 125, 506, 137 ], "score": 1.0, "content": "However, from a mechanical point of view discretization can simply be viewed as the first step of", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 136, 320, 148 ], "spans": [ { "bbox": [ 106, 136, 320, 148 ], "score": 1.0, "content": "the computation graph in the forward pass of an SSM.", "type": "text" } ], "index": 5 } ], "index": 2.5 }, { "type": "text", "bbox": [ 106, 153, 504, 175 ], "lines": [ { "bbox": [ 106, 152, 505, 165 ], "spans": [ { "bbox": [ 106, 152, 357, 165 ], "score": 1.0, "content": "Computation After the parameters have been transformed from", "type": "text" }, { "bbox": [ 357, 152, 461, 165 ], "score": 0.93, "content": "( \\Delta , A , B , C ) \\mapsto ( { \\overline { { A } } } , { \\overline { { B } } } , C )", "type": "inline_equation" }, { "bbox": [ 461, 152, 505, 165 ], "score": 1.0, "content": ", the model", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 163, 469, 176 ], "spans": [ { "bbox": [ 106, 163, 469, 176 ], "score": 1.0, "content": "can be computed in two ways, either as a linear recurrence (2) or a global convolution (3).", "type": "text" } ], "index": 7 } ], "index": 6.5 }, { "type": "text", "bbox": [ 108, 180, 503, 212 ], "lines": [ { "bbox": [ 106, 179, 505, 192 ], "spans": [ { "bbox": [ 106, 179, 505, 192 ], "score": 1.0, "content": "Commonly, the model uses the convolutional mode (3) for efficient parallelizable training (where", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 191, 505, 203 ], "spans": [ { "bbox": [ 106, 191, 505, 203 ], "score": 1.0, "content": "the whole input sequence is seen ahead of time), and switched into recurrent mode (2) for efficient", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 202, 402, 214 ], "spans": [ { "bbox": [ 106, 202, 402, 214 ], "score": 1.0, "content": "autoregressive inference (where the inputs are seen one timestep at a time).", "type": "text" } ], "index": 10 } ], "index": 9 }, { "type": "text", "bbox": [ 107, 216, 505, 272 ], "lines": [ { "bbox": [ 105, 216, 506, 230 ], "spans": [ { "bbox": [ 105, 216, 506, 230 ], "score": 1.0, "content": "Linear Time Invariance (LTI) An important property of equations (1) to (3) is that the model’s", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 227, 506, 241 ], "spans": [ { "bbox": [ 106, 227, 315, 241 ], "score": 1.0, "content": "dynamics are constant through time. In other words", "type": "text" }, { "bbox": [ 315, 228, 367, 240 ], "score": 0.91, "content": "( \\Delta , A , B , C )", "type": "inline_equation" }, { "bbox": [ 367, 227, 443, 241 ], "score": 1.0, "content": ", and consequently", "type": "text" }, { "bbox": [ 443, 227, 472, 240 ], "score": 0.9, "content": "( \\overline { { A } } , \\overline { { B } } )", "type": "inline_equation" }, { "bbox": [ 473, 227, 506, 241 ], "score": 1.0, "content": "as well,", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 237, 506, 253 ], "spans": [ { "bbox": [ 105, 237, 412, 253 ], "score": 1.0, "content": "are fixed for all time-steps. This property is called linear time invariance", "type": "text" }, { "bbox": [ 412, 240, 434, 250 ], "score": 0.4, "content": "( L T I )", "type": "inline_equation" }, { "bbox": [ 434, 237, 506, 253 ], "score": 1.0, "content": ", which is deeply", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 248, 506, 264 ], "spans": [ { "bbox": [ 105, 248, 506, 264 ], "score": 1.0, "content": "connected to recurrence and convolutions. Informally, we think of LTI SSMs as being equivalent to any", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 261, 506, 273 ], "spans": [ { "bbox": [ 106, 261, 506, 273 ], "score": 1.0, "content": "linear recurrence (2a) or convolution (3b), and use LTI as an umbrella term for these classes of models.", "type": "text" } ], "index": 15 } ], "index": 13 }, { "type": "text", "bbox": [ 107, 277, 505, 320 ], "lines": [ { "bbox": [ 105, 276, 506, 290 ], "spans": [ { "bbox": [ 105, 276, 506, 290 ], "score": 1.0, "content": "Thus far, all structured SSMs have been LTI (e.g. computed as convolutions) because of fundamental", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 287, 505, 299 ], "spans": [ { "bbox": [ 106, 287, 505, 299 ], "score": 1.0, "content": "efficiency constraints, discussed in Section 3.3. However, a core insight of this work is that LTI models", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 298, 504, 310 ], "spans": [ { "bbox": [ 106, 298, 504, 310 ], "score": 1.0, "content": "have fundamental limitations in modeling certain types of data, and our technical contributions involve", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 309, 397, 321 ], "spans": [ { "bbox": [ 105, 309, 397, 321 ], "score": 1.0, "content": "removing the LTI constraint while overcoming the efficiency bottlenecks.", "type": "text" } ], "index": 19 } ], "index": 17.5 }, { "type": "text", "bbox": [ 107, 325, 505, 357 ], "lines": [ { "bbox": [ 106, 324, 505, 337 ], "spans": [ { "bbox": [ 106, 324, 505, 337 ], "score": 1.0, "content": "Structure Finally, we note that structured SSMs are so named because computing them efficiently also", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 335, 507, 349 ], "spans": [ { "bbox": [ 105, 335, 244, 349 ], "score": 1.0, "content": "requires imposing structure on the", "type": "text" }, { "bbox": [ 245, 336, 254, 345 ], "score": 0.52, "content": "\\pmb { A }", "type": "inline_equation" }, { "bbox": [ 254, 335, 507, 349 ], "score": 1.0, "content": "matrix. The most popular form of structure is diagonal (Gupta,", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 345, 352, 358 ], "spans": [ { "bbox": [ 106, 345, 352, 358 ], "score": 1.0, "content": "2022; Gu et al., 2022b; Smith et al., 2023), which we also use.", "type": "text" } ], "index": 22 } ], "index": 21 }, { "type": "text", "bbox": [ 107, 360, 505, 416 ], "lines": [ { "bbox": [ 105, 359, 507, 375 ], "spans": [ { "bbox": [ 105, 359, 172, 375 ], "score": 1.0, "content": "In this case, the", "type": "text" }, { "bbox": [ 172, 360, 314, 373 ], "score": 0.32, "content": "\\pmb { A } \\in \\mathbb { R } ^ { N \\times N } , \\pmb { B } \\in \\mathbb { R } ^ { N \\times 1 } , \\pmb { C } \\in \\mathbb { R } ^ { 1 \\times N }", "type": "inline_equation" }, { "bbox": [ 315, 359, 455, 375 ], "score": 1.0, "content": "matrices can all be represented by", "type": "text" }, { "bbox": [ 456, 362, 466, 372 ], "score": 0.8, "content": "N", "type": "inline_equation" }, { "bbox": [ 466, 359, 507, 375 ], "score": 1.0, "content": "numbers.", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 372, 505, 385 ], "spans": [ { "bbox": [ 105, 372, 240, 385 ], "score": 1.0, "content": "To operate over an input sequence", "type": "text" }, { "bbox": [ 240, 375, 247, 383 ], "score": 0.79, "content": "x", "type": "inline_equation" }, { "bbox": [ 248, 372, 298, 385 ], "score": 1.0, "content": "of batch size", "type": "text" }, { "bbox": [ 299, 373, 308, 383 ], "score": 0.79, "content": "B", "type": "inline_equation" }, { "bbox": [ 308, 372, 351, 385 ], "score": 1.0, "content": "and length", "type": "text" }, { "bbox": [ 352, 373, 360, 383 ], "score": 0.8, "content": "L", "type": "inline_equation" }, { "bbox": [ 360, 372, 380, 385 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 380, 373, 389, 383 ], "score": 0.79, "content": "D", "type": "inline_equation" }, { "bbox": [ 390, 372, 505, 385 ], "score": 1.0, "content": "channels, the SSM is applied", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 383, 506, 397 ], "spans": [ { "bbox": [ 105, 383, 470, 397 ], "score": 1.0, "content": "independently to each channel. Note that in this case, the total hidden state has dimension", "type": "text" }, { "bbox": [ 470, 384, 489, 393 ], "score": 0.77, "content": "D N", "type": "inline_equation" }, { "bbox": [ 489, 383, 506, 397 ], "score": 1.0, "content": "per", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 394, 505, 407 ], "spans": [ { "bbox": [ 105, 394, 338, 407 ], "score": 1.0, "content": "input, and computing it over the sequence length requires", "type": "text" }, { "bbox": [ 339, 394, 387, 406 ], "score": 0.91, "content": "O ( B L D N )", "type": "inline_equation" }, { "bbox": [ 388, 394, 505, 407 ], "score": 1.0, "content": "time and memory; this is the", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 405, 385, 416 ], "spans": [ { "bbox": [ 105, 405, 385, 416 ], "score": 1.0, "content": "root of the fundamental efficiency bottleneck addressed in Section 3.3.", "type": "text" } ], "index": 27 } ], "index": 25 }, { "type": "text", "bbox": [ 107, 420, 505, 464 ], "lines": [ { "bbox": [ 106, 420, 505, 433 ], "spans": [ { "bbox": [ 106, 420, 505, 433 ], "score": 1.0, "content": "SSM Architectures SSMs are standalone sequence transformations that can be incorporated into", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 430, 507, 444 ], "spans": [ { "bbox": [ 105, 430, 507, 444 ], "score": 1.0, "content": "end-to-end neural network architectures. We discuss some of the most well-known SSM architectures,", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 443, 505, 455 ], "spans": [ { "bbox": [ 106, 443, 505, 455 ], "score": 1.0, "content": "many of which will also serve as our primary baselines. Other closely related SSMs and architectures", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 453, 358, 465 ], "spans": [ { "bbox": [ 106, 453, 358, 465 ], "score": 1.0, "content": "are discussed further in an extended related work (Appendix B).", "type": "text" } ], "index": 31 } ], "index": 29.5 }, { "type": "text", "bbox": [ 106, 468, 506, 567 ], "lines": [ { "bbox": [ 105, 466, 506, 480 ], "spans": [ { "bbox": [ 105, 466, 506, 480 ], "score": 1.0, "content": "• Linear attention (Katharopoulos et al., 2020) is an approximation of self-attention (Bahdanau et al.,", "type": "text" } ], "index": 32 }, { "bbox": [ 113, 478, 425, 491 ], "spans": [ { "bbox": [ 113, 478, 425, 491 ], "score": 1.0, "content": "2014) involving a recurrence which can be viewed as a degenerate linear SSM.", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 488, 505, 503 ], "spans": [ { "bbox": [ 106, 488, 505, 503 ], "score": 1.0, "content": "• H3 (Dao et al., 2023) generalized this recurrence to use S4; it can be viewed as an architecture", "type": "text" } ], "index": 34 }, { "bbox": [ 113, 500, 505, 513 ], "spans": [ { "bbox": [ 113, 500, 505, 513 ], "score": 1.0, "content": "with an SSM sandwiched by two gated connections (Figure 2). H3 also inserts a standard local", "type": "text" } ], "index": 35 }, { "bbox": [ 113, 511, 409, 523 ], "spans": [ { "bbox": [ 113, 511, 409, 523 ], "score": 1.0, "content": "convolution, which they frame as a shift-SSM, before the main SSM layer.", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 521, 506, 534 ], "spans": [ { "bbox": [ 106, 521, 506, 534 ], "score": 1.0, "content": "• Hyena (Poli et al., 2023) uses the same architecture but replaces the S4 layer with an MLP-", "type": "text" } ], "index": 37 }, { "bbox": [ 113, 533, 338, 545 ], "spans": [ { "bbox": [ 113, 533, 338, 545 ], "score": 1.0, "content": "parameterized global convolution (Romero et al., 2021).", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 542, 505, 556 ], "spans": [ { "bbox": [ 106, 542, 505, 556 ], "score": 1.0, "content": "• RetNet (Sun et al., 2023) adds an additional gate to the architecture and simplifies the SSM, allowing", "type": "text" } ], "index": 39 }, { "bbox": [ 114, 555, 503, 566 ], "spans": [ { "bbox": [ 114, 555, 503, 566 ], "score": 1.0, "content": "an alternative parallelizable computation path, using a variant of attention instead of convolutions.", "type": "text" } ], "index": 40 } ], "index": 36 }, { "type": "title", "bbox": [ 108, 578, 301, 591 ], "lines": [ { "bbox": [ 104, 576, 302, 593 ], "spans": [ { "bbox": [ 104, 576, 302, 593 ], "score": 1.0, "content": "3 SELECTIVE STATE SPACE MODELS", "type": "text" } ], "index": 41 } ], "index": 41 }, { "type": "text", "bbox": [ 106, 596, 505, 662 ], "lines": [ { "bbox": [ 106, 596, 505, 609 ], "spans": [ { "bbox": [ 106, 596, 505, 609 ], "score": 1.0, "content": "We motivate our selection mechanism using intuition from synthetic tasks (Section 3.1), then explain", "type": "text" } ], "index": 42 }, { "bbox": [ 104, 605, 507, 622 ], "spans": [ { "bbox": [ 104, 605, 507, 622 ], "score": 1.0, "content": "how to incorporate this mechanism into state space models (Section 3.2). The resulting time-varying", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 617, 506, 631 ], "spans": [ { "bbox": [ 105, 617, 506, 631 ], "score": 1.0, "content": "SSMs cannot use convolutions, presenting a technical challenge of how to compute them efficiently.", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 629, 506, 641 ], "spans": [ { "bbox": [ 105, 629, 506, 641 ], "score": 1.0, "content": "We overcome this with a hardware-aware algorithm that exploits the memory hierarchy on modern hard-", "type": "text" } ], "index": 45 }, { "bbox": [ 106, 640, 505, 651 ], "spans": [ { "bbox": [ 106, 640, 505, 651 ], "score": 1.0, "content": "ware (Section 3.3). We then describe a simple SSM architecture without attention or even MLP blocks", "type": "text" } ], "index": 46 }, { "bbox": [ 106, 650, 501, 663 ], "spans": [ { "bbox": [ 106, 650, 501, 663 ], "score": 1.0, "content": "(Section 3.4). Finally, we discuss some additional properties of selection mechanisms (Section 3.5).", "type": "text" } ], "index": 47 } ], "index": 44.5 }, { "type": "title", "bbox": [ 107, 672, 375, 684 ], "lines": [ { "bbox": [ 106, 672, 376, 685 ], "spans": [ { "bbox": [ 106, 672, 376, 685 ], "score": 1.0, "content": "3.1 MOTIVATION: SELECTION AS A MEANS OF COMPRESSION", "type": "text" } ], "index": 48 } ], "index": 48 }, { "type": "text", "bbox": [ 107, 689, 505, 732 ], "lines": [ { "bbox": [ 105, 687, 507, 702 ], "spans": [ { "bbox": [ 105, 687, 507, 702 ], "score": 1.0, "content": "We argue that a fundamental problem of sequence modeling is compressing context into a smaller state.", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 698, 507, 713 ], "spans": [ { "bbox": [ 105, 698, 507, 713 ], "score": 1.0, "content": "In fact, we can view the tradeoffs of popular sequence models from this point of view. For example,", "type": "text" } ], "index": 50 }, { "bbox": [ 106, 710, 505, 722 ], "spans": [ { "bbox": [ 106, 710, 505, 722 ], "score": 1.0, "content": "attention is both effective and inefficient because it explicitly does not compress context at all. This", "type": "text" } ], "index": 51 }, { "bbox": [ 105, 719, 506, 734 ], "spans": [ { "bbox": [ 105, 719, 506, 734 ], "score": 1.0, "content": "can be seen from the fact that autoregressive inference requires explicitly storing the entire context", "type": "text" } ], "index": 52 } ], "index": 50.5 } ], "page_idx": 2, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 304, 37 ], "lines": [ { "bbox": [ 106, 26, 304, 38 ], "spans": [ { "bbox": [ 106, 26, 304, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 309, 760 ], "lines": [ { "bbox": [ 301, 750, 310, 762 ], "spans": [ { "bbox": [ 301, 750, 310, 762 ], "score": 1.0, "content": "3", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 147 ], "lines": [ { "bbox": [ 105, 82, 505, 95 ], "spans": [ { "bbox": [ 105, 82, 505, 95 ], "score": 1.0, "content": "Discretization has deep connections to continuous-time systems which can endow them with additional", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 505, 105 ], "spans": [ { "bbox": [ 105, 93, 505, 105 ], "score": 1.0, "content": "properties such as resolution invariance (Nguyen et al., 2022) and automatically ensuring that the", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 103, 505, 117 ], "spans": [ { "bbox": [ 105, 103, 505, 117 ], "score": 1.0, "content": "model is properly normalized (Gu et al., 2023; Orvieto et al., 2023). It also has connections to gating", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 115, 506, 126 ], "spans": [ { "bbox": [ 106, 115, 506, 126 ], "score": 1.0, "content": "mechanisms of RNNs (Tallec & Ollivier, 2018; Gu et al., 2020b) which we will revisit in Section 3.5.", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 125, 506, 137 ], "spans": [ { "bbox": [ 105, 125, 506, 137 ], "score": 1.0, "content": "However, from a mechanical point of view discretization can simply be viewed as the first step of", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 136, 320, 148 ], "spans": [ { "bbox": [ 106, 136, 320, 148 ], "score": 1.0, "content": "the computation graph in the forward pass of an SSM.", "type": "text" } ], "index": 5 } ], "index": 2.5, "bbox_fs": [ 105, 82, 506, 148 ] }, { "type": "text", "bbox": [ 106, 153, 504, 175 ], "lines": [ { "bbox": [ 106, 152, 505, 165 ], "spans": [ { "bbox": [ 106, 152, 357, 165 ], "score": 1.0, "content": "Computation After the parameters have been transformed from", "type": "text" }, { "bbox": [ 357, 152, 461, 165 ], "score": 0.93, "content": "( \\Delta , A , B , C ) \\mapsto ( { \\overline { { A } } } , { \\overline { { B } } } , C )", "type": "inline_equation" }, { "bbox": [ 461, 152, 505, 165 ], "score": 1.0, "content": ", the model", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 163, 469, 176 ], "spans": [ { "bbox": [ 106, 163, 469, 176 ], "score": 1.0, "content": "can be computed in two ways, either as a linear recurrence (2) or a global convolution (3).", "type": "text" } ], "index": 7 } ], "index": 6.5, "bbox_fs": [ 106, 152, 505, 176 ] }, { "type": "text", "bbox": [ 108, 180, 503, 212 ], "lines": [ { "bbox": [ 106, 179, 505, 192 ], "spans": [ { "bbox": [ 106, 179, 505, 192 ], "score": 1.0, "content": "Commonly, the model uses the convolutional mode (3) for efficient parallelizable training (where", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 191, 505, 203 ], "spans": [ { "bbox": [ 106, 191, 505, 203 ], "score": 1.0, "content": "the whole input sequence is seen ahead of time), and switched into recurrent mode (2) for efficient", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 202, 402, 214 ], "spans": [ { "bbox": [ 106, 202, 402, 214 ], "score": 1.0, "content": "autoregressive inference (where the inputs are seen one timestep at a time).", "type": "text" } ], "index": 10 } ], "index": 9, "bbox_fs": [ 106, 179, 505, 214 ] }, { "type": "text", "bbox": [ 107, 216, 505, 272 ], "lines": [ { "bbox": [ 105, 216, 506, 230 ], "spans": [ { "bbox": [ 105, 216, 506, 230 ], "score": 1.0, "content": "Linear Time Invariance (LTI) An important property of equations (1) to (3) is that the model’s", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 227, 506, 241 ], "spans": [ { "bbox": [ 106, 227, 315, 241 ], "score": 1.0, "content": "dynamics are constant through time. In other words", "type": "text" }, { "bbox": [ 315, 228, 367, 240 ], "score": 0.91, "content": "( \\Delta , A , B , C )", "type": "inline_equation" }, { "bbox": [ 367, 227, 443, 241 ], "score": 1.0, "content": ", and consequently", "type": "text" }, { "bbox": [ 443, 227, 472, 240 ], "score": 0.9, "content": "( \\overline { { A } } , \\overline { { B } } )", "type": "inline_equation" }, { "bbox": [ 473, 227, 506, 241 ], "score": 1.0, "content": "as well,", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 237, 506, 253 ], "spans": [ { "bbox": [ 105, 237, 412, 253 ], "score": 1.0, "content": "are fixed for all time-steps. This property is called linear time invariance", "type": "text" }, { "bbox": [ 412, 240, 434, 250 ], "score": 0.4, "content": "( L T I )", "type": "inline_equation" }, { "bbox": [ 434, 237, 506, 253 ], "score": 1.0, "content": ", which is deeply", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 248, 506, 264 ], "spans": [ { "bbox": [ 105, 248, 506, 264 ], "score": 1.0, "content": "connected to recurrence and convolutions. Informally, we think of LTI SSMs as being equivalent to any", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 261, 506, 273 ], "spans": [ { "bbox": [ 106, 261, 506, 273 ], "score": 1.0, "content": "linear recurrence (2a) or convolution (3b), and use LTI as an umbrella term for these classes of models.", "type": "text" } ], "index": 15 } ], "index": 13, "bbox_fs": [ 105, 216, 506, 273 ] }, { "type": "text", "bbox": [ 107, 277, 505, 320 ], "lines": [ { "bbox": [ 105, 276, 506, 290 ], "spans": [ { "bbox": [ 105, 276, 506, 290 ], "score": 1.0, "content": "Thus far, all structured SSMs have been LTI (e.g. computed as convolutions) because of fundamental", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 287, 505, 299 ], "spans": [ { "bbox": [ 106, 287, 505, 299 ], "score": 1.0, "content": "efficiency constraints, discussed in Section 3.3. However, a core insight of this work is that LTI models", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 298, 504, 310 ], "spans": [ { "bbox": [ 106, 298, 504, 310 ], "score": 1.0, "content": "have fundamental limitations in modeling certain types of data, and our technical contributions involve", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 309, 397, 321 ], "spans": [ { "bbox": [ 105, 309, 397, 321 ], "score": 1.0, "content": "removing the LTI constraint while overcoming the efficiency bottlenecks.", "type": "text" } ], "index": 19 } ], "index": 17.5, "bbox_fs": [ 105, 276, 506, 321 ] }, { "type": "text", "bbox": [ 107, 325, 505, 357 ], "lines": [ { "bbox": [ 106, 324, 505, 337 ], "spans": [ { "bbox": [ 106, 324, 505, 337 ], "score": 1.0, "content": "Structure Finally, we note that structured SSMs are so named because computing them efficiently also", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 335, 507, 349 ], "spans": [ { "bbox": [ 105, 335, 244, 349 ], "score": 1.0, "content": "requires imposing structure on the", "type": "text" }, { "bbox": [ 245, 336, 254, 345 ], "score": 0.52, "content": "\\pmb { A }", "type": "inline_equation" }, { "bbox": [ 254, 335, 507, 349 ], "score": 1.0, "content": "matrix. The most popular form of structure is diagonal (Gupta,", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 345, 352, 358 ], "spans": [ { "bbox": [ 106, 345, 352, 358 ], "score": 1.0, "content": "2022; Gu et al., 2022b; Smith et al., 2023), which we also use.", "type": "text" } ], "index": 22 } ], "index": 21, "bbox_fs": [ 105, 324, 507, 358 ] }, { "type": "text", "bbox": [ 107, 360, 505, 416 ], "lines": [ { "bbox": [ 105, 359, 507, 375 ], "spans": [ { "bbox": [ 105, 359, 172, 375 ], "score": 1.0, "content": "In this case, the", "type": "text" }, { "bbox": [ 172, 360, 314, 373 ], "score": 0.32, "content": "\\pmb { A } \\in \\mathbb { R } ^ { N \\times N } , \\pmb { B } \\in \\mathbb { R } ^ { N \\times 1 } , \\pmb { C } \\in \\mathbb { R } ^ { 1 \\times N }", "type": "inline_equation" }, { "bbox": [ 315, 359, 455, 375 ], "score": 1.0, "content": "matrices can all be represented by", "type": "text" }, { "bbox": [ 456, 362, 466, 372 ], "score": 0.8, "content": "N", "type": "inline_equation" }, { "bbox": [ 466, 359, 507, 375 ], "score": 1.0, "content": "numbers.", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 372, 505, 385 ], "spans": [ { "bbox": [ 105, 372, 240, 385 ], "score": 1.0, "content": "To operate over an input sequence", "type": "text" }, { "bbox": [ 240, 375, 247, 383 ], "score": 0.79, "content": "x", "type": "inline_equation" }, { "bbox": [ 248, 372, 298, 385 ], "score": 1.0, "content": "of batch size", "type": "text" }, { "bbox": [ 299, 373, 308, 383 ], "score": 0.79, "content": "B", "type": "inline_equation" }, { "bbox": [ 308, 372, 351, 385 ], "score": 1.0, "content": "and length", "type": "text" }, { "bbox": [ 352, 373, 360, 383 ], "score": 0.8, "content": "L", "type": "inline_equation" }, { "bbox": [ 360, 372, 380, 385 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 380, 373, 389, 383 ], "score": 0.79, "content": "D", "type": "inline_equation" }, { "bbox": [ 390, 372, 505, 385 ], "score": 1.0, "content": "channels, the SSM is applied", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 383, 506, 397 ], "spans": [ { "bbox": [ 105, 383, 470, 397 ], "score": 1.0, "content": "independently to each channel. Note that in this case, the total hidden state has dimension", "type": "text" }, { "bbox": [ 470, 384, 489, 393 ], "score": 0.77, "content": "D N", "type": "inline_equation" }, { "bbox": [ 489, 383, 506, 397 ], "score": 1.0, "content": "per", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 394, 505, 407 ], "spans": [ { "bbox": [ 105, 394, 338, 407 ], "score": 1.0, "content": "input, and computing it over the sequence length requires", "type": "text" }, { "bbox": [ 339, 394, 387, 406 ], "score": 0.91, "content": "O ( B L D N )", "type": "inline_equation" }, { "bbox": [ 388, 394, 505, 407 ], "score": 1.0, "content": "time and memory; this is the", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 405, 385, 416 ], "spans": [ { "bbox": [ 105, 405, 385, 416 ], "score": 1.0, "content": "root of the fundamental efficiency bottleneck addressed in Section 3.3.", "type": "text" } ], "index": 27 } ], "index": 25, "bbox_fs": [ 105, 359, 507, 416 ] }, { "type": "text", "bbox": [ 107, 420, 505, 464 ], "lines": [ { "bbox": [ 106, 420, 505, 433 ], "spans": [ { "bbox": [ 106, 420, 505, 433 ], "score": 1.0, "content": "SSM Architectures SSMs are standalone sequence transformations that can be incorporated into", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 430, 507, 444 ], "spans": [ { "bbox": [ 105, 430, 507, 444 ], "score": 1.0, "content": "end-to-end neural network architectures. We discuss some of the most well-known SSM architectures,", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 443, 505, 455 ], "spans": [ { "bbox": [ 106, 443, 505, 455 ], "score": 1.0, "content": "many of which will also serve as our primary baselines. Other closely related SSMs and architectures", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 453, 358, 465 ], "spans": [ { "bbox": [ 106, 453, 358, 465 ], "score": 1.0, "content": "are discussed further in an extended related work (Appendix B).", "type": "text" } ], "index": 31 } ], "index": 29.5, "bbox_fs": [ 105, 420, 507, 465 ] }, { "type": "text", "bbox": [ 106, 468, 506, 567 ], "lines": [ { "bbox": [ 105, 466, 506, 480 ], "spans": [ { "bbox": [ 105, 466, 506, 480 ], "score": 1.0, "content": "• Linear attention (Katharopoulos et al., 2020) is an approximation of self-attention (Bahdanau et al.,", "type": "text" } ], "index": 32 }, { "bbox": [ 113, 478, 425, 491 ], "spans": [ { "bbox": [ 113, 478, 425, 491 ], "score": 1.0, "content": "2014) involving a recurrence which can be viewed as a degenerate linear SSM.", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 488, 505, 503 ], "spans": [ { "bbox": [ 106, 488, 505, 503 ], "score": 1.0, "content": "• H3 (Dao et al., 2023) generalized this recurrence to use S4; it can be viewed as an architecture", "type": "text" } ], "index": 34 }, { "bbox": [ 113, 500, 505, 513 ], "spans": [ { "bbox": [ 113, 500, 505, 513 ], "score": 1.0, "content": "with an SSM sandwiched by two gated connections (Figure 2). H3 also inserts a standard local", "type": "text" } ], "index": 35 }, { "bbox": [ 113, 511, 409, 523 ], "spans": [ { "bbox": [ 113, 511, 409, 523 ], "score": 1.0, "content": "convolution, which they frame as a shift-SSM, before the main SSM layer.", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 521, 506, 534 ], "spans": [ { "bbox": [ 106, 521, 506, 534 ], "score": 1.0, "content": "• Hyena (Poli et al., 2023) uses the same architecture but replaces the S4 layer with an MLP-", "type": "text" } ], "index": 37 }, { "bbox": [ 113, 533, 338, 545 ], "spans": [ { "bbox": [ 113, 533, 338, 545 ], "score": 1.0, "content": "parameterized global convolution (Romero et al., 2021).", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 542, 505, 556 ], "spans": [ { "bbox": [ 106, 542, 505, 556 ], "score": 1.0, "content": "• RetNet (Sun et al., 2023) adds an additional gate to the architecture and simplifies the SSM, allowing", "type": "text" } ], "index": 39 }, { "bbox": [ 114, 555, 503, 566 ], "spans": [ { "bbox": [ 114, 555, 503, 566 ], "score": 1.0, "content": "an alternative parallelizable computation path, using a variant of attention instead of convolutions.", "type": "text" } ], "index": 40 } ], "index": 36, "bbox_fs": [ 105, 466, 506, 566 ] }, { "type": "title", "bbox": [ 108, 578, 301, 591 ], "lines": [ { "bbox": [ 104, 576, 302, 593 ], "spans": [ { "bbox": [ 104, 576, 302, 593 ], "score": 1.0, "content": "3 SELECTIVE STATE SPACE MODELS", "type": "text" } ], "index": 41 } ], "index": 41 }, { "type": "text", "bbox": [ 106, 596, 505, 662 ], "lines": [ { "bbox": [ 106, 596, 505, 609 ], "spans": [ { "bbox": [ 106, 596, 505, 609 ], "score": 1.0, "content": "We motivate our selection mechanism using intuition from synthetic tasks (Section 3.1), then explain", "type": "text" } ], "index": 42 }, { "bbox": [ 104, 605, 507, 622 ], "spans": [ { "bbox": [ 104, 605, 507, 622 ], "score": 1.0, "content": "how to incorporate this mechanism into state space models (Section 3.2). The resulting time-varying", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 617, 506, 631 ], "spans": [ { "bbox": [ 105, 617, 506, 631 ], "score": 1.0, "content": "SSMs cannot use convolutions, presenting a technical challenge of how to compute them efficiently.", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 629, 506, 641 ], "spans": [ { "bbox": [ 105, 629, 506, 641 ], "score": 1.0, "content": "We overcome this with a hardware-aware algorithm that exploits the memory hierarchy on modern hard-", "type": "text" } ], "index": 45 }, { "bbox": [ 106, 640, 505, 651 ], "spans": [ { "bbox": [ 106, 640, 505, 651 ], "score": 1.0, "content": "ware (Section 3.3). We then describe a simple SSM architecture without attention or even MLP blocks", "type": "text" } ], "index": 46 }, { "bbox": [ 106, 650, 501, 663 ], "spans": [ { "bbox": [ 106, 650, 501, 663 ], "score": 1.0, "content": "(Section 3.4). Finally, we discuss some additional properties of selection mechanisms (Section 3.5).", "type": "text" } ], "index": 47 } ], "index": 44.5, "bbox_fs": [ 104, 596, 507, 663 ] }, { "type": "title", "bbox": [ 107, 672, 375, 684 ], "lines": [ { "bbox": [ 106, 672, 376, 685 ], "spans": [ { "bbox": [ 106, 672, 376, 685 ], "score": 1.0, "content": "3.1 MOTIVATION: SELECTION AS A MEANS OF COMPRESSION", "type": "text" } ], "index": 48 } ], "index": 48 }, { "type": "text", "bbox": [ 107, 689, 505, 732 ], "lines": [ { "bbox": [ 105, 687, 507, 702 ], "spans": [ { "bbox": [ 105, 687, 507, 702 ], "score": 1.0, "content": "We argue that a fundamental problem of sequence modeling is compressing context into a smaller state.", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 698, 507, 713 ], "spans": [ { "bbox": [ 105, 698, 507, 713 ], "score": 1.0, "content": "In fact, we can view the tradeoffs of popular sequence models from this point of view. For example,", "type": "text" } ], "index": 50 }, { "bbox": [ 106, 710, 505, 722 ], "spans": [ { "bbox": [ 106, 710, 505, 722 ], "score": 1.0, "content": "attention is both effective and inefficient because it explicitly does not compress context at all. This", "type": "text" } ], "index": 51 }, { "bbox": [ 105, 719, 506, 734 ], "spans": [ { "bbox": [ 105, 719, 506, 734 ], "score": 1.0, "content": "can be seen from the fact that autoregressive inference requires explicitly storing the entire context", "type": "text" } ], "index": 52 }, { "bbox": [ 104, 237, 506, 254 ], "spans": [ { "bbox": [ 104, 237, 506, 254 ], "score": 1.0, "content": "(i.e. the KV cache), which directly causes the slow linear-time inference and quadratic-time training", "type": "text", "cross_page": true } ], "index": 8 }, { "bbox": [ 105, 248, 506, 263 ], "spans": [ { "bbox": [ 105, 248, 506, 263 ], "score": 1.0, "content": "of Transformers. On the other hand, recurrent models are efficient because they have a finite state,", "type": "text", "cross_page": true } ], "index": 9 }, { "bbox": [ 106, 261, 505, 273 ], "spans": [ { "bbox": [ 106, 261, 505, 273 ], "score": 1.0, "content": "implying constant-time inference and linear-time training. However, their effectiveness is limited", "type": "text", "cross_page": true } ], "index": 10 }, { "bbox": [ 106, 271, 306, 284 ], "spans": [ { "bbox": [ 106, 271, 306, 284 ], "score": 1.0, "content": "by how well this state has compressed the context.", "type": "text", "cross_page": true } ], "index": 11 } ], "index": 50.5, "bbox_fs": [ 105, 687, 507, 734 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 107, 78, 504, 174 ], "blocks": [ { "type": "image_body", "bbox": [ 107, 78, 504, 174 ], "group_id": 0, "lines": [ { "bbox": [ 107, 78, 504, 174 ], "spans": [ { "bbox": [ 107, 78, 504, 174 ], "score": 0.967, "type": "image", "image_path": "b89480ae226b79d85c42d818542d10470c69bd199a250e4332dba21aa4f5cf97.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 107, 78, 504, 110.0 ], "spans": [], "index": 0 }, { "bbox": [ 107, 110.0, 504, 142.0 ], "spans": [], "index": 1 }, { "bbox": [ 107, 142.0, 504, 174.0 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 106, 182, 505, 232 ], "group_id": 0, "lines": [ { "bbox": [ 106, 182, 505, 193 ], "spans": [ { "bbox": [ 106, 182, 505, 193 ], "score": 1.0, "content": "Figure 1: (Left) The standard version of the Copying task involves constant spacing between input and output", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 191, 505, 203 ], "spans": [ { "bbox": [ 105, 191, 505, 203 ], "score": 1.0, "content": "elements and is easily solved by time-invariant models such as linear recurrences and global convolutions. (Right", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 201, 505, 213 ], "spans": [ { "bbox": [ 106, 201, 505, 213 ], "score": 1.0, "content": "Top) The Selective Copying task has random spacing in between inputs and requires time-varying models that can", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 211, 506, 223 ], "spans": [ { "bbox": [ 105, 211, 506, 223 ], "score": 1.0, "content": "selectively remember or ignore inputs depending on their content. (Right Bottom) The Induction Heads task is an", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 221, 478, 234 ], "spans": [ { "bbox": [ 105, 221, 478, 234 ], "score": 1.0, "content": "example of associative recall that requires retrieving an answer based on context, a key ability for LLMs.", "type": "text" } ], "index": 7 } ], "index": 5 } ], "index": 3.0 }, { "type": "text", "bbox": [ 106, 239, 506, 282 ], "lines": [ { "bbox": [ 104, 237, 506, 254 ], "spans": [ { "bbox": [ 104, 237, 506, 254 ], "score": 1.0, "content": "(i.e. the KV cache), which directly causes the slow linear-time inference and quadratic-time training", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 248, 506, 263 ], "spans": [ { "bbox": [ 105, 248, 506, 263 ], "score": 1.0, "content": "of Transformers. On the other hand, recurrent models are efficient because they have a finite state,", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 261, 505, 273 ], "spans": [ { "bbox": [ 106, 261, 505, 273 ], "score": 1.0, "content": "implying constant-time inference and linear-time training. However, their effectiveness is limited", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 271, 306, 284 ], "spans": [ { "bbox": [ 106, 271, 306, 284 ], "score": 1.0, "content": "by how well this state has compressed the context.", "type": "text" } ], "index": 11 } ], "index": 9.5 }, { "type": "text", "bbox": [ 105, 287, 472, 299 ], "lines": [ { "bbox": [ 105, 286, 474, 300 ], "spans": [ { "bbox": [ 105, 286, 474, 300 ], "score": 1.0, "content": "To understand this principle, we focus on two running examples of synthetic tasks (Figure 1).", "type": "text" } ], "index": 12 } ], "index": 12 }, { "type": "text", "bbox": [ 107, 303, 505, 370 ], "lines": [ { "bbox": [ 105, 301, 505, 316 ], "spans": [ { "bbox": [ 105, 301, 505, 316 ], "score": 1.0, "content": "• The Selective Copying task modifies the popular Copying task (Arjovsky et al., 2016) by varying", "type": "text" } ], "index": 13 }, { "bbox": [ 114, 313, 505, 326 ], "spans": [ { "bbox": [ 114, 313, 505, 326 ], "score": 1.0, "content": "the position of the tokens to memorize. It requires content-aware reasoning to be able to memorize", "type": "text" } ], "index": 14 }, { "bbox": [ 114, 324, 388, 336 ], "spans": [ { "bbox": [ 114, 324, 388, 336 ], "score": 1.0, "content": "the relevant tokens (colored) and filter out the irrelevant ones (white).", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 336, 506, 350 ], "spans": [ { "bbox": [ 105, 336, 506, 350 ], "score": 1.0, "content": "• The Induction Heads task is a well-known mechanism hypothesized to explain the majority of", "type": "text" } ], "index": 16 }, { "bbox": [ 113, 347, 506, 361 ], "spans": [ { "bbox": [ 113, 347, 506, 361 ], "score": 1.0, "content": "in-context learning abilities of LLMs (Olsson et al., 2022). It requires context-aware reasoning to", "type": "text" } ], "index": 17 }, { "bbox": [ 114, 358, 413, 371 ], "spans": [ { "bbox": [ 114, 358, 413, 371 ], "score": 1.0, "content": "know when to produce the correct output in the appropriate context (black).", "type": "text" } ], "index": 18 } ], "index": 15.5 }, { "type": "text", "bbox": [ 106, 374, 505, 451 ], "lines": [ { "bbox": [ 106, 374, 505, 387 ], "spans": [ { "bbox": [ 106, 374, 505, 387 ], "score": 1.0, "content": "These tasks reveal the failure mode of LTI models. From the recurrent view, their constant dynamics", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 385, 506, 399 ], "spans": [ { "bbox": [ 105, 385, 141, 399 ], "score": 1.0, "content": "(e.g. the", "type": "text" }, { "bbox": [ 142, 385, 171, 398 ], "score": 0.91, "content": "( \\overline { { A } } , \\overline { { B } } )", "type": "inline_equation" }, { "bbox": [ 171, 385, 506, 399 ], "score": 1.0, "content": "transitions in (2)) cannot let them select the correct information from their context,", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 397, 506, 410 ], "spans": [ { "bbox": [ 105, 397, 506, 410 ], "score": 1.0, "content": "or affect the hidden state passed along the sequence an in input-dependent way. From the convolutional", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 408, 505, 421 ], "spans": [ { "bbox": [ 106, 408, 505, 421 ], "score": 1.0, "content": "view, it is known that global convolutions can solve the vanilla Copying task (Romero et al., 2021)", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 417, 506, 433 ], "spans": [ { "bbox": [ 105, 417, 506, 433 ], "score": 1.0, "content": "because it only requires time-awareness, but that they have difficulty with the Selective Copying", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 429, 505, 442 ], "spans": [ { "bbox": [ 106, 429, 505, 442 ], "score": 1.0, "content": "task because of lack of content-awareness (Figure 1). More concretely, the spacing between", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 441, 427, 452 ], "spans": [ { "bbox": [ 106, 441, 427, 452 ], "score": 1.0, "content": "inputs-to-outputs is varying and cannot be modeled by static convolution kernels.", "type": "text" } ], "index": 25 } ], "index": 22 }, { "type": "text", "bbox": [ 107, 456, 505, 521 ], "lines": [ { "bbox": [ 106, 456, 505, 468 ], "spans": [ { "bbox": [ 106, 456, 505, 468 ], "score": 1.0, "content": "In summary, the efficiency vs. effectiveness tradeoff of sequence models is characterized by how well", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 466, 505, 478 ], "spans": [ { "bbox": [ 105, 466, 505, 478 ], "score": 1.0, "content": "they compress their state: efficient models must have a small state, while effective models must have", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 478, 506, 489 ], "spans": [ { "bbox": [ 105, 478, 506, 489 ], "score": 1.0, "content": "a state that contains all necessary information from the context. In turn, we propose that a fundamental", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 488, 506, 500 ], "spans": [ { "bbox": [ 105, 488, 506, 500 ], "score": 1.0, "content": "principle for building sequence models is selectivity: or the context-aware ability to focus on or", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 498, 506, 511 ], "spans": [ { "bbox": [ 105, 498, 506, 511 ], "score": 1.0, "content": "filter out inputs into a sequential state. In particular, a selection mechanism controls how information", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 510, 468, 522 ], "spans": [ { "bbox": [ 105, 510, 468, 522 ], "score": 1.0, "content": "propagates or interacts along the sequence dimension (see Section 3.5 for more discussion).", "type": "text" } ], "index": 31 } ], "index": 28.5 }, { "type": "title", "bbox": [ 108, 529, 285, 541 ], "lines": [ { "bbox": [ 106, 529, 286, 542 ], "spans": [ { "bbox": [ 106, 529, 286, 542 ], "score": 1.0, "content": "3.2 IMPROVING SSMS WITH SELECTION", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "text", "bbox": [ 107, 545, 505, 578 ], "lines": [ { "bbox": [ 106, 545, 505, 558 ], "spans": [ { "bbox": [ 106, 545, 505, 558 ], "score": 1.0, "content": "One method of incorporating a selection mechanism into models is by letting their parameters that", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 556, 506, 569 ], "spans": [ { "bbox": [ 105, 556, 506, 569 ], "score": 1.0, "content": "affect interactions along the sequence (e.g. the recurrent dynamics of an RNN or the convolution kernel", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 567, 231, 579 ], "spans": [ { "bbox": [ 106, 567, 231, 579 ], "score": 1.0, "content": "of a CNN) be input-dependent.", "type": "text" } ], "index": 35 } ], "index": 34 }, { "type": "text", "bbox": [ 107, 583, 505, 637 ], "lines": [ { "bbox": [ 105, 582, 505, 596 ], "spans": [ { "bbox": [ 105, 582, 505, 596 ], "score": 1.0, "content": "Algorithms 1 and 2 illustrates the main selection mechanism that we use. The main difference is simply", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 594, 506, 606 ], "spans": [ { "bbox": [ 105, 594, 213, 606 ], "score": 1.0, "content": "making several parameters", "type": "text" }, { "bbox": [ 213, 594, 246, 605 ], "score": 0.66, "content": "^ { \\Delta , B , C }", "type": "inline_equation" }, { "bbox": [ 246, 594, 506, 606 ], "score": 1.0, "content": "functions of the input, along with the associated changes to tensor", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 604, 505, 617 ], "spans": [ { "bbox": [ 105, 604, 505, 617 ], "score": 1.0, "content": "shapes throughout. In particular, we highlight that these parameters now have a length dimension", "type": "text" } ], "index": 38 }, { "bbox": [ 107, 613, 506, 629 ], "spans": [ { "bbox": [ 107, 615, 114, 625 ], "score": 0.74, "content": "L", "type": "inline_equation" }, { "bbox": [ 115, 613, 506, 629 ], "score": 1.0, "content": ", meaning that the model has changed from time-invariant to time-varying. This loses the equivalence", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 626, 383, 639 ], "spans": [ { "bbox": [ 105, 626, 383, 639 ], "score": 1.0, "content": "to convolutions (3) with implications for its efficiency, discussed next.", "type": "text" } ], "index": 40 } ], "index": 38 }, { "type": "text", "bbox": [ 107, 641, 505, 674 ], "lines": [ { "bbox": [ 105, 641, 506, 654 ], "spans": [ { "bbox": [ 105, 641, 201, 654 ], "score": 1.0, "content": "We specifically choose", "type": "text" }, { "bbox": [ 202, 641, 287, 653 ], "score": 0.82, "content": "s _ { B } ( x ) = \\mathsf { L i n e a r } _ { N } ( x )", "type": "inline_equation" }, { "bbox": [ 287, 641, 291, 654 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 291, 641, 375, 654 ], "score": 0.81, "content": "s _ { C } ( x ) = \\mathsf { L i n e a r } _ { N } ( x )", "type": "inline_equation" }, { "bbox": [ 376, 641, 379, 654 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 380, 641, 502, 654 ], "score": 0.8, "content": "s _ { \\Delta } ( x ) = \\mathsf { L i n e a r } _ { D } ( \\mathsf { L i n e a r } _ { 1 } ( x ) )", "type": "inline_equation" }, { "bbox": [ 502, 641, 506, 654 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 653, 506, 664 ], "spans": [ { "bbox": [ 106, 653, 123, 664 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 123, 653, 145, 663 ], "score": 0.72, "content": "\\tau _ { \\Delta } =", "type": "inline_equation" }, { "bbox": [ 146, 653, 232, 664 ], "score": 1.0, "content": "softplus, where Linea", "type": "text" }, { "bbox": [ 233, 655, 238, 663 ], "score": 0.35, "content": "\\dot { } d", "type": "inline_equation" }, { "bbox": [ 239, 653, 408, 664 ], "score": 1.0, "content": "is a parameterized projection to dimension", "type": "text" }, { "bbox": [ 408, 654, 415, 662 ], "score": 0.75, "content": "d", "type": "inline_equation" }, { "bbox": [ 415, 653, 474, 664 ], "score": 1.0, "content": ". The choice of", "type": "text" }, { "bbox": [ 475, 654, 487, 663 ], "score": 0.84, "content": "s _ { \\Delta }", "type": "inline_equation" }, { "bbox": [ 487, 653, 506, 664 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 663, 418, 675 ], "spans": [ { "bbox": [ 106, 664, 119, 674 ], "score": 0.85, "content": "\\tau _ { \\Delta }", "type": "inline_equation" }, { "bbox": [ 119, 663, 418, 675 ], "score": 1.0, "content": "is due to a connection to RNN gating mechanisms explained in Section 3.5.", "type": "text" } ], "index": 43 } ], "index": 42 }, { "type": "title", "bbox": [ 107, 683, 347, 694 ], "lines": [ { "bbox": [ 106, 683, 348, 696 ], "spans": [ { "bbox": [ 106, 683, 348, 696 ], "score": 1.0, "content": "3.3 EFFICIENT IMPLEMENTATION OF SELECTIVE SSMS", "type": "text" } ], "index": 44 } ], "index": 44 }, { "type": "text", "bbox": [ 108, 699, 505, 732 ], "lines": [ { "bbox": [ 105, 698, 506, 712 ], "spans": [ { "bbox": [ 105, 698, 506, 712 ], "score": 1.0, "content": "Hardware-friendly architectures such as convolutions (Krizhevsky et al., 2012) and Transform-", "type": "text" } ], "index": 45 }, { "bbox": [ 106, 710, 506, 722 ], "spans": [ { "bbox": [ 106, 710, 506, 722 ], "score": 1.0, "content": "ers (Vaswani et al., 2017) enjoy widespread application. Here we aim to make selective SSMs efficient", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 720, 505, 732 ], "spans": [ { "bbox": [ 105, 720, 505, 732 ], "score": 1.0, "content": "on modern hardware (GPU) as well. The selection mechanism is quite natural, and earlier works", "type": "text" } ], "index": 47 } ], "index": 46 } ], "page_idx": 3, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 304, 37 ], "lines": [ { "bbox": [ 106, 25, 305, 39 ], "spans": [ { "bbox": [ 106, 25, 305, 39 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 752, 308, 759 ], "lines": [] } ], "para_blocks": [ { "type": "image", "bbox": [ 107, 78, 504, 174 ], "blocks": [ { "type": "image_body", "bbox": [ 107, 78, 504, 174 ], "group_id": 0, "lines": [ { "bbox": [ 107, 78, 504, 174 ], "spans": [ { "bbox": [ 107, 78, 504, 174 ], "score": 0.967, "type": "image", "image_path": "b89480ae226b79d85c42d818542d10470c69bd199a250e4332dba21aa4f5cf97.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 107, 78, 504, 110.0 ], "spans": [], "index": 0 }, { "bbox": [ 107, 110.0, 504, 142.0 ], "spans": [], "index": 1 }, { "bbox": [ 107, 142.0, 504, 174.0 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 106, 182, 505, 232 ], "group_id": 0, "lines": [ { "bbox": [ 106, 182, 505, 193 ], "spans": [ { "bbox": [ 106, 182, 505, 193 ], "score": 1.0, "content": "Figure 1: (Left) The standard version of the Copying task involves constant spacing between input and output", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 191, 505, 203 ], "spans": [ { "bbox": [ 105, 191, 505, 203 ], "score": 1.0, "content": "elements and is easily solved by time-invariant models such as linear recurrences and global convolutions. (Right", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 201, 505, 213 ], "spans": [ { "bbox": [ 106, 201, 505, 213 ], "score": 1.0, "content": "Top) The Selective Copying task has random spacing in between inputs and requires time-varying models that can", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 211, 506, 223 ], "spans": [ { "bbox": [ 105, 211, 506, 223 ], "score": 1.0, "content": "selectively remember or ignore inputs depending on their content. (Right Bottom) The Induction Heads task is an", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 221, 478, 234 ], "spans": [ { "bbox": [ 105, 221, 478, 234 ], "score": 1.0, "content": "example of associative recall that requires retrieving an answer based on context, a key ability for LLMs.", "type": "text" } ], "index": 7 } ], "index": 5 } ], "index": 3.0 }, { "type": "text", "bbox": [ 106, 239, 506, 282 ], "lines": [], "index": 9.5, "bbox_fs": [ 104, 237, 506, 284 ], "lines_deleted": true }, { "type": "text", "bbox": [ 105, 287, 472, 299 ], "lines": [ { "bbox": [ 105, 286, 474, 300 ], "spans": [ { "bbox": [ 105, 286, 474, 300 ], "score": 1.0, "content": "To understand this principle, we focus on two running examples of synthetic tasks (Figure 1).", "type": "text" } ], "index": 12 } ], "index": 12, "bbox_fs": [ 105, 286, 474, 300 ] }, { "type": "text", "bbox": [ 107, 303, 505, 370 ], "lines": [ { "bbox": [ 105, 301, 505, 316 ], "spans": [ { "bbox": [ 105, 301, 505, 316 ], "score": 1.0, "content": "• The Selective Copying task modifies the popular Copying task (Arjovsky et al., 2016) by varying", "type": "text" } ], "index": 13 }, { "bbox": [ 114, 313, 505, 326 ], "spans": [ { "bbox": [ 114, 313, 505, 326 ], "score": 1.0, "content": "the position of the tokens to memorize. It requires content-aware reasoning to be able to memorize", "type": "text" } ], "index": 14 }, { "bbox": [ 114, 324, 388, 336 ], "spans": [ { "bbox": [ 114, 324, 388, 336 ], "score": 1.0, "content": "the relevant tokens (colored) and filter out the irrelevant ones (white).", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 336, 506, 350 ], "spans": [ { "bbox": [ 105, 336, 506, 350 ], "score": 1.0, "content": "• The Induction Heads task is a well-known mechanism hypothesized to explain the majority of", "type": "text" } ], "index": 16 }, { "bbox": [ 113, 347, 506, 361 ], "spans": [ { "bbox": [ 113, 347, 506, 361 ], "score": 1.0, "content": "in-context learning abilities of LLMs (Olsson et al., 2022). It requires context-aware reasoning to", "type": "text" } ], "index": 17 }, { "bbox": [ 114, 358, 413, 371 ], "spans": [ { "bbox": [ 114, 358, 413, 371 ], "score": 1.0, "content": "know when to produce the correct output in the appropriate context (black).", "type": "text" } ], "index": 18 } ], "index": 15.5, "bbox_fs": [ 105, 301, 506, 371 ] }, { "type": "text", "bbox": [ 106, 374, 505, 451 ], "lines": [ { "bbox": [ 106, 374, 505, 387 ], "spans": [ { "bbox": [ 106, 374, 505, 387 ], "score": 1.0, "content": "These tasks reveal the failure mode of LTI models. From the recurrent view, their constant dynamics", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 385, 506, 399 ], "spans": [ { "bbox": [ 105, 385, 141, 399 ], "score": 1.0, "content": "(e.g. the", "type": "text" }, { "bbox": [ 142, 385, 171, 398 ], "score": 0.91, "content": "( \\overline { { A } } , \\overline { { B } } )", "type": "inline_equation" }, { "bbox": [ 171, 385, 506, 399 ], "score": 1.0, "content": "transitions in (2)) cannot let them select the correct information from their context,", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 397, 506, 410 ], "spans": [ { "bbox": [ 105, 397, 506, 410 ], "score": 1.0, "content": "or affect the hidden state passed along the sequence an in input-dependent way. From the convolutional", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 408, 505, 421 ], "spans": [ { "bbox": [ 106, 408, 505, 421 ], "score": 1.0, "content": "view, it is known that global convolutions can solve the vanilla Copying task (Romero et al., 2021)", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 417, 506, 433 ], "spans": [ { "bbox": [ 105, 417, 506, 433 ], "score": 1.0, "content": "because it only requires time-awareness, but that they have difficulty with the Selective Copying", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 429, 505, 442 ], "spans": [ { "bbox": [ 106, 429, 505, 442 ], "score": 1.0, "content": "task because of lack of content-awareness (Figure 1). More concretely, the spacing between", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 441, 427, 452 ], "spans": [ { "bbox": [ 106, 441, 427, 452 ], "score": 1.0, "content": "inputs-to-outputs is varying and cannot be modeled by static convolution kernels.", "type": "text" } ], "index": 25 } ], "index": 22, "bbox_fs": [ 105, 374, 506, 452 ] }, { "type": "text", "bbox": [ 107, 456, 505, 521 ], "lines": [ { "bbox": [ 106, 456, 505, 468 ], "spans": [ { "bbox": [ 106, 456, 505, 468 ], "score": 1.0, "content": "In summary, the efficiency vs. effectiveness tradeoff of sequence models is characterized by how well", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 466, 505, 478 ], "spans": [ { "bbox": [ 105, 466, 505, 478 ], "score": 1.0, "content": "they compress their state: efficient models must have a small state, while effective models must have", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 478, 506, 489 ], "spans": [ { "bbox": [ 105, 478, 506, 489 ], "score": 1.0, "content": "a state that contains all necessary information from the context. In turn, we propose that a fundamental", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 488, 506, 500 ], "spans": [ { "bbox": [ 105, 488, 506, 500 ], "score": 1.0, "content": "principle for building sequence models is selectivity: or the context-aware ability to focus on or", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 498, 506, 511 ], "spans": [ { "bbox": [ 105, 498, 506, 511 ], "score": 1.0, "content": "filter out inputs into a sequential state. In particular, a selection mechanism controls how information", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 510, 468, 522 ], "spans": [ { "bbox": [ 105, 510, 468, 522 ], "score": 1.0, "content": "propagates or interacts along the sequence dimension (see Section 3.5 for more discussion).", "type": "text" } ], "index": 31 } ], "index": 28.5, "bbox_fs": [ 105, 456, 506, 522 ] }, { "type": "title", "bbox": [ 108, 529, 285, 541 ], "lines": [ { "bbox": [ 106, 529, 286, 542 ], "spans": [ { "bbox": [ 106, 529, 286, 542 ], "score": 1.0, "content": "3.2 IMPROVING SSMS WITH SELECTION", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "text", "bbox": [ 107, 545, 505, 578 ], "lines": [ { "bbox": [ 106, 545, 505, 558 ], "spans": [ { "bbox": [ 106, 545, 505, 558 ], "score": 1.0, "content": "One method of incorporating a selection mechanism into models is by letting their parameters that", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 556, 506, 569 ], "spans": [ { "bbox": [ 105, 556, 506, 569 ], "score": 1.0, "content": "affect interactions along the sequence (e.g. the recurrent dynamics of an RNN or the convolution kernel", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 567, 231, 579 ], "spans": [ { "bbox": [ 106, 567, 231, 579 ], "score": 1.0, "content": "of a CNN) be input-dependent.", "type": "text" } ], "index": 35 } ], "index": 34, "bbox_fs": [ 105, 545, 506, 579 ] }, { "type": "text", "bbox": [ 107, 583, 505, 637 ], "lines": [ { "bbox": [ 105, 582, 505, 596 ], "spans": [ { "bbox": [ 105, 582, 505, 596 ], "score": 1.0, "content": "Algorithms 1 and 2 illustrates the main selection mechanism that we use. The main difference is simply", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 594, 506, 606 ], "spans": [ { "bbox": [ 105, 594, 213, 606 ], "score": 1.0, "content": "making several parameters", "type": "text" }, { "bbox": [ 213, 594, 246, 605 ], "score": 0.66, "content": "^ { \\Delta , B , C }", "type": "inline_equation" }, { "bbox": [ 246, 594, 506, 606 ], "score": 1.0, "content": "functions of the input, along with the associated changes to tensor", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 604, 505, 617 ], "spans": [ { "bbox": [ 105, 604, 505, 617 ], "score": 1.0, "content": "shapes throughout. In particular, we highlight that these parameters now have a length dimension", "type": "text" } ], "index": 38 }, { "bbox": [ 107, 613, 506, 629 ], "spans": [ { "bbox": [ 107, 615, 114, 625 ], "score": 0.74, "content": "L", "type": "inline_equation" }, { "bbox": [ 115, 613, 506, 629 ], "score": 1.0, "content": ", meaning that the model has changed from time-invariant to time-varying. This loses the equivalence", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 626, 383, 639 ], "spans": [ { "bbox": [ 105, 626, 383, 639 ], "score": 1.0, "content": "to convolutions (3) with implications for its efficiency, discussed next.", "type": "text" } ], "index": 40 } ], "index": 38, "bbox_fs": [ 105, 582, 506, 639 ] }, { "type": "text", "bbox": [ 107, 641, 505, 674 ], "lines": [ { "bbox": [ 105, 641, 506, 654 ], "spans": [ { "bbox": [ 105, 641, 201, 654 ], "score": 1.0, "content": "We specifically choose", "type": "text" }, { "bbox": [ 202, 641, 287, 653 ], "score": 0.82, "content": "s _ { B } ( x ) = \\mathsf { L i n e a r } _ { N } ( x )", "type": "inline_equation" }, { "bbox": [ 287, 641, 291, 654 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 291, 641, 375, 654 ], "score": 0.81, "content": "s _ { C } ( x ) = \\mathsf { L i n e a r } _ { N } ( x )", "type": "inline_equation" }, { "bbox": [ 376, 641, 379, 654 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 380, 641, 502, 654 ], "score": 0.8, "content": "s _ { \\Delta } ( x ) = \\mathsf { L i n e a r } _ { D } ( \\mathsf { L i n e a r } _ { 1 } ( x ) )", "type": "inline_equation" }, { "bbox": [ 502, 641, 506, 654 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 653, 506, 664 ], "spans": [ { "bbox": [ 106, 653, 123, 664 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 123, 653, 145, 663 ], "score": 0.72, "content": "\\tau _ { \\Delta } =", "type": "inline_equation" }, { "bbox": [ 146, 653, 232, 664 ], "score": 1.0, "content": "softplus, where Linea", "type": "text" }, { "bbox": [ 233, 655, 238, 663 ], "score": 0.35, "content": "\\dot { } d", "type": "inline_equation" }, { "bbox": [ 239, 653, 408, 664 ], "score": 1.0, "content": "is a parameterized projection to dimension", "type": "text" }, { "bbox": [ 408, 654, 415, 662 ], "score": 0.75, "content": "d", "type": "inline_equation" }, { "bbox": [ 415, 653, 474, 664 ], "score": 1.0, "content": ". The choice of", "type": "text" }, { "bbox": [ 475, 654, 487, 663 ], "score": 0.84, "content": "s _ { \\Delta }", "type": "inline_equation" }, { "bbox": [ 487, 653, 506, 664 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 663, 418, 675 ], "spans": [ { "bbox": [ 106, 664, 119, 674 ], "score": 0.85, "content": "\\tau _ { \\Delta }", "type": "inline_equation" }, { "bbox": [ 119, 663, 418, 675 ], "score": 1.0, "content": "is due to a connection to RNN gating mechanisms explained in Section 3.5.", "type": "text" } ], "index": 43 } ], "index": 42, "bbox_fs": [ 105, 641, 506, 675 ] }, { "type": "title", "bbox": [ 107, 683, 347, 694 ], "lines": [ { "bbox": [ 106, 683, 348, 696 ], "spans": [ { "bbox": [ 106, 683, 348, 696 ], "score": 1.0, "content": "3.3 EFFICIENT IMPLEMENTATION OF SELECTIVE SSMS", "type": "text" } ], "index": 44 } ], "index": 44 }, { "type": "text", "bbox": [ 108, 699, 505, 732 ], "lines": [ { "bbox": [ 105, 698, 506, 712 ], "spans": [ { "bbox": [ 105, 698, 506, 712 ], "score": 1.0, "content": "Hardware-friendly architectures such as convolutions (Krizhevsky et al., 2012) and Transform-", "type": "text" } ], "index": 45 }, { "bbox": [ 106, 710, 506, 722 ], "spans": [ { "bbox": [ 106, 710, 506, 722 ], "score": 1.0, "content": "ers (Vaswani et al., 2017) enjoy widespread application. Here we aim to make selective SSMs efficient", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 720, 505, 732 ], "spans": [ { "bbox": [ 105, 720, 505, 732 ], "score": 1.0, "content": "on modern hardware (GPU) as well. The selection mechanism is quite natural, and earlier works", "type": "text" } ], "index": 47 }, { "bbox": [ 106, 231, 504, 243 ], "spans": [ { "bbox": [ 106, 231, 381, 243 ], "score": 1.0, "content": "attempted to incorporate special cases of selection, such as letting", "type": "text", "cross_page": true }, { "bbox": [ 381, 231, 391, 241 ], "score": 0.78, "content": "\\Delta", "type": "inline_equation", "cross_page": true }, { "bbox": [ 391, 231, 504, 243 ], "score": 1.0, "content": "vary over time in recurrent", "type": "text", "cross_page": true } ], "index": 3 }, { "bbox": [ 106, 240, 504, 254 ], "spans": [ { "bbox": [ 106, 240, 504, 254 ], "score": 1.0, "content": "SSMs (Gu et al., 2020a). However, this was computationally difficult, which was why S4 and all", "type": "text", "cross_page": true } ], "index": 4 }, { "bbox": [ 106, 252, 487, 265 ], "spans": [ { "bbox": [ 106, 252, 487, 265 ], "score": 1.0, "content": "derivatives used LTI (non-selective) models, most commonly in the form of global convolutions.", "type": "text", "cross_page": true } ], "index": 5 } ], "index": 46, "bbox_fs": [ 105, 698, 506, 732 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 106, 91, 501, 223 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 91, 501, 223 ], "group_id": 0, "lines": [ { "bbox": [ 106, 91, 501, 223 ], "spans": [ { "bbox": [ 106, 91, 501, 223 ], "score": 0.956, "html": "
Algorithm 1 SSM (S4)Algorithm 2 SSM + Selection (S6)
Input: x : (B,L,D)Input: x : (B,L,D)
Output: y:(B,L,D)Output: y: (B,L,D)
1:A:(D,N)←Parameter1:A:(D,N)←Parameter
Represents structured N × N matrix>Represents structured N × N matrix
2:B:(D,N)←Parameter2: B:(B,L,N)← SB(𝑥)
3: C:(D,N)←Parameter 3: C:(B,L,N)←sc(x)
4:△:(D)←T△(Parameter)4: △:(B,L,D)←T△(Parameter+s△(x))
5: A,B:(D,N)←discretize(△,A,B) 5: A,B:(B,L,D,N)←discretize(△,A,B)
6: y←SSM(A,B,C)(x)6: y←SSM(A,B,C)(x)
> Time-invariant: recurrence or convolution 7: return yTime-varying: recurrence (scan) only 7: return y
", "type": "table", "image_path": "1cc809589f51a34f27ff26fcb8db77db7b7e9bf7ddd2741fe757d13ad118b0b3.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 106, 91, 501, 135.0 ], "spans": [], "index": 0 }, { "bbox": [ 106, 135.0, 501, 179.0 ], "spans": [], "index": 1 }, { "bbox": [ 106, 179.0, 501, 223.0 ], "spans": [], "index": 2 } ] } ], "index": 1 }, { "type": "text", "bbox": [ 108, 231, 502, 264 ], "lines": [ { "bbox": [ 106, 231, 504, 243 ], "spans": [ { "bbox": [ 106, 231, 381, 243 ], "score": 1.0, "content": "attempted to incorporate special cases of selection, such as letting", "type": "text" }, { "bbox": [ 381, 231, 391, 241 ], "score": 0.78, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 391, 231, 504, 243 ], "score": 1.0, "content": "vary over time in recurrent", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 240, 504, 254 ], "spans": [ { "bbox": [ 106, 240, 504, 254 ], "score": 1.0, "content": "SSMs (Gu et al., 2020a). However, this was computationally difficult, which was why S4 and all", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 252, 487, 265 ], "spans": [ { "bbox": [ 106, 252, 487, 265 ], "score": 1.0, "content": "derivatives used LTI (non-selective) models, most commonly in the form of global convolutions.", "type": "text" } ], "index": 5 } ], "index": 4 }, { "type": "text", "bbox": [ 107, 268, 504, 301 ], "lines": [ { "bbox": [ 105, 267, 506, 281 ], "spans": [ { "bbox": [ 105, 267, 506, 281 ], "score": 1.0, "content": "The selection mechanism is designed to overcome the limitations of LTI models; at the same time,", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 279, 505, 291 ], "spans": [ { "bbox": [ 106, 279, 505, 291 ], "score": 1.0, "content": "we therefore need to revisit the computation problem of SSMs. We address this with three classical", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 290, 476, 302 ], "spans": [ { "bbox": [ 106, 290, 476, 302 ], "score": 1.0, "content": "techniques: kernel fusion, parallel scan, and recomputation. We make two main observations:", "type": "text" } ], "index": 8 } ], "index": 7 }, { "type": "text", "bbox": [ 106, 305, 505, 363 ], "lines": [ { "bbox": [ 105, 304, 505, 318 ], "spans": [ { "bbox": [ 105, 304, 271, 318 ], "score": 1.0, "content": "• The naive recurrent computation uses", "type": "text" }, { "bbox": [ 271, 305, 320, 317 ], "score": 0.91, "content": "O ( B L D N )", "type": "inline_equation" }, { "bbox": [ 320, 304, 505, 318 ], "score": 1.0, "content": "FLOPs while the convolutional computation", "type": "text" } ], "index": 9 }, { "bbox": [ 114, 316, 505, 329 ], "spans": [ { "bbox": [ 114, 316, 134, 329 ], "score": 1.0, "content": "uses", "type": "text" }, { "bbox": [ 135, 316, 201, 328 ], "score": 0.86, "content": "O ( B L D \\log ( L ) )", "type": "inline_equation" }, { "bbox": [ 201, 316, 505, 329 ], "score": 1.0, "content": "FLOPs, and the former has a lower constant factor. Thus for long sequences", "type": "text" } ], "index": 10 }, { "bbox": [ 114, 327, 460, 339 ], "spans": [ { "bbox": [ 114, 327, 249, 339 ], "score": 1.0, "content": "and not-too-large state dimension", "type": "text" }, { "bbox": [ 249, 327, 259, 337 ], "score": 0.78, "content": "N", "type": "inline_equation" }, { "bbox": [ 259, 327, 460, 339 ], "score": 1.0, "content": ", the recurrent mode can actually use fewer FLOPs.", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 339, 505, 353 ], "spans": [ { "bbox": [ 106, 339, 505, 353 ], "score": 1.0, "content": "• The two challenges are the sequential nature of recurrence, and the large memory usage. To address", "type": "text" } ], "index": 12 }, { "bbox": [ 114, 351, 503, 365 ], "spans": [ { "bbox": [ 114, 351, 496, 365 ], "score": 1.0, "content": "the latter, just like the convolutional mode, we can attempt to not actually materialize the full state", "type": "text" }, { "bbox": [ 496, 352, 503, 361 ], "score": 0.74, "content": "h", "type": "inline_equation" } ], "index": 13 } ], "index": 11 }, { "type": "text", "bbox": [ 107, 367, 505, 422 ], "lines": [ { "bbox": [ 106, 367, 504, 379 ], "spans": [ { "bbox": [ 106, 367, 496, 379 ], "score": 1.0, "content": "The main idea is to leverage properties of modern accelerators (GPUs) to materialize the state", "type": "text" }, { "bbox": [ 497, 367, 504, 377 ], "score": 0.71, "content": "h", "type": "inline_equation" } ], "index": 14 }, { "bbox": [ 106, 378, 505, 390 ], "spans": [ { "bbox": [ 106, 378, 505, 390 ], "score": 1.0, "content": "only in more efficient levels of the memory hierarchy. In particular, most operations (except matrix", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 388, 505, 400 ], "spans": [ { "bbox": [ 105, 388, 505, 400 ], "score": 1.0, "content": "multiplication) are bounded by memory bandwidth (Williams et al., 2009; Ivanov et al., 2021; Dao", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 398, 506, 411 ], "spans": [ { "bbox": [ 105, 398, 506, 411 ], "score": 1.0, "content": "et al., 2022). This includes our scan operation, and we use kernel fusion to reduce the amount of", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 410, 447, 422 ], "spans": [ { "bbox": [ 105, 410, 447, 422 ], "score": 1.0, "content": "memory IOs, leading to a significant speedup compared to a standard implementation.", "type": "text" } ], "index": 18 } ], "index": 16 }, { "type": "text", "bbox": [ 107, 426, 505, 460 ], "lines": [ { "bbox": [ 106, 426, 505, 439 ], "spans": [ { "bbox": [ 106, 426, 287, 439 ], "score": 1.0, "content": "Concretely, instead of preparing the scan input", "type": "text" }, { "bbox": [ 287, 426, 316, 438 ], "score": 0.87, "content": "( \\overline { { A } } , \\overline { { B } } )", "type": "inline_equation" }, { "bbox": [ 316, 426, 505, 439 ], "score": 1.0, "content": "of size (B,L,D,N) in GPU HBM (high-bandwidth", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 438, 505, 450 ], "spans": [ { "bbox": [ 106, 438, 261, 450 ], "score": 1.0, "content": "memory), we load the SSM parameters", "type": "text" }, { "bbox": [ 262, 438, 313, 450 ], "score": 0.88, "content": "( \\Delta , A , B , C )", "type": "inline_equation" }, { "bbox": [ 313, 438, 505, 450 ], "score": 1.0, "content": "directly from slow HBM to fast SRAM, perform", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 448, 507, 461 ], "spans": [ { "bbox": [ 105, 448, 507, 461 ], "score": 1.0, "content": "the discretization and recurrence in SRAM, and then write the final outputs of size (B,L,D) back to HBM.", "type": "text" } ], "index": 21 } ], "index": 20 }, { "type": "text", "bbox": [ 108, 464, 505, 486 ], "lines": [ { "bbox": [ 107, 464, 505, 475 ], "spans": [ { "bbox": [ 107, 464, 505, 475 ], "score": 1.0, "content": "To avoid the sequential recurrence, we observe that despite not being linear it can still be parallelized", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 473, 506, 487 ], "spans": [ { "bbox": [ 105, 473, 506, 487 ], "score": 1.0, "content": "with a work-efficient parallel scan algorithm (Blelloch, 1990; Martin & Cundy, 2018; Smith et al., 2023).", "type": "text" } ], "index": 23 } ], "index": 22.5 }, { "type": "text", "bbox": [ 107, 491, 505, 534 ], "lines": [ { "bbox": [ 105, 491, 507, 504 ], "spans": [ { "bbox": [ 105, 491, 507, 504 ], "score": 1.0, "content": "Finally, we must also avoid saving the intermediate states, which are necessary for backpropagation.", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 501, 506, 514 ], "spans": [ { "bbox": [ 105, 501, 506, 514 ], "score": 1.0, "content": "We carefully apply the classic technique of recomputation to reduce the memory requirements: the", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 513, 505, 525 ], "spans": [ { "bbox": [ 105, 513, 505, 525 ], "score": 1.0, "content": "intermediate states are not stored but recomputed in the backward pass when the inputs are loaded", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 523, 454, 536 ], "spans": [ { "bbox": [ 105, 523, 454, 536 ], "score": 1.0, "content": "from HBM to SRAM. Details of the fused kernel and recomputation are in Appendix D.", "type": "text" } ], "index": 27 } ], "index": 25.5 }, { "type": "title", "bbox": [ 108, 544, 284, 555 ], "lines": [ { "bbox": [ 106, 544, 286, 556 ], "spans": [ { "bbox": [ 106, 544, 286, 556 ], "score": 1.0, "content": "3.4 A SIMPLIFIED SSM ARCHITECTURE", "type": "text" } ], "index": 28 } ], "index": 28 }, { "type": "text", "bbox": [ 107, 560, 505, 614 ], "lines": [ { "bbox": [ 106, 560, 505, 572 ], "spans": [ { "bbox": [ 106, 560, 505, 572 ], "score": 1.0, "content": "As with structured SSMs, selective SSMs are standalone sequence transformations that can be flexibly", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 571, 505, 583 ], "spans": [ { "bbox": [ 106, 571, 505, 583 ], "score": 1.0, "content": "incorporated into neural networks. The H3 architecture is the basis for the most well-known SSM", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 582, 505, 595 ], "spans": [ { "bbox": [ 105, 582, 505, 595 ], "score": 1.0, "content": "architectures (Section 2), which are generally comprised of a block inspired by linear attention", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 590, 505, 606 ], "spans": [ { "bbox": [ 105, 590, 505, 606 ], "score": 1.0, "content": "interleaved with an MLP (multi-layer perceptron) block. We simplify this architecture by combining", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 603, 404, 617 ], "spans": [ { "bbox": [ 106, 603, 404, 617 ], "score": 1.0, "content": "these two components into one, which is stacked homogenously (Figure 2).", "type": "text" } ], "index": 33 } ], "index": 31 }, { "type": "text", "bbox": [ 107, 619, 505, 662 ], "lines": [ { "bbox": [ 105, 618, 503, 631 ], "spans": [ { "bbox": [ 105, 618, 344, 631 ], "score": 1.0, "content": "This architecture involves expanding the model dimension", "type": "text" }, { "bbox": [ 344, 619, 354, 629 ], "score": 0.75, "content": "D", "type": "inline_equation" }, { "bbox": [ 354, 618, 494, 631 ], "score": 1.0, "content": "by a controllable expansion factor", "type": "text" }, { "bbox": [ 494, 619, 503, 629 ], "score": 0.79, "content": "E", "type": "inline_equation" } ], "index": 34 }, { "bbox": [ 105, 629, 506, 642 ], "spans": [ { "bbox": [ 105, 629, 269, 642 ], "score": 1.0, "content": "For each block, most of the parameters", "type": "text" }, { "bbox": [ 269, 630, 302, 641 ], "score": 0.9, "content": "( 3 E D ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 302, 629, 506, 642 ], "score": 1.0, "content": "are in the linear projections while the inner SSM", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 641, 505, 653 ], "spans": [ { "bbox": [ 106, 641, 241, 653 ], "score": 1.0, "content": "contributes less. We always fix to", "type": "text" }, { "bbox": [ 241, 641, 266, 650 ], "score": 0.9, "content": "E = 2", "type": "inline_equation" }, { "bbox": [ 266, 641, 505, 653 ], "score": 1.0, "content": "in our experiments and use two stacks of the block to match", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 650, 496, 664 ], "spans": [ { "bbox": [ 105, 650, 120, 664 ], "score": 1.0, "content": "the", "type": "text" }, { "bbox": [ 121, 650, 145, 661 ], "score": 0.89, "content": "1 2 D ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 145, 650, 496, 664 ], "score": 1.0, "content": "parameters of a Transformer’s interleaved MHA (multi-head attention) and MLP blocks.", "type": "text" } ], "index": 37 } ], "index": 35.5 }, { "type": "title", "bbox": [ 107, 672, 310, 684 ], "lines": [ { "bbox": [ 105, 671, 312, 685 ], "spans": [ { "bbox": [ 105, 671, 312, 685 ], "score": 1.0, "content": "3.5 PROPERTIES OF SELECTION MECHANISMS", "type": "text" } ], "index": 38 } ], "index": 38 }, { "type": "text", "bbox": [ 108, 689, 505, 731 ], "lines": [ { "bbox": [ 106, 688, 506, 702 ], "spans": [ { "bbox": [ 106, 688, 506, 702 ], "score": 1.0, "content": "The selection mechanism is a broader concept that can be applied in different ways, such as to", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 699, 506, 712 ], "spans": [ { "bbox": [ 105, 699, 355, 712 ], "score": 1.0, "content": "more traditional RNNs or CNNs, to different parameters (e.g.", "type": "text" }, { "bbox": [ 356, 700, 365, 709 ], "score": 0.63, "content": "\\pmb { A }", "type": "inline_equation" }, { "bbox": [ 365, 699, 506, 712 ], "score": 1.0, "content": "in Algorithm 2), or using different", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 709, 505, 722 ], "spans": [ { "bbox": [ 106, 709, 170, 722 ], "score": 1.0, "content": "transformations", "type": "text" }, { "bbox": [ 171, 710, 189, 722 ], "score": 0.92, "content": "s ( x )", "type": "inline_equation" }, { "bbox": [ 190, 709, 505, 722 ], "score": 1.0, "content": ". We highlight the most important connection: the classical gating mechanism of", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 720, 505, 732 ], "spans": [ { "bbox": [ 106, 720, 505, 732 ], "score": 1.0, "content": "RNNs is an instance of our selection mechanism for SSMs. We note that the connection between RNN", "type": "text" } ], "index": 42 } ], "index": 40.5 } ], "page_idx": 4, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 303, 37 ], "lines": [ { "bbox": [ 106, 26, 305, 38 ], "spans": [ { "bbox": [ 106, 26, 305, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 308, 760 ], "lines": [ { "bbox": [ 302, 750, 309, 763 ], "spans": [ { "bbox": [ 302, 750, 309, 763 ], "score": 1.0, "content": "5", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 106, 91, 501, 223 ], "blocks": [ { "type": "table_body", "bbox": [ 106, 91, 501, 223 ], "group_id": 0, "lines": [ { "bbox": [ 106, 91, 501, 223 ], "spans": [ { "bbox": [ 106, 91, 501, 223 ], "score": 0.956, "html": "
Algorithm 1 SSM (S4)Algorithm 2 SSM + Selection (S6)
Input: x : (B,L,D)Input: x : (B,L,D)
Output: y:(B,L,D)Output: y: (B,L,D)
1:A:(D,N)←Parameter1:A:(D,N)←Parameter
Represents structured N × N matrix>Represents structured N × N matrix
2:B:(D,N)←Parameter2: B:(B,L,N)← SB(𝑥)
3: C:(D,N)←Parameter 3: C:(B,L,N)←sc(x)
4:△:(D)←T△(Parameter)4: △:(B,L,D)←T△(Parameter+s△(x))
5: A,B:(D,N)←discretize(△,A,B) 5: A,B:(B,L,D,N)←discretize(△,A,B)
6: y←SSM(A,B,C)(x)6: y←SSM(A,B,C)(x)
> Time-invariant: recurrence or convolution 7: return yTime-varying: recurrence (scan) only 7: return y
", "type": "table", "image_path": "1cc809589f51a34f27ff26fcb8db77db7b7e9bf7ddd2741fe757d13ad118b0b3.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 106, 91, 501, 135.0 ], "spans": [], "index": 0 }, { "bbox": [ 106, 135.0, 501, 179.0 ], "spans": [], "index": 1 }, { "bbox": [ 106, 179.0, 501, 223.0 ], "spans": [], "index": 2 } ] } ], "index": 1 }, { "type": "text", "bbox": [ 108, 231, 502, 264 ], "lines": [], "index": 4, "bbox_fs": [ 106, 231, 504, 265 ], "lines_deleted": true }, { "type": "text", "bbox": [ 107, 268, 504, 301 ], "lines": [ { "bbox": [ 105, 267, 506, 281 ], "spans": [ { "bbox": [ 105, 267, 506, 281 ], "score": 1.0, "content": "The selection mechanism is designed to overcome the limitations of LTI models; at the same time,", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 279, 505, 291 ], "spans": [ { "bbox": [ 106, 279, 505, 291 ], "score": 1.0, "content": "we therefore need to revisit the computation problem of SSMs. We address this with three classical", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 290, 476, 302 ], "spans": [ { "bbox": [ 106, 290, 476, 302 ], "score": 1.0, "content": "techniques: kernel fusion, parallel scan, and recomputation. We make two main observations:", "type": "text" } ], "index": 8 } ], "index": 7, "bbox_fs": [ 105, 267, 506, 302 ] }, { "type": "text", "bbox": [ 106, 305, 505, 363 ], "lines": [ { "bbox": [ 105, 304, 505, 318 ], "spans": [ { "bbox": [ 105, 304, 271, 318 ], "score": 1.0, "content": "• The naive recurrent computation uses", "type": "text" }, { "bbox": [ 271, 305, 320, 317 ], "score": 0.91, "content": "O ( B L D N )", "type": "inline_equation" }, { "bbox": [ 320, 304, 505, 318 ], "score": 1.0, "content": "FLOPs while the convolutional computation", "type": "text" } ], "index": 9 }, { "bbox": [ 114, 316, 505, 329 ], "spans": [ { "bbox": [ 114, 316, 134, 329 ], "score": 1.0, "content": "uses", "type": "text" }, { "bbox": [ 135, 316, 201, 328 ], "score": 0.86, "content": "O ( B L D \\log ( L ) )", "type": "inline_equation" }, { "bbox": [ 201, 316, 505, 329 ], "score": 1.0, "content": "FLOPs, and the former has a lower constant factor. Thus for long sequences", "type": "text" } ], "index": 10 }, { "bbox": [ 114, 327, 460, 339 ], "spans": [ { "bbox": [ 114, 327, 249, 339 ], "score": 1.0, "content": "and not-too-large state dimension", "type": "text" }, { "bbox": [ 249, 327, 259, 337 ], "score": 0.78, "content": "N", "type": "inline_equation" }, { "bbox": [ 259, 327, 460, 339 ], "score": 1.0, "content": ", the recurrent mode can actually use fewer FLOPs.", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 339, 505, 353 ], "spans": [ { "bbox": [ 106, 339, 505, 353 ], "score": 1.0, "content": "• The two challenges are the sequential nature of recurrence, and the large memory usage. To address", "type": "text" } ], "index": 12 }, { "bbox": [ 114, 351, 503, 365 ], "spans": [ { "bbox": [ 114, 351, 496, 365 ], "score": 1.0, "content": "the latter, just like the convolutional mode, we can attempt to not actually materialize the full state", "type": "text" }, { "bbox": [ 496, 352, 503, 361 ], "score": 0.74, "content": "h", "type": "inline_equation" } ], "index": 13 } ], "index": 11, "bbox_fs": [ 105, 304, 505, 365 ] }, { "type": "text", "bbox": [ 107, 367, 505, 422 ], "lines": [ { "bbox": [ 106, 367, 504, 379 ], "spans": [ { "bbox": [ 106, 367, 496, 379 ], "score": 1.0, "content": "The main idea is to leverage properties of modern accelerators (GPUs) to materialize the state", "type": "text" }, { "bbox": [ 497, 367, 504, 377 ], "score": 0.71, "content": "h", "type": "inline_equation" } ], "index": 14 }, { "bbox": [ 106, 378, 505, 390 ], "spans": [ { "bbox": [ 106, 378, 505, 390 ], "score": 1.0, "content": "only in more efficient levels of the memory hierarchy. In particular, most operations (except matrix", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 388, 505, 400 ], "spans": [ { "bbox": [ 105, 388, 505, 400 ], "score": 1.0, "content": "multiplication) are bounded by memory bandwidth (Williams et al., 2009; Ivanov et al., 2021; Dao", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 398, 506, 411 ], "spans": [ { "bbox": [ 105, 398, 506, 411 ], "score": 1.0, "content": "et al., 2022). This includes our scan operation, and we use kernel fusion to reduce the amount of", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 410, 447, 422 ], "spans": [ { "bbox": [ 105, 410, 447, 422 ], "score": 1.0, "content": "memory IOs, leading to a significant speedup compared to a standard implementation.", "type": "text" } ], "index": 18 } ], "index": 16, "bbox_fs": [ 105, 367, 506, 422 ] }, { "type": "text", "bbox": [ 107, 426, 505, 460 ], "lines": [ { "bbox": [ 106, 426, 505, 439 ], "spans": [ { "bbox": [ 106, 426, 287, 439 ], "score": 1.0, "content": "Concretely, instead of preparing the scan input", "type": "text" }, { "bbox": [ 287, 426, 316, 438 ], "score": 0.87, "content": "( \\overline { { A } } , \\overline { { B } } )", "type": "inline_equation" }, { "bbox": [ 316, 426, 505, 439 ], "score": 1.0, "content": "of size (B,L,D,N) in GPU HBM (high-bandwidth", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 438, 505, 450 ], "spans": [ { "bbox": [ 106, 438, 261, 450 ], "score": 1.0, "content": "memory), we load the SSM parameters", "type": "text" }, { "bbox": [ 262, 438, 313, 450 ], "score": 0.88, "content": "( \\Delta , A , B , C )", "type": "inline_equation" }, { "bbox": [ 313, 438, 505, 450 ], "score": 1.0, "content": "directly from slow HBM to fast SRAM, perform", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 448, 507, 461 ], "spans": [ { "bbox": [ 105, 448, 507, 461 ], "score": 1.0, "content": "the discretization and recurrence in SRAM, and then write the final outputs of size (B,L,D) back to HBM.", "type": "text" } ], "index": 21 } ], "index": 20, "bbox_fs": [ 105, 426, 507, 461 ] }, { "type": "text", "bbox": [ 108, 464, 505, 486 ], "lines": [ { "bbox": [ 107, 464, 505, 475 ], "spans": [ { "bbox": [ 107, 464, 505, 475 ], "score": 1.0, "content": "To avoid the sequential recurrence, we observe that despite not being linear it can still be parallelized", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 473, 506, 487 ], "spans": [ { "bbox": [ 105, 473, 506, 487 ], "score": 1.0, "content": "with a work-efficient parallel scan algorithm (Blelloch, 1990; Martin & Cundy, 2018; Smith et al., 2023).", "type": "text" } ], "index": 23 } ], "index": 22.5, "bbox_fs": [ 105, 464, 506, 487 ] }, { "type": "text", "bbox": [ 107, 491, 505, 534 ], "lines": [ { "bbox": [ 105, 491, 507, 504 ], "spans": [ { "bbox": [ 105, 491, 507, 504 ], "score": 1.0, "content": "Finally, we must also avoid saving the intermediate states, which are necessary for backpropagation.", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 501, 506, 514 ], "spans": [ { "bbox": [ 105, 501, 506, 514 ], "score": 1.0, "content": "We carefully apply the classic technique of recomputation to reduce the memory requirements: the", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 513, 505, 525 ], "spans": [ { "bbox": [ 105, 513, 505, 525 ], "score": 1.0, "content": "intermediate states are not stored but recomputed in the backward pass when the inputs are loaded", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 523, 454, 536 ], "spans": [ { "bbox": [ 105, 523, 454, 536 ], "score": 1.0, "content": "from HBM to SRAM. Details of the fused kernel and recomputation are in Appendix D.", "type": "text" } ], "index": 27 } ], "index": 25.5, "bbox_fs": [ 105, 491, 507, 536 ] }, { "type": "title", "bbox": [ 108, 544, 284, 555 ], "lines": [ { "bbox": [ 106, 544, 286, 556 ], "spans": [ { "bbox": [ 106, 544, 286, 556 ], "score": 1.0, "content": "3.4 A SIMPLIFIED SSM ARCHITECTURE", "type": "text" } ], "index": 28 } ], "index": 28 }, { "type": "text", "bbox": [ 107, 560, 505, 614 ], "lines": [ { "bbox": [ 106, 560, 505, 572 ], "spans": [ { "bbox": [ 106, 560, 505, 572 ], "score": 1.0, "content": "As with structured SSMs, selective SSMs are standalone sequence transformations that can be flexibly", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 571, 505, 583 ], "spans": [ { "bbox": [ 106, 571, 505, 583 ], "score": 1.0, "content": "incorporated into neural networks. The H3 architecture is the basis for the most well-known SSM", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 582, 505, 595 ], "spans": [ { "bbox": [ 105, 582, 505, 595 ], "score": 1.0, "content": "architectures (Section 2), which are generally comprised of a block inspired by linear attention", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 590, 505, 606 ], "spans": [ { "bbox": [ 105, 590, 505, 606 ], "score": 1.0, "content": "interleaved with an MLP (multi-layer perceptron) block. We simplify this architecture by combining", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 603, 404, 617 ], "spans": [ { "bbox": [ 106, 603, 404, 617 ], "score": 1.0, "content": "these two components into one, which is stacked homogenously (Figure 2).", "type": "text" } ], "index": 33 } ], "index": 31, "bbox_fs": [ 105, 560, 505, 617 ] }, { "type": "text", "bbox": [ 107, 619, 505, 662 ], "lines": [ { "bbox": [ 105, 618, 503, 631 ], "spans": [ { "bbox": [ 105, 618, 344, 631 ], "score": 1.0, "content": "This architecture involves expanding the model dimension", "type": "text" }, { "bbox": [ 344, 619, 354, 629 ], "score": 0.75, "content": "D", "type": "inline_equation" }, { "bbox": [ 354, 618, 494, 631 ], "score": 1.0, "content": "by a controllable expansion factor", "type": "text" }, { "bbox": [ 494, 619, 503, 629 ], "score": 0.79, "content": "E", "type": "inline_equation" } ], "index": 34 }, { "bbox": [ 105, 629, 506, 642 ], "spans": [ { "bbox": [ 105, 629, 269, 642 ], "score": 1.0, "content": "For each block, most of the parameters", "type": "text" }, { "bbox": [ 269, 630, 302, 641 ], "score": 0.9, "content": "( 3 E D ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 302, 629, 506, 642 ], "score": 1.0, "content": "are in the linear projections while the inner SSM", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 641, 505, 653 ], "spans": [ { "bbox": [ 106, 641, 241, 653 ], "score": 1.0, "content": "contributes less. We always fix to", "type": "text" }, { "bbox": [ 241, 641, 266, 650 ], "score": 0.9, "content": "E = 2", "type": "inline_equation" }, { "bbox": [ 266, 641, 505, 653 ], "score": 1.0, "content": "in our experiments and use two stacks of the block to match", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 650, 496, 664 ], "spans": [ { "bbox": [ 105, 650, 120, 664 ], "score": 1.0, "content": "the", "type": "text" }, { "bbox": [ 121, 650, 145, 661 ], "score": 0.89, "content": "1 2 D ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 145, 650, 496, 664 ], "score": 1.0, "content": "parameters of a Transformer’s interleaved MHA (multi-head attention) and MLP blocks.", "type": "text" } ], "index": 37 } ], "index": 35.5, "bbox_fs": [ 105, 618, 506, 664 ] }, { "type": "title", "bbox": [ 107, 672, 310, 684 ], "lines": [ { "bbox": [ 105, 671, 312, 685 ], "spans": [ { "bbox": [ 105, 671, 312, 685 ], "score": 1.0, "content": "3.5 PROPERTIES OF SELECTION MECHANISMS", "type": "text" } ], "index": 38 } ], "index": 38 }, { "type": "text", "bbox": [ 108, 689, 505, 731 ], "lines": [ { "bbox": [ 106, 688, 506, 702 ], "spans": [ { "bbox": [ 106, 688, 506, 702 ], "score": 1.0, "content": "The selection mechanism is a broader concept that can be applied in different ways, such as to", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 699, 506, 712 ], "spans": [ { "bbox": [ 105, 699, 355, 712 ], "score": 1.0, "content": "more traditional RNNs or CNNs, to different parameters (e.g.", "type": "text" }, { "bbox": [ 356, 700, 365, 709 ], "score": 0.63, "content": "\\pmb { A }", "type": "inline_equation" }, { "bbox": [ 365, 699, 506, 712 ], "score": 1.0, "content": "in Algorithm 2), or using different", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 709, 505, 722 ], "spans": [ { "bbox": [ 106, 709, 170, 722 ], "score": 1.0, "content": "transformations", "type": "text" }, { "bbox": [ 171, 710, 189, 722 ], "score": 0.92, "content": "s ( x )", "type": "inline_equation" }, { "bbox": [ 190, 709, 505, 722 ], "score": 1.0, "content": ". We highlight the most important connection: the classical gating mechanism of", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 720, 505, 732 ], "spans": [ { "bbox": [ 106, 720, 505, 732 ], "score": 1.0, "content": "RNNs is an instance of our selection mechanism for SSMs. We note that the connection between RNN", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 261, 507, 274 ], "spans": [ { "bbox": [ 105, 261, 507, 274 ], "score": 1.0, "content": "gating and the discretization of continuous-time systems is well established (Funahashi & Nakamura,", "type": "text", "cross_page": true } ], "index": 8 }, { "bbox": [ 105, 271, 506, 284 ], "spans": [ { "bbox": [ 105, 271, 506, 284 ], "score": 1.0, "content": "1993; Tallec & Ollivier, 2018). In fact, Theorem 1 is an improvement of Gu et al. (2021, Lemma", "type": "text", "cross_page": true } ], "index": 9 }, { "bbox": [ 105, 282, 505, 295 ], "spans": [ { "bbox": [ 105, 282, 505, 295 ], "score": 1.0, "content": "3.1) generalizing to the ZOH discretization and input-dependent gates (proof in Appendix C). More", "type": "text", "cross_page": true } ], "index": 10 }, { "bbox": [ 105, 293, 459, 306 ], "spans": [ { "bbox": [ 105, 293, 141, 306 ], "score": 1.0, "content": "broadly,", "type": "text", "cross_page": true }, { "bbox": [ 141, 294, 150, 303 ], "score": 0.8, "content": "\\Delta", "type": "inline_equation", "cross_page": true }, { "bbox": [ 151, 293, 459, 306 ], "score": 1.0, "content": "in SSMs can be seen to play a generalized role of the RNN gating mechanism.", "type": "text", "cross_page": true } ], "index": 11 } ], "index": 40.5, "bbox_fs": [ 105, 688, 506, 732 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 156, 80, 455, 198 ], "blocks": [ { "type": "image_body", "bbox": [ 156, 80, 455, 198 ], "group_id": 0, "lines": [ { "bbox": [ 156, 80, 455, 198 ], "spans": [ { "bbox": [ 156, 80, 455, 198 ], "score": 0.974, "type": "image", "image_path": "5fd969173aa404aa9688b66726cef87ddc5376286f900bcca5dcb0b9ddb454db.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 156, 80, 455, 119.33333333333334 ], "spans": [], "index": 0 }, { "bbox": [ 156, 119.33333333333334, 455, 158.66666666666669 ], "spans": [], "index": 1 }, { "bbox": [ 156, 158.66666666666669, 455, 198.00000000000003 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 106, 207, 505, 257 ], "group_id": 0, "lines": [ { "bbox": [ 106, 208, 505, 218 ], "spans": [ { "bbox": [ 106, 208, 505, 218 ], "score": 1.0, "content": "Figure 2: (Architecture.) Our simplified block design combines the H3 block, which is the basis of most SSM", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 216, 505, 228 ], "spans": [ { "bbox": [ 105, 216, 505, 228 ], "score": 1.0, "content": "architectures, with the ubiquitous MLP block of modern neural networks. Instead of interleaving these two", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 227, 506, 238 ], "spans": [ { "bbox": [ 106, 227, 506, 238 ], "score": 1.0, "content": "blocks, we simply repeat the Mamba block homogenously. Compared to the H3 block, Mamba replaces the first", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 237, 505, 248 ], "spans": [ { "bbox": [ 106, 237, 505, 248 ], "score": 1.0, "content": "multiplicative gate with an activation function. Compared to the MLP block, Mamba adds an SSM to the main", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 246, 488, 258 ], "spans": [ { "bbox": [ 106, 246, 149, 258 ], "score": 1.0, "content": "branch. For", "type": "text" }, { "bbox": [ 150, 249, 156, 256 ], "score": 0.76, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 157, 246, 488, 258 ], "score": 1.0, "content": "we use the SiLU / Swish activation (Hendrycks & Gimpel, 2016; Ramachandran et al., 2017).", "type": "text" } ], "index": 7 } ], "index": 5 } ], "index": 3.0 }, { "type": "text", "bbox": [ 107, 261, 505, 305 ], "lines": [ { "bbox": [ 105, 261, 507, 274 ], "spans": [ { "bbox": [ 105, 261, 507, 274 ], "score": 1.0, "content": "gating and the discretization of continuous-time systems is well established (Funahashi & Nakamura,", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 271, 506, 284 ], "spans": [ { "bbox": [ 105, 271, 506, 284 ], "score": 1.0, "content": "1993; Tallec & Ollivier, 2018). In fact, Theorem 1 is an improvement of Gu et al. (2021, Lemma", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 282, 505, 295 ], "spans": [ { "bbox": [ 105, 282, 505, 295 ], "score": 1.0, "content": "3.1) generalizing to the ZOH discretization and input-dependent gates (proof in Appendix C). More", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 293, 459, 306 ], "spans": [ { "bbox": [ 105, 293, 141, 306 ], "score": 1.0, "content": "broadly,", "type": "text" }, { "bbox": [ 141, 294, 150, 303 ], "score": 0.8, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 151, 293, 459, 306 ], "score": 1.0, "content": "in SSMs can be seen to play a generalized role of the RNN gating mechanism.", "type": "text" } ], "index": 11 } ], "index": 9.5 }, { "type": "text", "bbox": [ 105, 308, 505, 331 ], "lines": [ { "bbox": [ 105, 307, 506, 321 ], "spans": [ { "bbox": [ 105, 307, 185, 321 ], "score": 1.0, "content": "Theorem 1. When", "type": "text" }, { "bbox": [ 186, 308, 334, 319 ], "score": 0.77, "content": "N = 1 , \\pmb { A } = - 1 , \\pmb { B } = 1 , s _ { \\Delta } = \\mathsf { L i n e a r } ( x )", "type": "inline_equation" }, { "bbox": [ 335, 307, 355, 321 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 355, 309, 377, 319 ], "score": 0.87, "content": "\\tau _ { \\Delta } =", "type": "inline_equation" }, { "bbox": [ 377, 307, 506, 321 ], "score": 1.0, "content": "softplus, then the selective SSM", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 318, 503, 333 ], "spans": [ { "bbox": [ 105, 318, 265, 333 ], "score": 1.0, "content": "recurrence (Algorithm 2) takes the form", "type": "text" }, { "bbox": [ 266, 320, 342, 331 ], "score": 0.82, "content": "g _ { k } { = } \\sigma ( { \\mathsf { L i n e a r } } ( x _ { k } ) )", "type": "inline_equation" }, { "bbox": [ 343, 318, 400, 333 ], "score": 1.0, "content": "(the gate) and", "type": "text" }, { "bbox": [ 401, 319, 503, 331 ], "score": 0.93, "content": "h _ { k } = ( 1 - g _ { k } ) h _ { k - 1 } + g _ { k } x _ { k }", "type": "inline_equation" } ], "index": 13 } ], "index": 12.5 }, { "type": "text", "bbox": [ 107, 338, 505, 393 ], "lines": [ { "bbox": [ 106, 338, 506, 352 ], "spans": [ { "bbox": [ 106, 338, 321, 352 ], "score": 1.0, "content": "As mentioned in Section 3.2, our specific choices of", "type": "text" }, { "bbox": [ 322, 341, 349, 350 ], "score": 0.9, "content": "s \\Delta , \\tau _ { \\Delta }", "type": "inline_equation" }, { "bbox": [ 349, 338, 506, 352 ], "score": 1.0, "content": "is from this connection. In particular,", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 349, 505, 362 ], "spans": [ { "bbox": [ 105, 349, 207, 362 ], "score": 1.0, "content": "note that if a given input", "type": "text" }, { "bbox": [ 208, 351, 219, 361 ], "score": 0.86, "content": "x _ { k }", "type": "inline_equation" }, { "bbox": [ 220, 349, 505, 362 ], "score": 1.0, "content": "should be completely ignored (as necessary in the synthetic tasks), all", "type": "text" } ], "index": 15 }, { "bbox": [ 107, 360, 506, 373 ], "spans": [ { "bbox": [ 107, 361, 116, 370 ], "score": 0.78, "content": "D", "type": "inline_equation" }, { "bbox": [ 117, 360, 506, 373 ], "score": 1.0, "content": "channels should ignore it, and so we project the input down to 1 dimension before broadcasting with", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 370, 507, 384 ], "spans": [ { "bbox": [ 106, 372, 115, 381 ], "score": 0.72, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 115, 370, 261, 384 ], "score": 1.0, "content": ". Finally, we observe that the function", "type": "text" }, { "bbox": [ 261, 371, 380, 383 ], "score": 0.85, "content": "s \\bar { \\Delta ( x ) } = \\mathsf { L i n e a r } _ { D } ( \\mathsf { L i n e a r } _ { 1 } ( x ) )", "type": "inline_equation" }, { "bbox": [ 381, 370, 507, 384 ], "score": 1.0, "content": "is simply a low-rank projection,", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 381, 461, 394 ], "spans": [ { "bbox": [ 105, 381, 461, 394 ], "score": 1.0, "content": "and relax the 1 dimension to a larger dimension R, generally set to be a small fraction of D.", "type": "text" } ], "index": 18 } ], "index": 16 }, { "type": "text", "bbox": [ 106, 397, 356, 408 ], "lines": [ { "bbox": [ 106, 397, 356, 410 ], "spans": [ { "bbox": [ 106, 397, 356, 410 ], "score": 1.0, "content": "We elaborate on two particular mechanistic effects of selection.", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "text", "bbox": [ 106, 413, 505, 468 ], "lines": [ { "bbox": [ 106, 414, 505, 425 ], "spans": [ { "bbox": [ 106, 414, 505, 425 ], "score": 1.0, "content": "Variable spacing. Selectivity allows filtering out irrelevant noise tokens that may occur between inputs", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 423, 505, 437 ], "spans": [ { "bbox": [ 105, 423, 505, 437 ], "score": 1.0, "content": "of interest. This is exemplified by the Selective Copying task, but occurs ubiquitously in common", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 434, 506, 447 ], "spans": [ { "bbox": [ 105, 434, 506, 447 ], "score": 1.0, "content": "data modalities, particularly for discrete data – for example the presence of language fillers such as", "type": "text" } ], "index": 22 }, { "bbox": [ 104, 443, 507, 460 ], "spans": [ { "bbox": [ 104, 443, 490, 460 ], "score": 1.0, "content": "“um”. This property arises because the model can mechanistically filter out any particular input", "type": "text" }, { "bbox": [ 491, 447, 502, 457 ], "score": 0.85, "content": "x _ { k }", "type": "inline_equation" }, { "bbox": [ 502, 443, 507, 460 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 456, 353, 469 ], "spans": [ { "bbox": [ 105, 456, 320, 469 ], "score": 1.0, "content": "for example in the gated RNN case (Theorem 1) when", "type": "text" }, { "bbox": [ 321, 457, 348, 468 ], "score": 0.9, "content": "g _ { k } \\to 0", "type": "inline_equation" }, { "bbox": [ 349, 456, 353, 469 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 24 } ], "index": 22 }, { "type": "text", "bbox": [ 106, 472, 505, 537 ], "lines": [ { "bbox": [ 105, 471, 505, 484 ], "spans": [ { "bbox": [ 105, 471, 505, 484 ], "score": 1.0, "content": "Filtering context. It has been empirically observed that many sequence models do not improve with", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 482, 506, 495 ], "spans": [ { "bbox": [ 106, 482, 506, 495 ], "score": 1.0, "content": "longer context (Shi et al., 2023a), despite the principle that more context should lead to strictly better", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 493, 506, 506 ], "spans": [ { "bbox": [ 105, 493, 506, 506 ], "score": 1.0, "content": "performance. An explanation is that many sequence models cannot effectively ignore irrelevant context", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 504, 506, 516 ], "spans": [ { "bbox": [ 105, 504, 506, 516 ], "score": 1.0, "content": "when necessary; an intuitive example are global convolutions (and general LTI models). On the other", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 514, 505, 527 ], "spans": [ { "bbox": [ 105, 514, 505, 527 ], "score": 1.0, "content": "hand, selective models can simply reset their state at any time to remove extraneous history, and thus", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 525, 485, 538 ], "spans": [ { "bbox": [ 106, 525, 485, 538 ], "score": 1.0, "content": "their performance in principle improves monotonicly with context length (e.g. Appendix E.2.2).", "type": "text" } ], "index": 30 } ], "index": 27.5 }, { "type": "text", "bbox": [ 107, 541, 505, 588 ], "lines": [ { "bbox": [ 106, 542, 505, 554 ], "spans": [ { "bbox": [ 106, 542, 210, 554 ], "score": 1.0, "content": "We remark that while the", "type": "text" }, { "bbox": [ 211, 542, 221, 551 ], "score": 0.68, "content": "\\pmb { A }", "type": "inline_equation" }, { "bbox": [ 221, 542, 505, 554 ], "score": 1.0, "content": "parameter could also be selective, it ultimately affects the model only", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 552, 505, 566 ], "spans": [ { "bbox": [ 105, 553, 215, 566 ], "score": 1.0, "content": "through its interaction with", "type": "text" }, { "bbox": [ 216, 554, 225, 563 ], "score": 0.8, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 225, 553, 240, 566 ], "score": 1.0, "content": "via", "type": "text" }, { "bbox": [ 240, 552, 299, 565 ], "score": 0.93, "content": "\\overline { { A } } = \\exp ( \\Delta A )", "type": "inline_equation" }, { "bbox": [ 300, 553, 454, 566 ], "score": 1.0, "content": "(the discretization). Thus selectivity in", "type": "text" }, { "bbox": [ 454, 554, 464, 564 ], "score": 0.77, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 464, 553, 505, 566 ], "score": 1.0, "content": "is enough", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 564, 504, 578 ], "spans": [ { "bbox": [ 106, 566, 195, 578 ], "score": 1.0, "content": "to ensure selectivity in", "type": "text" }, { "bbox": [ 196, 564, 224, 578 ], "score": 0.9, "content": "( \\overline { { A } } , \\overline { { B } } )", "type": "inline_equation" }, { "bbox": [ 225, 566, 494, 578 ], "score": 1.0, "content": ", and is the main source of improvement. We hypothesize that making", "type": "text" }, { "bbox": [ 494, 566, 504, 576 ], "score": 0.56, "content": "\\pmb { A }", "type": "inline_equation" } ], "index": 33 }, { "bbox": [ 105, 575, 507, 590 ], "spans": [ { "bbox": [ 105, 575, 251, 590 ], "score": 1.0, "content": "selective in addition to (or instead of)", "type": "text" }, { "bbox": [ 252, 577, 261, 586 ], "score": 0.79, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 261, 575, 507, 590 ], "score": 1.0, "content": "would have similar performance, and leave it out for simplicity.", "type": "text" } ], "index": 34 } ], "index": 32.5 }, { "type": "text", "bbox": [ 106, 591, 504, 613 ], "lines": [ { "bbox": [ 106, 590, 505, 604 ], "spans": [ { "bbox": [ 106, 590, 505, 604 ], "score": 1.0, "content": "Remark 3.1. For brevity in our experimental results, we sometimes abbreviate selective SSMs as", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 601, 478, 615 ], "spans": [ { "bbox": [ 105, 601, 219, 615 ], "score": 1.0, "content": "S6 models, because they are", "type": "text" }, { "bbox": [ 219, 603, 231, 612 ], "score": 0.38, "content": "S 4", "type": "inline_equation" }, { "bbox": [ 231, 601, 478, 615 ], "score": 1.0, "content": "models with a selection mechanism and computed with a scan.", "type": "text" } ], "index": 36 } ], "index": 35.5 }, { "type": "title", "bbox": [ 108, 623, 254, 636 ], "lines": [ { "bbox": [ 105, 622, 255, 638 ], "spans": [ { "bbox": [ 105, 622, 255, 638 ], "score": 1.0, "content": "4 EMPIRICAL EVALUATION", "type": "text" } ], "index": 37 } ], "index": 37 }, { "type": "text", "bbox": [ 107, 642, 505, 675 ], "lines": [ { "bbox": [ 105, 641, 505, 654 ], "spans": [ { "bbox": [ 105, 641, 505, 654 ], "score": 1.0, "content": "Mamba achieves state-of-the-art results on the synthetic tasks (Section 4.1) and three different domains", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 651, 506, 666 ], "spans": [ { "bbox": [ 105, 651, 506, 666 ], "score": 1.0, "content": "(language, DNA, audio) (Sections 4.2 to 4.4) on both pretraining and downstream tasks, while being", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 663, 283, 675 ], "spans": [ { "bbox": [ 105, 663, 283, 675 ], "score": 1.0, "content": "very computationally efficient (Section 4.5).", "type": "text" } ], "index": 40 } ], "index": 39 }, { "type": "title", "bbox": [ 107, 683, 212, 694 ], "lines": [ { "bbox": [ 105, 681, 213, 696 ], "spans": [ { "bbox": [ 105, 681, 213, 696 ], "score": 1.0, "content": "4.1 SYNTHETIC TASKS", "type": "text" } ], "index": 41 } ], "index": 41 }, { "type": "text", "bbox": [ 107, 699, 505, 732 ], "lines": [ { "bbox": [ 105, 698, 505, 712 ], "spans": [ { "bbox": [ 105, 698, 505, 712 ], "score": 1.0, "content": "Table 1 and Figure 3 show results for the synthetic tasks. On Selective Copying, the selective SSM layer", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 711, 505, 721 ], "spans": [ { "bbox": [ 106, 711, 505, 721 ], "score": 1.0, "content": "is enough to solve the task independently of the architecture used, while previous LTI SSMs cannot even", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 721, 505, 732 ], "spans": [ { "bbox": [ 106, 721, 505, 732 ], "score": 1.0, "content": "when combined with more powerful architectures. On Induction Heads, Mamba learns the task perfectly", "type": "text" } ], "index": 44 } ], "index": 43 } ], "page_idx": 5, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 304, 37 ], "lines": [ { "bbox": [ 106, 25, 305, 39 ], "spans": [ { "bbox": [ 106, 25, 305, 39 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 752, 308, 760 ], "lines": [ { "bbox": [ 302, 751, 309, 762 ], "spans": [ { "bbox": [ 302, 751, 309, 762 ], "score": 1.0, "content": "6", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "image", "bbox": [ 156, 80, 455, 198 ], "blocks": [ { "type": "image_body", "bbox": [ 156, 80, 455, 198 ], "group_id": 0, "lines": [ { "bbox": [ 156, 80, 455, 198 ], "spans": [ { "bbox": [ 156, 80, 455, 198 ], "score": 0.974, "type": "image", "image_path": "5fd969173aa404aa9688b66726cef87ddc5376286f900bcca5dcb0b9ddb454db.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 156, 80, 455, 119.33333333333334 ], "spans": [], "index": 0 }, { "bbox": [ 156, 119.33333333333334, 455, 158.66666666666669 ], "spans": [], "index": 1 }, { "bbox": [ 156, 158.66666666666669, 455, 198.00000000000003 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 106, 207, 505, 257 ], "group_id": 0, "lines": [ { "bbox": [ 106, 208, 505, 218 ], "spans": [ { "bbox": [ 106, 208, 505, 218 ], "score": 1.0, "content": "Figure 2: (Architecture.) Our simplified block design combines the H3 block, which is the basis of most SSM", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 216, 505, 228 ], "spans": [ { "bbox": [ 105, 216, 505, 228 ], "score": 1.0, "content": "architectures, with the ubiquitous MLP block of modern neural networks. Instead of interleaving these two", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 227, 506, 238 ], "spans": [ { "bbox": [ 106, 227, 506, 238 ], "score": 1.0, "content": "blocks, we simply repeat the Mamba block homogenously. Compared to the H3 block, Mamba replaces the first", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 237, 505, 248 ], "spans": [ { "bbox": [ 106, 237, 505, 248 ], "score": 1.0, "content": "multiplicative gate with an activation function. Compared to the MLP block, Mamba adds an SSM to the main", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 246, 488, 258 ], "spans": [ { "bbox": [ 106, 246, 149, 258 ], "score": 1.0, "content": "branch. For", "type": "text" }, { "bbox": [ 150, 249, 156, 256 ], "score": 0.76, "content": "\\sigma", "type": "inline_equation" }, { "bbox": [ 157, 246, 488, 258 ], "score": 1.0, "content": "we use the SiLU / Swish activation (Hendrycks & Gimpel, 2016; Ramachandran et al., 2017).", "type": "text" } ], "index": 7 } ], "index": 5 } ], "index": 3.0 }, { "type": "text", "bbox": [ 107, 261, 505, 305 ], "lines": [], "index": 9.5, "bbox_fs": [ 105, 261, 507, 306 ], "lines_deleted": true }, { "type": "text", "bbox": [ 105, 308, 505, 331 ], "lines": [ { "bbox": [ 105, 307, 506, 321 ], "spans": [ { "bbox": [ 105, 307, 185, 321 ], "score": 1.0, "content": "Theorem 1. When", "type": "text" }, { "bbox": [ 186, 308, 334, 319 ], "score": 0.77, "content": "N = 1 , \\pmb { A } = - 1 , \\pmb { B } = 1 , s _ { \\Delta } = \\mathsf { L i n e a r } ( x )", "type": "inline_equation" }, { "bbox": [ 335, 307, 355, 321 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 355, 309, 377, 319 ], "score": 0.87, "content": "\\tau _ { \\Delta } =", "type": "inline_equation" }, { "bbox": [ 377, 307, 506, 321 ], "score": 1.0, "content": "softplus, then the selective SSM", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 318, 503, 333 ], "spans": [ { "bbox": [ 105, 318, 265, 333 ], "score": 1.0, "content": "recurrence (Algorithm 2) takes the form", "type": "text" }, { "bbox": [ 266, 320, 342, 331 ], "score": 0.82, "content": "g _ { k } { = } \\sigma ( { \\mathsf { L i n e a r } } ( x _ { k } ) )", "type": "inline_equation" }, { "bbox": [ 343, 318, 400, 333 ], "score": 1.0, "content": "(the gate) and", "type": "text" }, { "bbox": [ 401, 319, 503, 331 ], "score": 0.93, "content": "h _ { k } = ( 1 - g _ { k } ) h _ { k - 1 } + g _ { k } x _ { k }", "type": "inline_equation" } ], "index": 13 } ], "index": 12.5, "bbox_fs": [ 105, 307, 506, 333 ] }, { "type": "text", "bbox": [ 107, 338, 505, 393 ], "lines": [ { "bbox": [ 106, 338, 506, 352 ], "spans": [ { "bbox": [ 106, 338, 321, 352 ], "score": 1.0, "content": "As mentioned in Section 3.2, our specific choices of", "type": "text" }, { "bbox": [ 322, 341, 349, 350 ], "score": 0.9, "content": "s \\Delta , \\tau _ { \\Delta }", "type": "inline_equation" }, { "bbox": [ 349, 338, 506, 352 ], "score": 1.0, "content": "is from this connection. In particular,", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 349, 505, 362 ], "spans": [ { "bbox": [ 105, 349, 207, 362 ], "score": 1.0, "content": "note that if a given input", "type": "text" }, { "bbox": [ 208, 351, 219, 361 ], "score": 0.86, "content": "x _ { k }", "type": "inline_equation" }, { "bbox": [ 220, 349, 505, 362 ], "score": 1.0, "content": "should be completely ignored (as necessary in the synthetic tasks), all", "type": "text" } ], "index": 15 }, { "bbox": [ 107, 360, 506, 373 ], "spans": [ { "bbox": [ 107, 361, 116, 370 ], "score": 0.78, "content": "D", "type": "inline_equation" }, { "bbox": [ 117, 360, 506, 373 ], "score": 1.0, "content": "channels should ignore it, and so we project the input down to 1 dimension before broadcasting with", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 370, 507, 384 ], "spans": [ { "bbox": [ 106, 372, 115, 381 ], "score": 0.72, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 115, 370, 261, 384 ], "score": 1.0, "content": ". Finally, we observe that the function", "type": "text" }, { "bbox": [ 261, 371, 380, 383 ], "score": 0.85, "content": "s \\bar { \\Delta ( x ) } = \\mathsf { L i n e a r } _ { D } ( \\mathsf { L i n e a r } _ { 1 } ( x ) )", "type": "inline_equation" }, { "bbox": [ 381, 370, 507, 384 ], "score": 1.0, "content": "is simply a low-rank projection,", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 381, 461, 394 ], "spans": [ { "bbox": [ 105, 381, 461, 394 ], "score": 1.0, "content": "and relax the 1 dimension to a larger dimension R, generally set to be a small fraction of D.", "type": "text" } ], "index": 18 } ], "index": 16, "bbox_fs": [ 105, 338, 507, 394 ] }, { "type": "text", "bbox": [ 106, 397, 356, 408 ], "lines": [ { "bbox": [ 106, 397, 356, 410 ], "spans": [ { "bbox": [ 106, 397, 356, 410 ], "score": 1.0, "content": "We elaborate on two particular mechanistic effects of selection.", "type": "text" } ], "index": 19 } ], "index": 19, "bbox_fs": [ 106, 397, 356, 410 ] }, { "type": "text", "bbox": [ 106, 413, 505, 468 ], "lines": [ { "bbox": [ 106, 414, 505, 425 ], "spans": [ { "bbox": [ 106, 414, 505, 425 ], "score": 1.0, "content": "Variable spacing. Selectivity allows filtering out irrelevant noise tokens that may occur between inputs", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 423, 505, 437 ], "spans": [ { "bbox": [ 105, 423, 505, 437 ], "score": 1.0, "content": "of interest. This is exemplified by the Selective Copying task, but occurs ubiquitously in common", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 434, 506, 447 ], "spans": [ { "bbox": [ 105, 434, 506, 447 ], "score": 1.0, "content": "data modalities, particularly for discrete data – for example the presence of language fillers such as", "type": "text" } ], "index": 22 }, { "bbox": [ 104, 443, 507, 460 ], "spans": [ { "bbox": [ 104, 443, 490, 460 ], "score": 1.0, "content": "“um”. This property arises because the model can mechanistically filter out any particular input", "type": "text" }, { "bbox": [ 491, 447, 502, 457 ], "score": 0.85, "content": "x _ { k }", "type": "inline_equation" }, { "bbox": [ 502, 443, 507, 460 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 456, 353, 469 ], "spans": [ { "bbox": [ 105, 456, 320, 469 ], "score": 1.0, "content": "for example in the gated RNN case (Theorem 1) when", "type": "text" }, { "bbox": [ 321, 457, 348, 468 ], "score": 0.9, "content": "g _ { k } \\to 0", "type": "inline_equation" }, { "bbox": [ 349, 456, 353, 469 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 24 } ], "index": 22, "bbox_fs": [ 104, 414, 507, 469 ] }, { "type": "text", "bbox": [ 106, 472, 505, 537 ], "lines": [ { "bbox": [ 105, 471, 505, 484 ], "spans": [ { "bbox": [ 105, 471, 505, 484 ], "score": 1.0, "content": "Filtering context. It has been empirically observed that many sequence models do not improve with", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 482, 506, 495 ], "spans": [ { "bbox": [ 106, 482, 506, 495 ], "score": 1.0, "content": "longer context (Shi et al., 2023a), despite the principle that more context should lead to strictly better", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 493, 506, 506 ], "spans": [ { "bbox": [ 105, 493, 506, 506 ], "score": 1.0, "content": "performance. An explanation is that many sequence models cannot effectively ignore irrelevant context", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 504, 506, 516 ], "spans": [ { "bbox": [ 105, 504, 506, 516 ], "score": 1.0, "content": "when necessary; an intuitive example are global convolutions (and general LTI models). On the other", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 514, 505, 527 ], "spans": [ { "bbox": [ 105, 514, 505, 527 ], "score": 1.0, "content": "hand, selective models can simply reset their state at any time to remove extraneous history, and thus", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 525, 485, 538 ], "spans": [ { "bbox": [ 106, 525, 485, 538 ], "score": 1.0, "content": "their performance in principle improves monotonicly with context length (e.g. Appendix E.2.2).", "type": "text" } ], "index": 30 } ], "index": 27.5, "bbox_fs": [ 105, 471, 506, 538 ] }, { "type": "text", "bbox": [ 107, 541, 505, 588 ], "lines": [ { "bbox": [ 106, 542, 505, 554 ], "spans": [ { "bbox": [ 106, 542, 210, 554 ], "score": 1.0, "content": "We remark that while the", "type": "text" }, { "bbox": [ 211, 542, 221, 551 ], "score": 0.68, "content": "\\pmb { A }", "type": "inline_equation" }, { "bbox": [ 221, 542, 505, 554 ], "score": 1.0, "content": "parameter could also be selective, it ultimately affects the model only", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 552, 505, 566 ], "spans": [ { "bbox": [ 105, 553, 215, 566 ], "score": 1.0, "content": "through its interaction with", "type": "text" }, { "bbox": [ 216, 554, 225, 563 ], "score": 0.8, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 225, 553, 240, 566 ], "score": 1.0, "content": "via", "type": "text" }, { "bbox": [ 240, 552, 299, 565 ], "score": 0.93, "content": "\\overline { { A } } = \\exp ( \\Delta A )", "type": "inline_equation" }, { "bbox": [ 300, 553, 454, 566 ], "score": 1.0, "content": "(the discretization). Thus selectivity in", "type": "text" }, { "bbox": [ 454, 554, 464, 564 ], "score": 0.77, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 464, 553, 505, 566 ], "score": 1.0, "content": "is enough", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 564, 504, 578 ], "spans": [ { "bbox": [ 106, 566, 195, 578 ], "score": 1.0, "content": "to ensure selectivity in", "type": "text" }, { "bbox": [ 196, 564, 224, 578 ], "score": 0.9, "content": "( \\overline { { A } } , \\overline { { B } } )", "type": "inline_equation" }, { "bbox": [ 225, 566, 494, 578 ], "score": 1.0, "content": ", and is the main source of improvement. We hypothesize that making", "type": "text" }, { "bbox": [ 494, 566, 504, 576 ], "score": 0.56, "content": "\\pmb { A }", "type": "inline_equation" } ], "index": 33 }, { "bbox": [ 105, 575, 507, 590 ], "spans": [ { "bbox": [ 105, 575, 251, 590 ], "score": 1.0, "content": "selective in addition to (or instead of)", "type": "text" }, { "bbox": [ 252, 577, 261, 586 ], "score": 0.79, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 261, 575, 507, 590 ], "score": 1.0, "content": "would have similar performance, and leave it out for simplicity.", "type": "text" } ], "index": 34 } ], "index": 32.5, "bbox_fs": [ 105, 542, 507, 590 ] }, { "type": "text", "bbox": [ 106, 591, 504, 613 ], "lines": [ { "bbox": [ 106, 590, 505, 604 ], "spans": [ { "bbox": [ 106, 590, 505, 604 ], "score": 1.0, "content": "Remark 3.1. For brevity in our experimental results, we sometimes abbreviate selective SSMs as", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 601, 478, 615 ], "spans": [ { "bbox": [ 105, 601, 219, 615 ], "score": 1.0, "content": "S6 models, because they are", "type": "text" }, { "bbox": [ 219, 603, 231, 612 ], "score": 0.38, "content": "S 4", "type": "inline_equation" }, { "bbox": [ 231, 601, 478, 615 ], "score": 1.0, "content": "models with a selection mechanism and computed with a scan.", "type": "text" } ], "index": 36 } ], "index": 35.5, "bbox_fs": [ 105, 590, 505, 615 ] }, { "type": "title", "bbox": [ 108, 623, 254, 636 ], "lines": [ { "bbox": [ 105, 622, 255, 638 ], "spans": [ { "bbox": [ 105, 622, 255, 638 ], "score": 1.0, "content": "4 EMPIRICAL EVALUATION", "type": "text" } ], "index": 37 } ], "index": 37 }, { "type": "text", "bbox": [ 107, 642, 505, 675 ], "lines": [ { "bbox": [ 105, 641, 505, 654 ], "spans": [ { "bbox": [ 105, 641, 505, 654 ], "score": 1.0, "content": "Mamba achieves state-of-the-art results on the synthetic tasks (Section 4.1) and three different domains", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 651, 506, 666 ], "spans": [ { "bbox": [ 105, 651, 506, 666 ], "score": 1.0, "content": "(language, DNA, audio) (Sections 4.2 to 4.4) on both pretraining and downstream tasks, while being", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 663, 283, 675 ], "spans": [ { "bbox": [ 105, 663, 283, 675 ], "score": 1.0, "content": "very computationally efficient (Section 4.5).", "type": "text" } ], "index": 40 } ], "index": 39, "bbox_fs": [ 105, 641, 506, 675 ] }, { "type": "title", "bbox": [ 107, 683, 212, 694 ], "lines": [ { "bbox": [ 105, 681, 213, 696 ], "spans": [ { "bbox": [ 105, 681, 213, 696 ], "score": 1.0, "content": "4.1 SYNTHETIC TASKS", "type": "text" } ], "index": 41 } ], "index": 41 }, { "type": "text", "bbox": [ 107, 699, 505, 732 ], "lines": [ { "bbox": [ 105, 698, 505, 712 ], "spans": [ { "bbox": [ 105, 698, 505, 712 ], "score": 1.0, "content": "Table 1 and Figure 3 show results for the synthetic tasks. On Selective Copying, the selective SSM layer", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 711, 505, 721 ], "spans": [ { "bbox": [ 106, 711, 505, 721 ], "score": 1.0, "content": "is enough to solve the task independently of the architecture used, while previous LTI SSMs cannot even", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 721, 505, 732 ], "spans": [ { "bbox": [ 106, 721, 505, 732 ], "score": 1.0, "content": "when combined with more powerful architectures. On Induction Heads, Mamba learns the task perfectly", "type": "text" } ], "index": 44 } ], "index": 43, "bbox_fs": [ 105, 698, 505, 732 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 109, 80, 234, 191 ], "blocks": [ { "type": "table_body", "bbox": [ 109, 80, 234, 191 ], "group_id": 0, "lines": [ { "bbox": [ 109, 80, 234, 191 ], "spans": [ { "bbox": [ 109, 80, 234, 191 ], "score": 0.908, "html": "
MODEL ARCH.LAYER ACC.
S4No gate S4 No gate S618.3 97.0
H3H3S457.0
HyenaH3Hyena30.1
1H3S699.7
Mamba S456.4
Mamba Hyena28.4
MambaMamba S699.8
", "type": "table", "image_path": "277588017c3983620171c24625a0eb409c312b2f46bdda72ce562793b1faa647.jpg" } ] } ], "index": 0.5, "virtual_lines": [ { "bbox": [ 109, 80, 234, 135.5 ], "spans": [], "index": 0 }, { "bbox": [ 109, 135.5, 234, 191.0 ], "spans": [], "index": 1 } ] } ], "index": 0.5 }, { "type": "text", "bbox": [ 106, 201, 239, 231 ], "lines": [ { "bbox": [ 105, 199, 215, 214 ], "spans": [ { "bbox": [ 105, 199, 215, 214 ], "score": 1.0, "content": "Table 1: (Selective Copying.)", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 211, 241, 221 ], "spans": [ { "bbox": [ 106, 211, 241, 221 ], "score": 1.0, "content": "Accuracy for combinations of archi-", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 221, 232, 232 ], "spans": [ { "bbox": [ 106, 221, 232, 232 ], "score": 1.0, "content": "tectures and inner sequence layers.", "type": "text" } ], "index": 4 } ], "index": 3 }, { "type": "image", "bbox": [ 256, 85, 501, 197 ], "blocks": [ { "type": "image_body", "bbox": [ 256, 85, 501, 197 ], "group_id": 0, "lines": [ { "bbox": [ 256, 85, 501, 197 ], "spans": [ { "bbox": [ 256, 85, 501, 197 ], "score": 0.965, "type": "image", "image_path": "f69b759105d19813c86e3230b2579ef8e0c264126b0cc0b2657bf26e763023a6.jpg" } ] } ], "index": 6, "virtual_lines": [ { "bbox": [ 256, 85, 501, 122.33333333333334 ], "spans": [], "index": 5 }, { "bbox": [ 256, 122.33333333333334, 501, 159.66666666666669 ], "spans": [], "index": 6 }, { "bbox": [ 256, 159.66666666666669, 501, 197.00000000000003 ], "spans": [], "index": 7 } ] }, { "type": "image_caption", "bbox": [ 252, 198, 504, 228 ], "group_id": 0, "lines": [ { "bbox": [ 252, 198, 505, 209 ], "spans": [ { "bbox": [ 252, 198, 505, 209 ], "score": 1.0, "content": "Figure 3: (Induction Heads.) Models are trained on sequence length", "type": "text" } ], "index": 8 }, { "bbox": [ 253, 205, 506, 220 ], "spans": [ { "bbox": [ 253, 207, 287, 217 ], "score": 0.89, "content": "2 ^ { 8 } = 2 5 6", "type": "inline_equation" }, { "bbox": [ 287, 205, 453, 220 ], "score": 1.0, "content": ", and tested on increasing sequence lengths of", "type": "text" }, { "bbox": [ 454, 207, 484, 217 ], "score": 0.91, "content": "2 ^ { 6 } = 6 4", "type": "inline_equation" }, { "bbox": [ 484, 205, 506, 220 ], "score": 1.0, "content": "up to", "type": "text" } ], "index": 9 }, { "bbox": [ 253, 215, 405, 230 ], "spans": [ { "bbox": [ 253, 217, 308, 228 ], "score": 0.85, "content": "2 ^ { 2 0 } = 1 0 4 8 5 7 6", "type": "inline_equation" }, { "bbox": [ 308, 215, 405, 230 ], "score": 1.0, "content": ". Full numbers in Table 10.", "type": "text" } ], "index": 10 } ], "index": 9 } ], "index": 7.5 }, { "type": "image", "bbox": [ 111, 261, 500, 350 ], "blocks": [ { "type": "image_body", "bbox": [ 111, 261, 500, 350 ], "group_id": 1, "lines": [ { "bbox": [ 111, 261, 500, 350 ], "spans": [ { "bbox": [ 111, 261, 500, 350 ], "score": 0.963, "type": "image", "image_path": "9db8fba5b9f20e0b9c38d467bc8134fd8fb8345b39ed7a657d05892ed382ad40.jpg" } ] } ], "index": 12, "virtual_lines": [ { "bbox": [ 111, 261, 500, 290.6666666666667 ], "spans": [], "index": 11 }, { "bbox": [ 111, 290.6666666666667, 500, 320.33333333333337 ], "spans": [], "index": 12 }, { "bbox": [ 111, 320.33333333333337, 500, 350.00000000000006 ], "spans": [], "index": 13 } ] }, { "type": "image_caption", "bbox": [ 106, 360, 505, 390 ], "group_id": 1, "lines": [ { "bbox": [ 105, 360, 505, 371 ], "spans": [ { "bbox": [ 105, 360, 291, 371 ], "score": 1.0, "content": "Figure 4: (Chinchilla scaling laws.) Models of size", "type": "text" }, { "bbox": [ 292, 360, 325, 369 ], "score": 0.88, "content": "\\approx 1 2 5 M", "type": "inline_equation" }, { "bbox": [ 325, 360, 334, 371 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 334, 360, 363, 369 ], "score": 0.88, "content": "{ \\approx } 1 . 3 B", "type": "inline_equation" }, { "bbox": [ 363, 360, 505, 371 ], "score": 1.0, "content": "parameters, trained on the Pile. Mamba", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 368, 506, 382 ], "spans": [ { "bbox": [ 105, 368, 506, 382 ], "score": 1.0, "content": "scales better than all other attention-free models and is the first to match the performance of a very strong", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 378, 454, 391 ], "spans": [ { "bbox": [ 105, 378, 154, 391 ], "score": 1.0, "content": "“Transformer", "type": "text" }, { "bbox": [ 155, 380, 166, 389 ], "score": 0.46, "content": "+ + ^ { \\prime }", "type": "inline_equation" }, { "bbox": [ 167, 378, 454, 391 ], "score": 1.0, "content": "” recipe that has now become standard, particularly as the sequence length grows.", "type": "text" } ], "index": 16 } ], "index": 15 } ], "index": 13.5 }, { "type": "text", "bbox": [ 106, 413, 505, 435 ], "lines": [ { "bbox": [ 105, 411, 507, 427 ], "spans": [ { "bbox": [ 105, 411, 336, 427 ], "score": 1.0, "content": "and can even extrapolate to million-length sequences, or", "type": "text" }, { "bbox": [ 336, 414, 365, 424 ], "score": 0.88, "content": "4 0 0 0 \\times", "type": "inline_equation" }, { "bbox": [ 366, 411, 507, 427 ], "score": 1.0, "content": "longer than it saw during training,", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 423, 484, 436 ], "spans": [ { "bbox": [ 106, 423, 248, 436 ], "score": 1.0, "content": "while no other method goes beyond", "type": "text" }, { "bbox": [ 248, 424, 262, 434 ], "score": 0.86, "content": "2 \\times", "type": "inline_equation" }, { "bbox": [ 262, 423, 484, 436 ], "score": 1.0, "content": ". 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MODEL ARCH.LAYER ACC.
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Full numbers in Table 10.", "type": "text" } ], "index": 10 } ], "index": 9 } ], "index": 7.5 }, { "type": "image", "bbox": [ 111, 261, 500, 350 ], "blocks": [ { "type": "image_body", "bbox": [ 111, 261, 500, 350 ], "group_id": 1, "lines": [ { "bbox": [ 111, 261, 500, 350 ], "spans": [ { "bbox": [ 111, 261, 500, 350 ], "score": 0.963, "type": "image", "image_path": "9db8fba5b9f20e0b9c38d467bc8134fd8fb8345b39ed7a657d05892ed382ad40.jpg" } ] } ], "index": 12, "virtual_lines": [ { "bbox": [ 111, 261, 500, 290.6666666666667 ], "spans": [], "index": 11 }, { "bbox": [ 111, 290.6666666666667, 500, 320.33333333333337 ], "spans": [], "index": 12 }, { "bbox": [ 111, 320.33333333333337, 500, 350.00000000000006 ], "spans": [], "index": 13 } ] }, { "type": "image_caption", "bbox": [ 106, 360, 505, 390 ], "group_id": 1, "lines": [ { "bbox": [ 105, 360, 505, 371 ], "spans": [ { "bbox": [ 105, 360, 291, 371 ], "score": 1.0, "content": "Figure 4: (Chinchilla scaling laws.) 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MODELPPL↓PPL↓TOKEN. PILE LAMBADA LAMBADA HellSwag PIQA Arc-E Arc-C WinoGrande Average ACC个ACC个ACC个ACC↑ACC↑ACC个ACC↑
Pythia-160MNeoX29.64 38.1033.030.261.443.224.151.940.6
Mamba-130M NeoX10.5616.0744.335.364.548.024.351.944.7
Pythia-410MNeoX9.9510.8451.440.666.952.124.653.848.2
Mamba-370M NeoX8.288.1455.646.569.555.128.055.350.0
Pythia-1BNeoX7.827.9256.147.270.757.027.153.551.9
Mamba-790M NeoX7.336.0262.755.172.161.229.556.157.1
GPT-Neo 1.3BGPT2一7.5057.248.971.156.225.954.952.4
OPT-1.3BOPT6.6458.053.772.456.729.659.555.0
Pythia-1.4BNeoX7.516.0861.752.171.060.528.557.255.2
RWKV-1.5BNeoX7.707.0456.452.572.460.529.454.654.3
Mamba-1.4BNeoX6.805.0465.059.174.265.532.861.559.7
GPT-Neo 2.7BGPT25.6362.255.872.161.130.257.656.5
OPT-2.7BOPT5.1263.660.674.860.831.361.058.7
Pythia-2.8BNeoX6.735.0464.759.374.064.132.959.759.1
RWKV-3BNeoX7.005.2463.959.673.767.833.159.659.6
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Best results for each size in bold. We compare against open source LMs with", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 90, 505, 101 ], "spans": [ { "bbox": [ 106, 90, 505, 101 ], "score": 1.0, "content": "various tokenizers, trained for up to 300B tokens. Pile refers to the validation split, comparing only against models", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 97, 505, 113 ], "spans": [ { "bbox": [ 105, 97, 505, 113 ], "score": 1.0, "content": "trained on the same dataset and tokenizer (GPT-NeoX-20B). 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MODELPPL↓PPL↓TOKEN. PILE LAMBADA LAMBADA HellSwag PIQA Arc-E Arc-C WinoGrande Average ACC个ACC个ACC个ACC↑ACC↑ACC个ACC↑
Pythia-160MNeoX29.64 38.1033.030.261.443.224.151.940.6
Mamba-130M NeoX10.5616.0744.335.364.548.024.351.944.7
Pythia-410MNeoX9.9510.8451.440.666.952.124.653.848.2
Mamba-370M NeoX8.288.1455.646.569.555.128.055.350.0
Pythia-1BNeoX7.827.9256.147.270.757.027.153.551.9
Mamba-790M NeoX7.336.0262.755.172.161.229.556.157.1
GPT-Neo 1.3BGPT2一7.5057.248.971.156.225.954.952.4
OPT-1.3BOPT6.6458.053.772.456.729.659.555.0
Pythia-1.4BNeoX7.516.0861.752.171.060.528.557.255.2
RWKV-1.5BNeoX7.707.0456.452.572.460.529.454.654.3
Mamba-1.4BNeoX6.805.0465.059.174.265.532.861.559.7
GPT-Neo 2.7BGPT25.6362.255.872.161.130.257.656.5
OPT-2.7BOPT5.1263.660.674.860.831.361.058.7
Pythia-2.8BNeoX6.735.0464.759.374.064.132.959.759.1
RWKV-3BNeoX7.005.2463.959.673.767.833.159.659.6
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Pretraining on the HG38 (human genome) dataset. 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It can also be viewed as related to “fast weights” (Ba et al., 2016), which connects", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 137, 505, 149 ], "spans": [ { "bbox": [ 106, 137, 505, 149 ], "score": 1.0, "content": "classical RNNs with the mechanism of linear attention (Schlag et al., 2021). However, we believe", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 147, 305, 162 ], "spans": [ { "bbox": [ 105, 147, 305, 162 ], "score": 1.0, "content": "that it is a distinct concept that is worth clarifying.", "type": "text" } ], "index": 5 } ], "index": 3.5 }, { "type": "text", "bbox": [ 107, 175, 505, 229 ], "lines": [ { "bbox": [ 106, 176, 506, 187 ], "spans": [ { "bbox": [ 106, 176, 506, 187 ], "score": 1.0, "content": "Gating Gating originally referred to the gating mechanisms of RNNs such as the LSTM (Hochreiter", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 185, 505, 198 ], "spans": [ { "bbox": [ 105, 185, 505, 198 ], "score": 1.0, "content": "& Schmidhuber, 1997) and GRU (Chung et al., 2014), or the gated equation in Theorem 1. This was", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 196, 505, 208 ], "spans": [ { "bbox": [ 105, 196, 505, 208 ], "score": 1.0, "content": "interpreted as a particular mechanism for controlling whether to let an input into the hidden state of", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 207, 506, 220 ], "spans": [ { "bbox": [ 105, 207, 506, 220 ], "score": 1.0, "content": "an RNN. In particular, this affects the propagation of signal through time and causes inputs to interact", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 219, 256, 230 ], "spans": [ { "bbox": [ 106, 219, 256, 230 ], "score": 1.0, "content": "along the sequence length dimension.", "type": "text" } ], "index": 10 } ], "index": 8 }, { "type": "text", "bbox": [ 107, 234, 505, 299 ], "lines": [ { "bbox": [ 105, 233, 505, 247 ], "spans": [ { "bbox": [ 105, 233, 505, 247 ], "score": 1.0, "content": "However, the concept of gating has since been relaxed in popular usage to simply mean any", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 244, 505, 257 ], "spans": [ { "bbox": [ 105, 244, 505, 257 ], "score": 1.0, "content": "multiplicative interaction (often with an activation function). For example, elementwise multiplicative", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 255, 505, 267 ], "spans": [ { "bbox": [ 105, 255, 505, 267 ], "score": 1.0, "content": "components of neural network architectures (that do not interact along sequence length) are now", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 264, 505, 280 ], "spans": [ { "bbox": [ 105, 264, 505, 280 ], "score": 1.0, "content": "commonly referred to as gated architectures (Hua et al., 2022; Mehta et al., 2023), despite a very", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 275, 505, 291 ], "spans": [ { "bbox": [ 105, 275, 505, 291 ], "score": 1.0, "content": "different meaning than the original RNN sense. Thus we believe the original concept of RNN gating", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 287, 492, 301 ], "spans": [ { "bbox": [ 105, 287, 492, 301 ], "score": 1.0, "content": "versus the popular usage of multiplicative gating actually have a very different semantic meaning.", "type": "text" } ], "index": 16 } ], "index": 13.5 }, { "type": "text", "bbox": [ 107, 315, 505, 347 ], "lines": [ { "bbox": [ 106, 315, 505, 327 ], "spans": [ { "bbox": [ 106, 315, 505, 327 ], "score": 1.0, "content": "Hypernetworks Hypernetworks refer to neural networks whose parameters are themselves", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 326, 506, 338 ], "spans": [ { "bbox": [ 105, 326, 506, 338 ], "score": 1.0, "content": "generated by smaller neural networks. The original idea (Ha et al., 2017) used it in a narrow sense", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 336, 435, 349 ], "spans": [ { "bbox": [ 105, 336, 435, 349 ], "score": 1.0, "content": "to define a large RNN whose recurrent parameters are generated by a smaller RNN.", "type": "text" } ], "index": 19 } ], "index": 18 }, { "type": "text", "bbox": [ 108, 364, 502, 385 ], "lines": [ { "bbox": [ 105, 362, 504, 377 ], "spans": [ { "bbox": [ 105, 362, 504, 377 ], "score": 1.0, "content": "Data-dependence Similar to hypernetworks, data-dependence can refer to any notion where some", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 374, 352, 387 ], "spans": [ { "bbox": [ 105, 374, 352, 387 ], "score": 1.0, "content": "parameters of the model depend on the data (Poli et al., 2023).", "type": "text" } ], "index": 21 } ], "index": 20.5 }, { "type": "text", "bbox": [ 108, 390, 505, 434 ], "lines": [ { "bbox": [ 105, 389, 505, 402 ], "spans": [ { "bbox": [ 105, 389, 442, 402 ], "score": 1.0, "content": "To illustrate the issues with these concepts, consider a simple diagonal linear layer", "type": "text" }, { "bbox": [ 442, 390, 474, 401 ], "score": 0.92, "content": "y = D x", "type": "inline_equation" }, { "bbox": [ 474, 389, 505, 402 ], "score": 1.0, "content": ", where", "type": "text" } ], "index": 22 }, { "bbox": [ 107, 401, 505, 412 ], "spans": [ { "bbox": [ 107, 401, 116, 411 ], "score": 0.81, "content": "D", "type": "inline_equation" }, { "bbox": [ 117, 401, 271, 412 ], "score": 1.0, "content": "is a diagonal matrix. Now suppose that", "type": "text" }, { "bbox": [ 271, 401, 281, 411 ], "score": 0.83, "content": "D", "type": "inline_equation" }, { "bbox": [ 281, 401, 475, 412 ], "score": 1.0, "content": "is itself generated from a linear transformation of", "type": "text" }, { "bbox": [ 475, 403, 482, 411 ], "score": 0.74, "content": "x", "type": "inline_equation" }, { "bbox": [ 482, 401, 505, 412 ], "score": 1.0, "content": ", with", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 411, 505, 424 ], "spans": [ { "bbox": [ 105, 411, 204, 424 ], "score": 1.0, "content": "an optional nonlinearity:", "type": "text" }, { "bbox": [ 205, 412, 254, 424 ], "score": 0.93, "content": "D = \\sigma ( W x )", "type": "inline_equation" }, { "bbox": [ 254, 411, 505, 424 ], "score": 1.0, "content": ". Since it is diagonal, the multiplication becomes an elementwise", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 422, 205, 435 ], "spans": [ { "bbox": [ 106, 423, 142, 435 ], "score": 1.0, "content": "product:", "type": "text" }, { "bbox": [ 142, 422, 201, 434 ], "score": 0.93, "content": "y = \\sigma ( W x ) \\circ \\dot { x }", "type": "inline_equation" }, { "bbox": [ 201, 423, 205, 435 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 25 } ], "index": 23.5 }, { "type": "text", "bbox": [ 107, 438, 505, 492 ], "lines": [ { "bbox": [ 105, 437, 505, 450 ], "spans": [ { "bbox": [ 105, 437, 505, 450 ], "score": 1.0, "content": "This is a rather trivial transformation, yet it technically satisfies the common meanings of gating (since", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 448, 505, 461 ], "spans": [ { "bbox": [ 105, 448, 376, 461 ], "score": 1.0, "content": "it has a multiplicative “branch”), hypernetworks (since the parameter", "type": "text" }, { "bbox": [ 376, 449, 385, 459 ], "score": 0.79, "content": "D", "type": "inline_equation" }, { "bbox": [ 386, 448, 505, 461 ], "score": 1.0, "content": "is generated by another layer),", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 459, 505, 471 ], "spans": [ { "bbox": [ 105, 459, 215, 471 ], "score": 1.0, "content": "and data-dependent (since", "type": "text" }, { "bbox": [ 216, 460, 225, 469 ], "score": 0.77, "content": "D", "type": "inline_equation" }, { "bbox": [ 226, 459, 309, 471 ], "score": 1.0, "content": "depends on the data", "type": "text" }, { "bbox": [ 310, 462, 317, 470 ], "score": 0.75, "content": "x", "type": "inline_equation" }, { "bbox": [ 317, 459, 505, 471 ], "score": 1.0, "content": "). However, this in fact simply defines a GLU", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 469, 506, 483 ], "spans": [ { "bbox": [ 105, 469, 506, 483 ], "score": 1.0, "content": "function, which is so simple that it is often considered just an activation function (Dauphin et al., 2017;", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 480, 286, 494 ], "spans": [ { "bbox": [ 105, 480, 286, 494 ], "score": 1.0, "content": "Shazeer, 2020) instead of a meaningful layer.", "type": "text" } ], "index": 30 } ], "index": 28 }, { "type": "text", "bbox": [ 108, 497, 505, 540 ], "lines": [ { "bbox": [ 106, 496, 506, 509 ], "spans": [ { "bbox": [ 106, 496, 506, 509 ], "score": 1.0, "content": "Thus, while selection mechanisms could be considered a special case of ideas such as architectural gat-", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 507, 505, 519 ], "spans": [ { "bbox": [ 105, 507, 505, 519 ], "score": 1.0, "content": "ing, hypernetworks, or data-dependence, so can an enormous range of other constructions—essentially", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 518, 506, 529 ], "spans": [ { "bbox": [ 105, 518, 506, 529 ], "score": 1.0, "content": "anything with a multiplication, including standard attention mechanisms (Bahdanau et al., 2014;", "type": "text" } ], "index": 33 }, { "bbox": [ 107, 529, 438, 540 ], "spans": [ { "bbox": [ 107, 529, 438, 540 ], "score": 1.0, "content": "Vaswani et al., 2017) as well—and we find it uninformative to think of them as such.", "type": "text" } ], "index": 34 } ], "index": 32.5 }, { "type": "text", "bbox": [ 107, 545, 505, 631 ], "lines": [ { "bbox": [ 105, 544, 506, 557 ], "spans": [ { "bbox": [ 105, 544, 506, 557 ], "score": 1.0, "content": "Instead, we view it as most closely related to the gating mechanism of traditional RNNs (LSTM and", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 554, 506, 568 ], "spans": [ { "bbox": [ 105, 554, 506, 568 ], "score": 1.0, "content": "GRU), which is a special case (Theorem 1) and also has a deeper history of connections to SSMs", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 566, 506, 578 ], "spans": [ { "bbox": [ 105, 566, 322, 578 ], "score": 1.0, "content": "through variable (input-dependent) discretization of", "type": "text" }, { "bbox": [ 323, 567, 332, 576 ], "score": 0.77, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 333, 566, 506, 578 ], "score": 1.0, "content": "(Funahashi & Nakamura, 1993; Tallec &", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 576, 505, 590 ], "spans": [ { "bbox": [ 106, 576, 505, 590 ], "score": 1.0, "content": "Ollivier, 2018; Gu et al., 2020a). We also eschew the term “gating” in favor of selection to clarify the", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 587, 505, 599 ], "spans": [ { "bbox": [ 105, 587, 505, 599 ], "score": 1.0, "content": "overloaded use of former. More narrowly, we use selection to refer to the mechanistic action of a model", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 599, 504, 609 ], "spans": [ { "bbox": [ 106, 599, 504, 609 ], "score": 1.0, "content": "to select or ignore inputs and facilitate data interaction along the sequence length (Section 3.1). Beyond", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 608, 505, 622 ], "spans": [ { "bbox": [ 105, 608, 505, 622 ], "score": 1.0, "content": "selective SSMs and gated RNNs, other examples may include input-dependent convolutions (Yang", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 619, 309, 632 ], "spans": [ { "bbox": [ 105, 619, 309, 632 ], "score": 1.0, "content": "et al., 2019; Kosma et al., 2023) and even attention.", "type": "text" } ], "index": 42 } ], "index": 38.5 }, { "type": "title", "bbox": [ 108, 648, 212, 661 ], "lines": [ { "bbox": [ 105, 647, 214, 663 ], "spans": [ { "bbox": [ 105, 647, 214, 663 ], "score": 1.0, "content": "B RELATED WORK", "type": "text" } ], "index": 43 } ], "index": 43 }, { "type": "text", "bbox": [ 109, 667, 327, 678 ], "lines": [ { "bbox": [ 107, 667, 329, 680 ], "spans": [ { "bbox": [ 107, 667, 329, 680 ], "score": 1.0, "content": "We highlight several prior works related to our methods.", "type": "text" } ], "index": 44 } ], "index": 44 }, { "type": "title", "bbox": [ 108, 694, 271, 705 ], "lines": [ { "bbox": [ 105, 693, 273, 707 ], "spans": [ { "bbox": [ 105, 693, 273, 707 ], "score": 1.0, "content": "B.1 S4 VARIANTS AND DERIVATIVES", "type": "text" } ], "index": 45 } ], "index": 45 }, { "type": "text", "bbox": [ 107, 710, 504, 732 ], "lines": [ { "bbox": [ 105, 709, 505, 723 ], "spans": [ { "bbox": [ 105, 709, 505, 723 ], "score": 1.0, "content": "We describe a brief overview of some structured SSMs from past work, particularly those that have", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 721, 205, 731 ], "spans": [ { "bbox": [ 105, 721, 205, 731 ], "score": 1.0, "content": "a relation to our method.", "type": "text" } ], "index": 47 } ], "index": 46.5 } ], "page_idx": 14, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 304, 37 ], "lines": [ { "bbox": [ 106, 26, 304, 38 ], "spans": [ { "bbox": [ 106, 26, 304, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 310, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "", "type": "text", "height": 14, "width": 13 } ] } ] } ], "para_blocks": [ { "type": "title", "bbox": [ 108, 81, 193, 94 ], "lines": [ { "bbox": [ 105, 80, 195, 96 ], "spans": [ { "bbox": [ 105, 80, 195, 96 ], "score": 1.0, "content": "A DISCUSSION", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "title", "bbox": [ 107, 100, 362, 111 ], "lines": [ { "bbox": [ 106, 100, 363, 113 ], "spans": [ { "bbox": [ 106, 100, 363, 113 ], "score": 1.0, "content": "A.1 RELATED CONCEPTS TO THE SELECTION MECHANISM", "type": "text" } ], "index": 1 } ], "index": 1 }, { "type": "text", "bbox": [ 107, 116, 505, 159 ], "lines": [ { "bbox": [ 105, 115, 505, 128 ], "spans": [ { "bbox": [ 105, 115, 505, 128 ], "score": 1.0, "content": "Our selection mechanism is inspired by and related to concepts such as gating, hypernetworks, and", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 126, 506, 139 ], "spans": [ { "bbox": [ 105, 126, 506, 139 ], "score": 1.0, "content": "data-dependence. It can also be viewed as related to “fast weights” (Ba et al., 2016), which connects", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 137, 505, 149 ], "spans": [ { "bbox": [ 106, 137, 505, 149 ], "score": 1.0, "content": "classical RNNs with the mechanism of linear attention (Schlag et al., 2021). However, we believe", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 147, 305, 162 ], "spans": [ { "bbox": [ 105, 147, 305, 162 ], "score": 1.0, "content": "that it is a distinct concept that is worth clarifying.", "type": "text" } ], "index": 5 } ], "index": 3.5, "bbox_fs": [ 105, 115, 506, 162 ] }, { "type": "text", "bbox": [ 107, 175, 505, 229 ], "lines": [ { "bbox": [ 106, 176, 506, 187 ], "spans": [ { "bbox": [ 106, 176, 506, 187 ], "score": 1.0, "content": "Gating Gating originally referred to the gating mechanisms of RNNs such as the LSTM (Hochreiter", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 185, 505, 198 ], "spans": [ { "bbox": [ 105, 185, 505, 198 ], "score": 1.0, "content": "& Schmidhuber, 1997) and GRU (Chung et al., 2014), or the gated equation in Theorem 1. This was", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 196, 505, 208 ], "spans": [ { "bbox": [ 105, 196, 505, 208 ], "score": 1.0, "content": "interpreted as a particular mechanism for controlling whether to let an input into the hidden state of", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 207, 506, 220 ], "spans": [ { "bbox": [ 105, 207, 506, 220 ], "score": 1.0, "content": "an RNN. In particular, this affects the propagation of signal through time and causes inputs to interact", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 219, 256, 230 ], "spans": [ { "bbox": [ 106, 219, 256, 230 ], "score": 1.0, "content": "along the sequence length dimension.", "type": "text" } ], "index": 10 } ], "index": 8, "bbox_fs": [ 105, 176, 506, 230 ] }, { "type": "text", "bbox": [ 107, 234, 505, 299 ], "lines": [ { "bbox": [ 105, 233, 505, 247 ], "spans": [ { "bbox": [ 105, 233, 505, 247 ], "score": 1.0, "content": "However, the concept of gating has since been relaxed in popular usage to simply mean any", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 244, 505, 257 ], "spans": [ { "bbox": [ 105, 244, 505, 257 ], "score": 1.0, "content": "multiplicative interaction (often with an activation function). For example, elementwise multiplicative", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 255, 505, 267 ], "spans": [ { "bbox": [ 105, 255, 505, 267 ], "score": 1.0, "content": "components of neural network architectures (that do not interact along sequence length) are now", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 264, 505, 280 ], "spans": [ { "bbox": [ 105, 264, 505, 280 ], "score": 1.0, "content": "commonly referred to as gated architectures (Hua et al., 2022; Mehta et al., 2023), despite a very", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 275, 505, 291 ], "spans": [ { "bbox": [ 105, 275, 505, 291 ], "score": 1.0, "content": "different meaning than the original RNN sense. Thus we believe the original concept of RNN gating", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 287, 492, 301 ], "spans": [ { "bbox": [ 105, 287, 492, 301 ], "score": 1.0, "content": "versus the popular usage of multiplicative gating actually have a very different semantic meaning.", "type": "text" } ], "index": 16 } ], "index": 13.5, "bbox_fs": [ 105, 233, 505, 301 ] }, { "type": "text", "bbox": [ 107, 315, 505, 347 ], "lines": [ { "bbox": [ 106, 315, 505, 327 ], "spans": [ { "bbox": [ 106, 315, 505, 327 ], "score": 1.0, "content": "Hypernetworks Hypernetworks refer to neural networks whose parameters are themselves", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 326, 506, 338 ], "spans": [ { "bbox": [ 105, 326, 506, 338 ], "score": 1.0, "content": "generated by smaller neural networks. The original idea (Ha et al., 2017) used it in a narrow sense", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 336, 435, 349 ], "spans": [ { "bbox": [ 105, 336, 435, 349 ], "score": 1.0, "content": "to define a large RNN whose recurrent parameters are generated by a smaller RNN.", "type": "text" } ], "index": 19 } ], "index": 18, "bbox_fs": [ 105, 315, 506, 349 ] }, { "type": "text", "bbox": [ 108, 364, 502, 385 ], "lines": [ { "bbox": [ 105, 362, 504, 377 ], "spans": [ { "bbox": [ 105, 362, 504, 377 ], "score": 1.0, "content": "Data-dependence Similar to hypernetworks, data-dependence can refer to any notion where some", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 374, 352, 387 ], "spans": [ { "bbox": [ 105, 374, 352, 387 ], "score": 1.0, "content": "parameters of the model depend on the data (Poli et al., 2023).", "type": "text" } ], "index": 21 } ], "index": 20.5, "bbox_fs": [ 105, 362, 504, 387 ] }, { "type": "text", "bbox": [ 108, 390, 505, 434 ], "lines": [ { "bbox": [ 105, 389, 505, 402 ], "spans": [ { "bbox": [ 105, 389, 442, 402 ], "score": 1.0, "content": "To illustrate the issues with these concepts, consider a simple diagonal linear layer", "type": "text" }, { "bbox": [ 442, 390, 474, 401 ], "score": 0.92, "content": "y = D x", "type": "inline_equation" }, { "bbox": [ 474, 389, 505, 402 ], "score": 1.0, "content": ", where", "type": "text" } ], "index": 22 }, { "bbox": [ 107, 401, 505, 412 ], "spans": [ { "bbox": [ 107, 401, 116, 411 ], "score": 0.81, "content": "D", "type": "inline_equation" }, { "bbox": [ 117, 401, 271, 412 ], "score": 1.0, "content": "is a diagonal matrix. Now suppose that", "type": "text" }, { "bbox": [ 271, 401, 281, 411 ], "score": 0.83, "content": "D", "type": "inline_equation" }, { "bbox": [ 281, 401, 475, 412 ], "score": 1.0, "content": "is itself generated from a linear transformation of", "type": "text" }, { "bbox": [ 475, 403, 482, 411 ], "score": 0.74, "content": "x", "type": "inline_equation" }, { "bbox": [ 482, 401, 505, 412 ], "score": 1.0, "content": ", with", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 411, 505, 424 ], "spans": [ { "bbox": [ 105, 411, 204, 424 ], "score": 1.0, "content": "an optional nonlinearity:", "type": "text" }, { "bbox": [ 205, 412, 254, 424 ], "score": 0.93, "content": "D = \\sigma ( W x )", "type": "inline_equation" }, { "bbox": [ 254, 411, 505, 424 ], "score": 1.0, "content": ". Since it is diagonal, the multiplication becomes an elementwise", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 422, 205, 435 ], "spans": [ { "bbox": [ 106, 423, 142, 435 ], "score": 1.0, "content": "product:", "type": "text" }, { "bbox": [ 142, 422, 201, 434 ], "score": 0.93, "content": "y = \\sigma ( W x ) \\circ \\dot { x }", "type": "inline_equation" }, { "bbox": [ 201, 423, 205, 435 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 25 } ], "index": 23.5, "bbox_fs": [ 105, 389, 505, 435 ] }, { "type": "text", "bbox": [ 107, 438, 505, 492 ], "lines": [ { "bbox": [ 105, 437, 505, 450 ], "spans": [ { "bbox": [ 105, 437, 505, 450 ], "score": 1.0, "content": "This is a rather trivial transformation, yet it technically satisfies the common meanings of gating (since", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 448, 505, 461 ], "spans": [ { "bbox": [ 105, 448, 376, 461 ], "score": 1.0, "content": "it has a multiplicative “branch”), hypernetworks (since the parameter", "type": "text" }, { "bbox": [ 376, 449, 385, 459 ], "score": 0.79, "content": "D", "type": "inline_equation" }, { "bbox": [ 386, 448, 505, 461 ], "score": 1.0, "content": "is generated by another layer),", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 459, 505, 471 ], "spans": [ { "bbox": [ 105, 459, 215, 471 ], "score": 1.0, "content": "and data-dependent (since", "type": "text" }, { "bbox": [ 216, 460, 225, 469 ], "score": 0.77, "content": "D", "type": "inline_equation" }, { "bbox": [ 226, 459, 309, 471 ], "score": 1.0, "content": "depends on the data", "type": "text" }, { "bbox": [ 310, 462, 317, 470 ], "score": 0.75, "content": "x", "type": "inline_equation" }, { "bbox": [ 317, 459, 505, 471 ], "score": 1.0, "content": "). However, this in fact simply defines a GLU", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 469, 506, 483 ], "spans": [ { "bbox": [ 105, 469, 506, 483 ], "score": 1.0, "content": "function, which is so simple that it is often considered just an activation function (Dauphin et al., 2017;", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 480, 286, 494 ], "spans": [ { "bbox": [ 105, 480, 286, 494 ], "score": 1.0, "content": "Shazeer, 2020) instead of a meaningful layer.", "type": "text" } ], "index": 30 } ], "index": 28, "bbox_fs": [ 105, 437, 506, 494 ] }, { "type": "text", "bbox": [ 108, 497, 505, 540 ], "lines": [ { "bbox": [ 106, 496, 506, 509 ], "spans": [ { "bbox": [ 106, 496, 506, 509 ], "score": 1.0, "content": "Thus, while selection mechanisms could be considered a special case of ideas such as architectural gat-", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 507, 505, 519 ], "spans": [ { "bbox": [ 105, 507, 505, 519 ], "score": 1.0, "content": "ing, hypernetworks, or data-dependence, so can an enormous range of other constructions—essentially", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 518, 506, 529 ], "spans": [ { "bbox": [ 105, 518, 506, 529 ], "score": 1.0, "content": "anything with a multiplication, including standard attention mechanisms (Bahdanau et al., 2014;", "type": "text" } ], "index": 33 }, { "bbox": [ 107, 529, 438, 540 ], "spans": [ { "bbox": [ 107, 529, 438, 540 ], "score": 1.0, "content": "Vaswani et al., 2017) as well—and we find it uninformative to think of them as such.", "type": "text" } ], "index": 34 } ], "index": 32.5, "bbox_fs": [ 105, 496, 506, 540 ] }, { "type": "text", "bbox": [ 107, 545, 505, 631 ], "lines": [ { "bbox": [ 105, 544, 506, 557 ], "spans": [ { "bbox": [ 105, 544, 506, 557 ], "score": 1.0, "content": "Instead, we view it as most closely related to the gating mechanism of traditional RNNs (LSTM and", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 554, 506, 568 ], "spans": [ { "bbox": [ 105, 554, 506, 568 ], "score": 1.0, "content": "GRU), which is a special case (Theorem 1) and also has a deeper history of connections to SSMs", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 566, 506, 578 ], "spans": [ { "bbox": [ 105, 566, 322, 578 ], "score": 1.0, "content": "through variable (input-dependent) discretization of", "type": "text" }, { "bbox": [ 323, 567, 332, 576 ], "score": 0.77, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 333, 566, 506, 578 ], "score": 1.0, "content": "(Funahashi & Nakamura, 1993; Tallec &", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 576, 505, 590 ], "spans": [ { "bbox": [ 106, 576, 505, 590 ], "score": 1.0, "content": "Ollivier, 2018; Gu et al., 2020a). We also eschew the term “gating” in favor of selection to clarify the", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 587, 505, 599 ], "spans": [ { "bbox": [ 105, 587, 505, 599 ], "score": 1.0, "content": "overloaded use of former. More narrowly, we use selection to refer to the mechanistic action of a model", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 599, 504, 609 ], "spans": [ { "bbox": [ 106, 599, 504, 609 ], "score": 1.0, "content": "to select or ignore inputs and facilitate data interaction along the sequence length (Section 3.1). Beyond", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 608, 505, 622 ], "spans": [ { "bbox": [ 105, 608, 505, 622 ], "score": 1.0, "content": "selective SSMs and gated RNNs, other examples may include input-dependent convolutions (Yang", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 619, 309, 632 ], "spans": [ { "bbox": [ 105, 619, 309, 632 ], "score": 1.0, "content": "et al., 2019; Kosma et al., 2023) and even attention.", "type": "text" } ], "index": 42 } ], "index": 38.5, "bbox_fs": [ 105, 544, 506, 632 ] }, { "type": "title", "bbox": [ 108, 648, 212, 661 ], "lines": [ { "bbox": [ 105, 647, 214, 663 ], "spans": [ { "bbox": [ 105, 647, 214, 663 ], "score": 1.0, "content": "B RELATED WORK", "type": "text" } ], "index": 43 } ], "index": 43 }, { "type": "text", "bbox": [ 109, 667, 327, 678 ], "lines": [ { "bbox": [ 107, 667, 329, 680 ], "spans": [ { "bbox": [ 107, 667, 329, 680 ], "score": 1.0, "content": "We highlight several prior works related to our methods.", "type": "text" } ], "index": 44 } ], "index": 44, "bbox_fs": [ 107, 667, 329, 680 ] }, { "type": "title", "bbox": [ 108, 694, 271, 705 ], "lines": [ { "bbox": [ 105, 693, 273, 707 ], "spans": [ { "bbox": [ 105, 693, 273, 707 ], "score": 1.0, "content": "B.1 S4 VARIANTS AND DERIVATIVES", "type": "text" } ], "index": 45 } ], "index": 45 }, { "type": "text", "bbox": [ 107, 710, 504, 732 ], "lines": [ { "bbox": [ 105, 709, 505, 723 ], "spans": [ { "bbox": [ 105, 709, 505, 723 ], "score": 1.0, "content": "We describe a brief overview of some structured SSMs from past work, particularly those that have", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 721, 205, 731 ], "spans": [ { "bbox": [ 105, 721, 205, 731 ], "score": 1.0, "content": "a relation to our method.", "type": "text" } ], "index": 47 } ], "index": 46.5, "bbox_fs": [ 105, 709, 505, 731 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 82, 506, 320 ], "lines": [ { "bbox": [ 105, 81, 505, 95 ], "spans": [ { "bbox": [ 105, 81, 505, 95 ], "score": 1.0, "content": "• S4 (Gu et al., 2021; 2022a) introduced the first structured SSM, describing diagonal structure and", "type": "text" } ], "index": 0 }, { "bbox": [ 114, 93, 505, 105 ], "spans": [ { "bbox": [ 114, 93, 505, 105 ], "score": 1.0, "content": "diagonal plus low-rank (DPLR). It focused on efficient convolutional algorithms for DPLR SSMs", "type": "text" } ], "index": 1 }, { "bbox": [ 113, 104, 464, 116 ], "spans": [ { "bbox": [ 113, 104, 464, 116 ], "score": 1.0, "content": "due to a connection to continuous-time online memorization (HIPPO) (Gu et al., 2020a).", "type": "text" } ], "index": 2 }, { "bbox": [ 107, 116, 505, 129 ], "spans": [ { "bbox": [ 107, 116, 505, 129 ], "score": 1.0, "content": "• DSS (Gupta, 2022) first discovered the empirical effectiveness of diagonal structured SSMs by", "type": "text" } ], "index": 3 }, { "bbox": [ 112, 127, 506, 140 ], "spans": [ { "bbox": [ 112, 127, 506, 140 ], "score": 1.0, "content": "approximating the HIPPO initialization. This was expanded on theoretically in S4D (Gu et al., 2022b).", "type": "text" } ], "index": 4 }, { "bbox": [ 107, 140, 505, 153 ], "spans": [ { "bbox": [ 107, 140, 505, 153 ], "score": 1.0, "content": "• S5 (Smith et al., 2023) independently discovered the diagonal SSM approximation, and is the first", "type": "text" } ], "index": 5 }, { "bbox": [ 114, 151, 505, 164 ], "spans": [ { "bbox": [ 114, 151, 505, 164 ], "score": 1.0, "content": "S4 model to be computed recurrently with the parallel scan. However, this required lowering the", "type": "text" } ], "index": 6 }, { "bbox": [ 114, 162, 505, 174 ], "spans": [ { "bbox": [ 114, 162, 505, 174 ], "score": 1.0, "content": "effective state dimension, which they accomplished by switching the SSM dimensions from a SISO", "type": "text" } ], "index": 7 }, { "bbox": [ 114, 173, 423, 185 ], "spans": [ { "bbox": [ 114, 173, 423, 185 ], "score": 1.0, "content": "(single-input single-output) to MIMO (multi-input multi-output) formulation.", "type": "text" } ], "index": 8 }, { "bbox": [ 107, 185, 506, 198 ], "spans": [ { "bbox": [ 107, 185, 506, 198 ], "score": 1.0, "content": "• Mega (Ma et al., 2023) introduced a simplification of S4 to be real- instead of complex- valued,", "type": "text" } ], "index": 9 }, { "bbox": [ 114, 196, 505, 209 ], "spans": [ { "bbox": [ 114, 196, 505, 209 ], "score": 1.0, "content": "giving it an interpretation of being an exponential moving average (EMA). They additionally make", "type": "text" } ], "index": 10 }, { "bbox": [ 113, 206, 505, 220 ], "spans": [ { "bbox": [ 113, 206, 505, 220 ], "score": 1.0, "content": "an interesting connection of the discretization step of SSMs to an EMA damping term. Contrary", "type": "text" } ], "index": 11 }, { "bbox": [ 113, 217, 505, 231 ], "spans": [ { "bbox": [ 113, 217, 505, 231 ], "score": 1.0, "content": "to findings in the original S4 papers, this was the first model to show that real-valued SSMs are", "type": "text" } ], "index": 12 }, { "bbox": [ 114, 229, 505, 241 ], "spans": [ { "bbox": [ 114, 229, 505, 241 ], "score": 1.0, "content": "empirically effective in certain settings or when combined with different architectural components.", "type": "text" } ], "index": 13 }, { "bbox": [ 107, 241, 505, 254 ], "spans": [ { "bbox": [ 107, 241, 505, 254 ], "score": 1.0, "content": "• Liquid S4 (Hasani et al., 2023) is also motivated by augmenting S4 with an input-dependent state", "type": "text" } ], "index": 14 }, { "bbox": [ 114, 253, 505, 264 ], "spans": [ { "bbox": [ 114, 253, 505, 264 ], "score": 1.0, "content": "transition. From this perspective it shares similarity to selection mechanisms, although in a limited", "type": "text" } ], "index": 15 }, { "bbox": [ 114, 263, 361, 275 ], "spans": [ { "bbox": [ 114, 263, 361, 275 ], "score": 1.0, "content": "form which is still computed convolutionally and close to LTI.", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 274, 506, 289 ], "spans": [ { "bbox": [ 106, 274, 506, 289 ], "score": 1.0, "content": "• SGConv (Li et al., 2023), Hyena (Poli et al., 2023), LongConv (Fu et al., 2023), MultiresConv (Shi", "type": "text" } ], "index": 17 }, { "bbox": [ 114, 287, 505, 298 ], "spans": [ { "bbox": [ 114, 287, 505, 298 ], "score": 1.0, "content": "et al., 2023b) all focus on the convolutional representation of S4 and create global or long convolution", "type": "text" } ], "index": 18 }, { "bbox": [ 114, 297, 505, 311 ], "spans": [ { "bbox": [ 114, 297, 505, 311 ], "score": 1.0, "content": "kernels with different parameterizations. However, these methods cannot do fast autoregressive", "type": "text" } ], "index": 19 }, { "bbox": [ 114, 308, 189, 320 ], "spans": [ { "bbox": [ 114, 308, 189, 320 ], "score": 1.0, "content": "inference directly.", "type": "text" } ], "index": 20 } ], "index": 10 }, { "type": "text", "bbox": [ 107, 325, 504, 347 ], "lines": [ { "bbox": [ 105, 324, 505, 337 ], "spans": [ { "bbox": [ 105, 324, 505, 337 ], "score": 1.0, "content": "Notably, all of these methods, and all other structured SSMs that we are aware of, have been", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 335, 346, 348 ], "spans": [ { "bbox": [ 105, 335, 346, 348 ], "score": 1.0, "content": "non-selective and usually strictly LTI (linear time invariant).", "type": "text" } ], "index": 22 } ], "index": 21.5 }, { "type": "title", "bbox": [ 107, 356, 230, 367 ], "lines": [ { "bbox": [ 105, 355, 231, 369 ], "spans": [ { "bbox": [ 105, 355, 231, 369 ], "score": 1.0, "content": "B.2 SSM ARCHITECTURES", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 107, 372, 504, 394 ], "lines": [ { "bbox": [ 106, 371, 505, 384 ], "spans": [ { "bbox": [ 106, 371, 505, 384 ], "score": 1.0, "content": "We use SSM architectures or state space neural networks (SSNN) to refer to deep neural network", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 382, 399, 395 ], "spans": [ { "bbox": [ 106, 382, 399, 395 ], "score": 1.0, "content": "architectures incorporating one of the previous SSMs as a black box layer.", "type": "text" } ], "index": 25 } ], "index": 24.5 }, { "type": "text", "bbox": [ 106, 400, 506, 657 ], "lines": [ { "bbox": [ 107, 399, 505, 411 ], "spans": [ { "bbox": [ 107, 399, 505, 411 ], "score": 1.0, "content": "• GSS (Mehta et al., 2023) was the first gated neural network architecture incorporating SSMs. It is", "type": "text" } ], "index": 26 }, { "bbox": [ 114, 410, 505, 421 ], "spans": [ { "bbox": [ 114, 410, 505, 421 ], "score": 1.0, "content": "motivated by the gated attention unit (GAU) of Hua et al. (2022) and looks quite similar to our block,", "type": "text" } ], "index": 27 }, { "bbox": [ 114, 420, 505, 432 ], "spans": [ { "bbox": [ 114, 420, 505, 432 ], "score": 1.0, "content": "except with additional projections. Most importantly, its projection contracts the model dimension", "type": "text" } ], "index": 28 }, { "bbox": [ 114, 431, 505, 443 ], "spans": [ { "bbox": [ 114, 431, 505, 443 ], "score": 1.0, "content": "to reduce the state size of the SSM, while ours expands the model dimension in order to increase", "type": "text" } ], "index": 29 }, { "bbox": [ 114, 442, 323, 453 ], "spans": [ { "bbox": [ 114, 442, 323, 453 ], "score": 1.0, "content": "the state size, based on the motivation in Section 3.1.", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 454, 505, 467 ], "spans": [ { "bbox": [ 106, 454, 505, 467 ], "score": 1.0, "content": "• Mega (Ma et al., 2023) combined the EMA simplification of S4 described above into a hybrid", "type": "text" } ], "index": 31 }, { "bbox": [ 112, 466, 333, 478 ], "spans": [ { "bbox": [ 112, 466, 333, 478 ], "score": 1.0, "content": "architecture using an efficient attention approximation.", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 477, 506, 491 ], "spans": [ { "bbox": [ 105, 477, 506, 491 ], "score": 1.0, "content": "• H3 (Dao et al., 2023) is motivated by combining S4 with linear attention (Katharopoulos et al., 2020).", "type": "text" } ], "index": 33 }, { "bbox": [ 113, 488, 505, 502 ], "spans": [ { "bbox": [ 113, 488, 505, 502 ], "score": 1.0, "content": "It is the first to generalize this formulation of linear attention to more general recurrences, which", "type": "text" } ], "index": 34 }, { "bbox": [ 113, 500, 264, 512 ], "spans": [ { "bbox": [ 113, 500, 264, 512 ], "score": 1.0, "content": "is also the basis of later architectures.", "type": "text" } ], "index": 35 }, { "bbox": [ 107, 513, 505, 525 ], "spans": [ { "bbox": [ 107, 513, 505, 525 ], "score": 1.0, "content": "• Selective S4 (Wang et al., 2023) incorporates S4 as a black box to generate a binary mask which", "type": "text" } ], "index": 36 }, { "bbox": [ 114, 523, 505, 535 ], "spans": [ { "bbox": [ 114, 523, 505, 535 ], "score": 1.0, "content": "is multiplied on the input. While sharing the “selection” name, we consider this an architectural", "type": "text" } ], "index": 37 }, { "bbox": [ 114, 534, 505, 546 ], "spans": [ { "bbox": [ 114, 534, 505, 546 ], "score": 1.0, "content": "modification that is closer to architectural gating than a selection mechanism (Appendix A.1). For", "type": "text" } ], "index": 38 }, { "bbox": [ 114, 544, 506, 558 ], "spans": [ { "bbox": [ 114, 544, 506, 558 ], "score": 1.0, "content": "example, we hypothesize that it would not solve the Selective Copying task because simply masking", "type": "text" } ], "index": 39 }, { "bbox": [ 114, 555, 506, 568 ], "spans": [ { "bbox": [ 114, 555, 506, 568 ], "score": 1.0, "content": "out the irrelevant inputs does not affect the spacing between the relevant ones (indeed, the Selective", "type": "text" } ], "index": 40 }, { "bbox": [ 114, 567, 490, 578 ], "spans": [ { "bbox": [ 114, 567, 490, 578 ], "score": 1.0, "content": "Copying task can even be viewed as coming pre-masked if the noise tokens are embedded to 0).", "type": "text" } ], "index": 41 }, { "bbox": [ 108, 579, 505, 591 ], "spans": [ { "bbox": [ 108, 579, 505, 591 ], "score": 1.0, "content": "• RetNet (Sun et al., 2023) is also based on Linear Attention and very similar to H3, but reduces the inner", "type": "text" } ], "index": 42 }, { "bbox": [ 113, 590, 506, 603 ], "spans": [ { "bbox": [ 113, 590, 322, 603 ], "score": 1.0, "content": "S4 layer to a special case where the state dimension is", "type": "text" }, { "bbox": [ 322, 591, 348, 601 ], "score": 0.9, "content": "N = 1", "type": "inline_equation" }, { "bbox": [ 348, 590, 506, 603 ], "score": 1.0, "content": ". This simplification leads to an alternate", "type": "text" } ], "index": 43 }, { "bbox": [ 113, 601, 507, 613 ], "spans": [ { "bbox": [ 113, 601, 507, 613 ], "score": 1.0, "content": "way to parallelize the computation with a variant of standard multi-head attention instead of convo-", "type": "text" } ], "index": 44 }, { "bbox": [ 113, 611, 506, 624 ], "spans": [ { "bbox": [ 113, 611, 506, 624 ], "score": 1.0, "content": "lutions. Although not framed as such, its recurrence can be viewed as a special case of a linear SSM.", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 624, 505, 637 ], "spans": [ { "bbox": [ 105, 624, 505, 637 ], "score": 1.0, "content": "• RWKV (Peng et al., 2023) is another recent RNN designed for language modeling. It is based on", "type": "text" } ], "index": 46 }, { "bbox": [ 113, 635, 505, 648 ], "spans": [ { "bbox": [ 113, 635, 505, 648 ], "score": 1.0, "content": "AFT (attention-free Transformer (Zhai et al., 2021)), another variant of linear attention. Its main", "type": "text" } ], "index": 47 }, { "bbox": [ 113, 645, 460, 658 ], "spans": [ { "bbox": [ 113, 645, 460, 658 ], "score": 1.0, "content": "“WKV” mechanism involves LTI recurrences and can be seen as the ratio of two SSMs.", "type": "text" } ], "index": 48 } ], "index": 37 }, { "type": "text", "bbox": [ 107, 663, 505, 696 ], "lines": [ { "bbox": [ 106, 663, 505, 675 ], "spans": [ { "bbox": [ 106, 663, 505, 675 ], "score": 1.0, "content": "We also highlight the gated attention unit (GAU) from Hua et al. (2022), which was motivated", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 673, 506, 687 ], "spans": [ { "bbox": [ 105, 673, 506, 687 ], "score": 1.0, "content": "by combining the Transformer’s MHA and MLP blocks together and was an inspiration for our", "type": "text" } ], "index": 50 }, { "bbox": [ 106, 685, 351, 697 ], "spans": [ { "bbox": [ 106, 685, 351, 697 ], "score": 1.0, "content": "architecture (Section 3.4) combining the H3 and MLP blocks.", "type": "text" } ], "index": 51 } ], "index": 50 }, { "type": "title", "bbox": [ 108, 705, 240, 716 ], "lines": [ { "bbox": [ 105, 704, 241, 718 ], "spans": [ { "bbox": [ 105, 704, 241, 718 ], "score": 1.0, "content": "B.3 RELATIONSHIP TO RNNS", "type": "text" } ], "index": 52 } ], "index": 52 }, { "type": "text", "bbox": [ 108, 721, 504, 732 ], "lines": [ { "bbox": [ 106, 719, 506, 733 ], "spans": [ { "bbox": [ 106, 719, 506, 733 ], "score": 1.0, "content": "RNNs and SSMs are broadly related, as they both involve the concepts of recurrence on a latent state.", "type": "text" } ], "index": 53 } ], "index": 53 } ], "page_idx": 15, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 304, 37 ], "lines": [ { "bbox": [ 106, 26, 304, 38 ], "spans": [ { "bbox": [ 106, 26, 304, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 752, 310, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "", "type": "text", "height": 14, "width": 13 } ] } ] } ], "para_blocks": [ { "type": "list", "bbox": [ 106, 82, 506, 320 ], "lines": [ { "bbox": [ 105, 81, 505, 95 ], "spans": [ { "bbox": [ 105, 81, 505, 95 ], "score": 1.0, "content": "• S4 (Gu et al., 2021; 2022a) introduced the first structured SSM, describing diagonal structure and", "type": "text" } ], "index": 0, "is_list_start_line": true }, { "bbox": [ 114, 93, 505, 105 ], "spans": [ { "bbox": [ 114, 93, 505, 105 ], "score": 1.0, "content": "diagonal plus low-rank (DPLR). It focused on efficient convolutional algorithms for DPLR SSMs", "type": "text" } ], "index": 1 }, { "bbox": [ 113, 104, 464, 116 ], "spans": [ { "bbox": [ 113, 104, 464, 116 ], "score": 1.0, "content": "due to a connection to continuous-time online memorization (HIPPO) (Gu et al., 2020a).", "type": "text" } ], "index": 2, "is_list_end_line": true }, { "bbox": [ 107, 116, 505, 129 ], "spans": [ { "bbox": [ 107, 116, 505, 129 ], "score": 1.0, "content": "• DSS (Gupta, 2022) first discovered the empirical effectiveness of diagonal structured SSMs by", "type": "text" } ], "index": 3, "is_list_start_line": true }, { "bbox": [ 112, 127, 506, 140 ], "spans": [ { "bbox": [ 112, 127, 506, 140 ], "score": 1.0, "content": "approximating the HIPPO initialization. This was expanded on theoretically in S4D (Gu et al., 2022b).", "type": "text" } ], "index": 4 }, { "bbox": [ 107, 140, 505, 153 ], "spans": [ { "bbox": [ 107, 140, 505, 153 ], "score": 1.0, "content": "• S5 (Smith et al., 2023) independently discovered the diagonal SSM approximation, and is the first", "type": "text" } ], "index": 5, "is_list_start_line": true }, { "bbox": [ 114, 151, 505, 164 ], "spans": [ { "bbox": [ 114, 151, 505, 164 ], "score": 1.0, "content": "S4 model to be computed recurrently with the parallel scan. However, this required lowering the", "type": "text" } ], "index": 6 }, { "bbox": [ 114, 162, 505, 174 ], "spans": [ { "bbox": [ 114, 162, 505, 174 ], "score": 1.0, "content": "effective state dimension, which they accomplished by switching the SSM dimensions from a SISO", "type": "text" } ], "index": 7 }, { "bbox": [ 114, 173, 423, 185 ], "spans": [ { "bbox": [ 114, 173, 423, 185 ], "score": 1.0, "content": "(single-input single-output) to MIMO (multi-input multi-output) formulation.", "type": "text" } ], "index": 8, "is_list_end_line": true }, { "bbox": [ 107, 185, 506, 198 ], "spans": [ { "bbox": [ 107, 185, 506, 198 ], "score": 1.0, "content": "• Mega (Ma et al., 2023) introduced a simplification of S4 to be real- instead of complex- valued,", "type": "text" } ], "index": 9, "is_list_start_line": true }, { "bbox": [ 114, 196, 505, 209 ], "spans": [ { "bbox": [ 114, 196, 505, 209 ], "score": 1.0, "content": "giving it an interpretation of being an exponential moving average (EMA). They additionally make", "type": "text" } ], "index": 10 }, { "bbox": [ 113, 206, 505, 220 ], "spans": [ { "bbox": [ 113, 206, 505, 220 ], "score": 1.0, "content": "an interesting connection of the discretization step of SSMs to an EMA damping term. Contrary", "type": "text" } ], "index": 11 }, { "bbox": [ 113, 217, 505, 231 ], "spans": [ { "bbox": [ 113, 217, 505, 231 ], "score": 1.0, "content": "to findings in the original S4 papers, this was the first model to show that real-valued SSMs are", "type": "text" } ], "index": 12 }, { "bbox": [ 114, 229, 505, 241 ], "spans": [ { "bbox": [ 114, 229, 505, 241 ], "score": 1.0, "content": "empirically effective in certain settings or when combined with different architectural components.", "type": "text" } ], "index": 13 }, { "bbox": [ 107, 241, 505, 254 ], "spans": [ { "bbox": [ 107, 241, 505, 254 ], "score": 1.0, "content": "• Liquid S4 (Hasani et al., 2023) is also motivated by augmenting S4 with an input-dependent state", "type": "text" } ], "index": 14, "is_list_start_line": true }, { "bbox": [ 114, 253, 505, 264 ], "spans": [ { "bbox": [ 114, 253, 505, 264 ], "score": 1.0, "content": "transition. From this perspective it shares similarity to selection mechanisms, although in a limited", "type": "text" } ], "index": 15 }, { "bbox": [ 114, 263, 361, 275 ], "spans": [ { "bbox": [ 114, 263, 361, 275 ], "score": 1.0, "content": "form which is still computed convolutionally and close to LTI.", "type": "text" } ], "index": 16, "is_list_end_line": true }, { "bbox": [ 106, 274, 506, 289 ], "spans": [ { "bbox": [ 106, 274, 506, 289 ], "score": 1.0, "content": "• SGConv (Li et al., 2023), Hyena (Poli et al., 2023), LongConv (Fu et al., 2023), MultiresConv (Shi", "type": "text" } ], "index": 17, "is_list_start_line": true }, { "bbox": [ 114, 287, 505, 298 ], "spans": [ { "bbox": [ 114, 287, 505, 298 ], "score": 1.0, "content": "et al., 2023b) all focus on the convolutional representation of S4 and create global or long convolution", "type": "text" } ], "index": 18 }, { "bbox": [ 114, 297, 505, 311 ], "spans": [ { "bbox": [ 114, 297, 505, 311 ], "score": 1.0, "content": "kernels with different parameterizations. However, these methods cannot do fast autoregressive", "type": "text" } ], "index": 19 }, { "bbox": [ 114, 308, 189, 320 ], "spans": [ { "bbox": [ 114, 308, 189, 320 ], "score": 1.0, "content": "inference directly.", "type": "text" } ], "index": 20, "is_list_end_line": true } ], "index": 10, "bbox_fs": [ 105, 81, 506, 320 ] }, { "type": "text", "bbox": [ 107, 325, 504, 347 ], "lines": [ { "bbox": [ 105, 324, 505, 337 ], "spans": [ { "bbox": [ 105, 324, 505, 337 ], "score": 1.0, "content": "Notably, all of these methods, and all other structured SSMs that we are aware of, have been", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 335, 346, 348 ], "spans": [ { "bbox": [ 105, 335, 346, 348 ], "score": 1.0, "content": "non-selective and usually strictly LTI (linear time invariant).", "type": "text" } ], "index": 22 } ], "index": 21.5, "bbox_fs": [ 105, 324, 505, 348 ] }, { "type": "title", "bbox": [ 107, 356, 230, 367 ], "lines": [ { "bbox": [ 105, 355, 231, 369 ], "spans": [ { "bbox": [ 105, 355, 231, 369 ], "score": 1.0, "content": "B.2 SSM ARCHITECTURES", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 107, 372, 504, 394 ], "lines": [ { "bbox": [ 106, 371, 505, 384 ], "spans": [ { "bbox": [ 106, 371, 505, 384 ], "score": 1.0, "content": "We use SSM architectures or state space neural networks (SSNN) to refer to deep neural network", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 382, 399, 395 ], "spans": [ { "bbox": [ 106, 382, 399, 395 ], "score": 1.0, "content": "architectures incorporating one of the previous SSMs as a black box layer.", "type": "text" } ], "index": 25 } ], "index": 24.5, "bbox_fs": [ 106, 371, 505, 395 ] }, { "type": "list", "bbox": [ 106, 400, 506, 657 ], "lines": [ { "bbox": [ 107, 399, 505, 411 ], "spans": [ { "bbox": [ 107, 399, 505, 411 ], "score": 1.0, "content": "• GSS (Mehta et al., 2023) was the first gated neural network architecture incorporating SSMs. It is", "type": "text" } ], "index": 26, "is_list_start_line": true }, { "bbox": [ 114, 410, 505, 421 ], "spans": [ { "bbox": [ 114, 410, 505, 421 ], "score": 1.0, "content": "motivated by the gated attention unit (GAU) of Hua et al. (2022) and looks quite similar to our block,", "type": "text" } ], "index": 27 }, { "bbox": [ 114, 420, 505, 432 ], "spans": [ { "bbox": [ 114, 420, 505, 432 ], "score": 1.0, "content": "except with additional projections. Most importantly, its projection contracts the model dimension", "type": "text" } ], "index": 28 }, { "bbox": [ 114, 431, 505, 443 ], "spans": [ { "bbox": [ 114, 431, 505, 443 ], "score": 1.0, "content": "to reduce the state size of the SSM, while ours expands the model dimension in order to increase", "type": "text" } ], "index": 29 }, { "bbox": [ 114, 442, 323, 453 ], "spans": [ { "bbox": [ 114, 442, 323, 453 ], "score": 1.0, "content": "the state size, based on the motivation in Section 3.1.", "type": "text" } ], "index": 30, "is_list_end_line": true }, { "bbox": [ 106, 454, 505, 467 ], "spans": [ { "bbox": [ 106, 454, 505, 467 ], "score": 1.0, "content": "• Mega (Ma et al., 2023) combined the EMA simplification of S4 described above into a hybrid", "type": "text" } ], "index": 31, "is_list_start_line": true }, { "bbox": [ 112, 466, 333, 478 ], "spans": [ { "bbox": [ 112, 466, 333, 478 ], "score": 1.0, "content": "architecture using an efficient attention approximation.", "type": "text" } ], "index": 32, "is_list_end_line": true }, { "bbox": [ 105, 477, 506, 491 ], "spans": [ { "bbox": [ 105, 477, 506, 491 ], "score": 1.0, "content": "• H3 (Dao et al., 2023) is motivated by combining S4 with linear attention (Katharopoulos et al., 2020).", "type": "text" } ], "index": 33, "is_list_start_line": true }, { "bbox": [ 113, 488, 505, 502 ], "spans": [ { "bbox": [ 113, 488, 505, 502 ], "score": 1.0, "content": "It is the first to generalize this formulation of linear attention to more general recurrences, which", "type": "text" } ], "index": 34 }, { "bbox": [ 113, 500, 264, 512 ], "spans": [ { "bbox": [ 113, 500, 264, 512 ], "score": 1.0, "content": "is also the basis of later architectures.", "type": "text" } ], "index": 35, "is_list_end_line": true }, { "bbox": [ 107, 513, 505, 525 ], "spans": [ { "bbox": [ 107, 513, 505, 525 ], "score": 1.0, "content": "• Selective S4 (Wang et al., 2023) incorporates S4 as a black box to generate a binary mask which", "type": "text" } ], "index": 36, "is_list_start_line": true }, { "bbox": [ 114, 523, 505, 535 ], "spans": [ { "bbox": [ 114, 523, 505, 535 ], "score": 1.0, "content": "is multiplied on the input. While sharing the “selection” name, we consider this an architectural", "type": "text" } ], "index": 37 }, { "bbox": [ 114, 534, 505, 546 ], "spans": [ { "bbox": [ 114, 534, 505, 546 ], "score": 1.0, "content": "modification that is closer to architectural gating than a selection mechanism (Appendix A.1). For", "type": "text" } ], "index": 38 }, { "bbox": [ 114, 544, 506, 558 ], "spans": [ { "bbox": [ 114, 544, 506, 558 ], "score": 1.0, "content": "example, we hypothesize that it would not solve the Selective Copying task because simply masking", "type": "text" } ], "index": 39 }, { "bbox": [ 114, 555, 506, 568 ], "spans": [ { "bbox": [ 114, 555, 506, 568 ], "score": 1.0, "content": "out the irrelevant inputs does not affect the spacing between the relevant ones (indeed, the Selective", "type": "text" } ], "index": 40 }, { "bbox": [ 114, 567, 490, 578 ], "spans": [ { "bbox": [ 114, 567, 490, 578 ], "score": 1.0, "content": "Copying task can even be viewed as coming pre-masked if the noise tokens are embedded to 0).", "type": "text" } ], "index": 41, "is_list_end_line": true }, { "bbox": [ 108, 579, 505, 591 ], "spans": [ { "bbox": [ 108, 579, 505, 591 ], "score": 1.0, "content": "• RetNet (Sun et al., 2023) is also based on Linear Attention and very similar to H3, but reduces the inner", "type": "text" } ], "index": 42, "is_list_start_line": true }, { "bbox": [ 113, 590, 506, 603 ], "spans": [ { "bbox": [ 113, 590, 322, 603 ], "score": 1.0, "content": "S4 layer to a special case where the state dimension is", "type": "text" }, { "bbox": [ 322, 591, 348, 601 ], "score": 0.9, "content": "N = 1", "type": "inline_equation" }, { "bbox": [ 348, 590, 506, 603 ], "score": 1.0, "content": ". This simplification leads to an alternate", "type": "text" } ], "index": 43 }, { "bbox": [ 113, 601, 507, 613 ], "spans": [ { "bbox": [ 113, 601, 507, 613 ], "score": 1.0, "content": "way to parallelize the computation with a variant of standard multi-head attention instead of convo-", "type": "text" } ], "index": 44 }, { "bbox": [ 113, 611, 506, 624 ], "spans": [ { "bbox": [ 113, 611, 506, 624 ], "score": 1.0, "content": "lutions. Although not framed as such, its recurrence can be viewed as a special case of a linear SSM.", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 624, 505, 637 ], "spans": [ { "bbox": [ 105, 624, 505, 637 ], "score": 1.0, "content": "• RWKV (Peng et al., 2023) is another recent RNN designed for language modeling. It is based on", "type": "text" } ], "index": 46, "is_list_start_line": true }, { "bbox": [ 113, 635, 505, 648 ], "spans": [ { "bbox": [ 113, 635, 505, 648 ], "score": 1.0, "content": "AFT (attention-free Transformer (Zhai et al., 2021)), another variant of linear attention. Its main", "type": "text" } ], "index": 47 }, { "bbox": [ 113, 645, 460, 658 ], "spans": [ { "bbox": [ 113, 645, 460, 658 ], "score": 1.0, "content": "“WKV” mechanism involves LTI recurrences and can be seen as the ratio of two SSMs.", "type": "text" } ], "index": 48, "is_list_end_line": true } ], "index": 37, "bbox_fs": [ 105, 399, 507, 658 ] }, { "type": "text", "bbox": [ 107, 663, 505, 696 ], "lines": [ { "bbox": [ 106, 663, 505, 675 ], "spans": [ { "bbox": [ 106, 663, 505, 675 ], "score": 1.0, "content": "We also highlight the gated attention unit (GAU) from Hua et al. (2022), which was motivated", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 673, 506, 687 ], "spans": [ { "bbox": [ 105, 673, 506, 687 ], "score": 1.0, "content": "by combining the Transformer’s MHA and MLP blocks together and was an inspiration for our", "type": "text" } ], "index": 50 }, { "bbox": [ 106, 685, 351, 697 ], "spans": [ { "bbox": [ 106, 685, 351, 697 ], "score": 1.0, "content": "architecture (Section 3.4) combining the H3 and MLP blocks.", "type": "text" } ], "index": 51 } ], "index": 50, "bbox_fs": [ 105, 663, 506, 697 ] }, { "type": "title", "bbox": [ 108, 705, 240, 716 ], "lines": [ { "bbox": [ 105, 704, 241, 718 ], "spans": [ { "bbox": [ 105, 704, 241, 718 ], "score": 1.0, "content": "B.3 RELATIONSHIP TO RNNS", "type": "text" } ], "index": 52 } ], "index": 52 }, { "type": "text", "bbox": [ 108, 721, 504, 732 ], "lines": [ { "bbox": [ 106, 719, 506, 733 ], "spans": [ { "bbox": [ 106, 719, 506, 733 ], "score": 1.0, "content": "RNNs and SSMs are broadly related, as they both involve the concepts of recurrence on a latent state.", "type": "text" } ], "index": 53 } ], "index": 53, "bbox_fs": [ 106, 719, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 148 ], "lines": [ { "bbox": [ 105, 81, 506, 95 ], "spans": [ { "bbox": [ 105, 81, 506, 95 ], "score": 1.0, "content": "Several older RNNs such as the strongly typed RNN (Balduzzi & Ghifary, 2016), quasi RNN (QRNN)", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 506, 105 ], "spans": [ { "bbox": [ 105, 93, 506, 105 ], "score": 1.0, "content": "(Bradbury et al., 2016), and simple recurrent unit (SRU) (Lei et al., 2017; Lei, 2021) involve forms", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 104, 505, 116 ], "spans": [ { "bbox": [ 105, 104, 505, 116 ], "score": 1.0, "content": "of gated RNNs without time-wise nonlinearities. Because of the connections of gating mechanisms", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 114, 505, 127 ], "spans": [ { "bbox": [ 105, 114, 505, 127 ], "score": 1.0, "content": "and selection mechanisms, these can be viewed as cases of selective SSMs, and are thus more powerful", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 126, 505, 137 ], "spans": [ { "bbox": [ 105, 126, 505, 137 ], "score": 1.0, "content": "in a sense than the family of LTI structured SSMs above. However, they do not use state expansion", "type": "text" } ], "index": 4 }, { "bbox": [ 109, 135, 489, 149 ], "spans": [ { "bbox": [ 109, 136, 136, 147 ], "score": 0.85, "content": "N { = } 1", "type": "inline_equation" }, { "bbox": [ 136, 135, 185, 149 ], "score": 1.0, "content": ") or selective", "type": "text" }, { "bbox": [ 186, 136, 208, 147 ], "score": 0.69, "content": "^ { _ { B , C } }", "type": "inline_equation" }, { "bbox": [ 208, 135, 489, 149 ], "score": 1.0, "content": "parameters, both of which are important for performance (Section 4.6).", "type": "text" } ], "index": 5 } ], "index": 2.5 }, { "type": "text", "bbox": [ 107, 151, 505, 217 ], "lines": [ { "bbox": [ 105, 151, 505, 164 ], "spans": [ { "bbox": [ 105, 151, 505, 164 ], "score": 1.0, "content": "Additionally, older RNNs famously suffered from efficiency issues and the vanishing gradients", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 163, 505, 175 ], "spans": [ { "bbox": [ 105, 163, 505, 175 ], "score": 1.0, "content": "problem (Pascanu et al., 2013), both caused by their sequential nature. The latter could be solved", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 173, 506, 186 ], "spans": [ { "bbox": [ 105, 173, 506, 186 ], "score": 1.0, "content": "for some of the above RNNs by leveraging the parallel scan (Martin & Cundy, 2018), but the former", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 184, 506, 196 ], "spans": [ { "bbox": [ 105, 184, 506, 196 ], "score": 1.0, "content": "was difficult without theory later developed for SSMs. For example, modern structured SSMs differ", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 193, 507, 209 ], "spans": [ { "bbox": [ 105, 193, 507, 209 ], "score": 1.0, "content": "in more careful parameterization of the recurrent dynamics inspired by classical SSM theory (e.g.", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 206, 451, 217 ], "spans": [ { "bbox": [ 106, 206, 451, 217 ], "score": 1.0, "content": "through discretization (Gu et al., 2021; 2023)), or direct analysis (Orvieto et al., 2023)).", "type": "text" } ], "index": 11 } ], "index": 8.5 }, { "type": "text", "bbox": [ 107, 221, 505, 298 ], "lines": [ { "bbox": [ 106, 221, 506, 233 ], "spans": [ { "bbox": [ 106, 221, 506, 233 ], "score": 1.0, "content": "We also note that there is a long line of work on orthogonal RNNs (Arjovsky et al., 2016; Henaff", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 231, 506, 244 ], "spans": [ { "bbox": [ 105, 231, 506, 244 ], "score": 1.0, "content": "et al., 2016; Mhammedi et al., 2017; Vorontsov et al., 2017; Lezcano-Casado & Mart´ınez-Rubio,", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 241, 506, 255 ], "spans": [ { "bbox": [ 105, 241, 299, 255 ], "score": 1.0, "content": "2019) which are motivated by constraining the", "type": "text" }, { "bbox": [ 299, 243, 309, 253 ], "score": 0.84, "content": "\\overline { { A } }", "type": "inline_equation" }, { "bbox": [ 309, 241, 506, 255 ], "score": 1.0, "content": "transition matrix to be orthogonal or unitary, in", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 253, 506, 266 ], "spans": [ { "bbox": [ 106, 253, 506, 266 ], "score": 1.0, "content": "order to control its eigenvalues and prevent the vanishing gradient problem. However, these had other", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 264, 505, 275 ], "spans": [ { "bbox": [ 106, 264, 505, 275 ], "score": 1.0, "content": "limitations; we believe that these stem from the fact that orthogonal/unitary RNNs are also LTI. For", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 276, 505, 287 ], "spans": [ { "bbox": [ 106, 276, 505, 287 ], "score": 1.0, "content": "example, they are almost always evaluated on the Copying task which they can solve perfectly, but", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 286, 379, 298 ], "spans": [ { "bbox": [ 106, 286, 379, 298 ], "score": 1.0, "content": "observed to struggle on the Selective Copying task (Jing et al., 2019).", "type": "text" } ], "index": 18 } ], "index": 15 }, { "type": "title", "bbox": [ 107, 307, 242, 318 ], "lines": [ { "bbox": [ 105, 306, 244, 320 ], "spans": [ { "bbox": [ 105, 306, 244, 320 ], "score": 1.0, "content": "B.4 LONG CONTEXT MODELS", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "text", "bbox": [ 107, 323, 505, 356 ], "lines": [ { "bbox": [ 105, 322, 505, 336 ], "spans": [ { "bbox": [ 105, 322, 505, 336 ], "score": 1.0, "content": "Long context has become a popular subject, and several recent models have claimed to scale to longer", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 334, 505, 346 ], "spans": [ { "bbox": [ 106, 334, 505, 346 ], "score": 1.0, "content": "and longer sequences. However, these are often from a computational standpoint and have not been", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 345, 255, 356 ], "spans": [ { "bbox": [ 105, 345, 255, 356 ], "score": 1.0, "content": "extensively validated. These include:", "type": "text" } ], "index": 22 } ], "index": 21 }, { "type": "text", "bbox": [ 106, 360, 505, 453 ], "lines": [ { "bbox": [ 105, 359, 505, 373 ], "spans": [ { "bbox": [ 105, 359, 505, 373 ], "score": 1.0, "content": "• Recurrent Memory Transformer (Bulatov et al., 2023), a lightweight wrapper around a Transformer", "type": "text" } ], "index": 23 }, { "bbox": [ 114, 370, 505, 383 ], "spans": [ { "bbox": [ 114, 370, 505, 383 ], "score": 1.0, "content": "backbone. It showed ability to generalize up to 1M sequences but only on synthetic memorization", "type": "text" } ], "index": 24 }, { "bbox": [ 114, 381, 456, 393 ], "spans": [ { "bbox": [ 114, 381, 456, 393 ], "score": 1.0, "content": "tasks; this result is similar to our Induction Heads extrapolation experiment (Figure 3).", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 394, 505, 408 ], "spans": [ { "bbox": [ 105, 394, 505, 408 ], "score": 1.0, "content": "• LongNet (Ding et al., 2023), which claimed to scale to 1B length but only evaluated on length", "type": "text" } ], "index": 26 }, { "bbox": [ 114, 406, 214, 418 ], "spans": [ { "bbox": [ 114, 407, 149, 417 ], "score": 0.87, "content": "< 1 0 0 K", "type": "inline_equation" }, { "bbox": [ 150, 406, 214, 418 ], "score": 1.0, "content": "for actual tasks.", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 420, 505, 433 ], "spans": [ { "bbox": [ 106, 420, 505, 433 ], "score": 1.0, "content": "• Hyena and HyenaDNA (Poli et al., 2023; Nguyen et al., 2023), which claimed to leverage up to 1M", "type": "text" } ], "index": 28 }, { "bbox": [ 114, 430, 505, 444 ], "spans": [ { "bbox": [ 114, 430, 505, 444 ], "score": 1.0, "content": "context, but did not control for computation time. In fact, its claims about efficiency and performance", "type": "text" } ], "index": 29 }, { "bbox": [ 114, 442, 361, 453 ], "spans": [ { "bbox": [ 114, 442, 361, 453 ], "score": 1.0, "content": "would be largely matched by any of the LTI S4 variants above.", "type": "text" } ], "index": 30 } ], "index": 26.5 }, { "type": "text", "bbox": [ 105, 457, 504, 479 ], "lines": [ { "bbox": [ 105, 457, 505, 470 ], "spans": [ { "bbox": [ 105, 457, 505, 470 ], "score": 1.0, "content": "In contrast, we believe this work presents one of the first approaches to meaningfully demonstrate", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 468, 283, 480 ], "spans": [ { "bbox": [ 106, 468, 283, 480 ], "score": 1.0, "content": "increasing performance with longer context.", "type": "text" } ], "index": 32 } ], "index": 31.5 }, { "type": "title", "bbox": [ 107, 491, 303, 504 ], "lines": [ { "bbox": [ 106, 490, 303, 505 ], "spans": [ { "bbox": [ 106, 490, 303, 505 ], "score": 1.0, "content": "C MECHANICS OF SELECTIVE SSMS", "type": "text" } ], "index": 33 } ], "index": 33 }, { "type": "text", "bbox": [ 108, 509, 504, 532 ], "lines": [ { "bbox": [ 105, 507, 505, 523 ], "spans": [ { "bbox": [ 105, 507, 379, 523 ], "score": 1.0, "content": "Proof of Theorem 1. Consider a selective SSM (Algorithm 2) with", "type": "text" }, { "bbox": [ 380, 509, 505, 521 ], "score": 0.91, "content": "N = 1 , A = - 1 , B = 1 , s _ { \\Delta } =", "type": "inline_equation" } ], "index": 34 }, { "bbox": [ 106, 520, 395, 533 ], "spans": [ { "bbox": [ 106, 520, 132, 533 ], "score": 1.0, "content": "Linear", "type": "text" }, { "bbox": [ 133, 521, 146, 532 ], "score": 0.58, "content": "( x )", "type": "inline_equation" }, { "bbox": [ 147, 520, 148, 533 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 148, 521, 170, 532 ], "score": 0.62, "content": "\\tau _ { \\Delta } =", "type": "inline_equation" }, { "bbox": [ 171, 520, 395, 533 ], "score": 1.0, "content": "softplus. The corresponding continuous-time SSM (1) is", "type": "text" } ], "index": 35 } ], "index": 34.5 }, { "type": "interline_equation", "bbox": [ 265, 533, 345, 547 ], "lines": [ { "bbox": [ 265, 533, 345, 547 ], "spans": [ { "bbox": [ 265, 533, 345, 547 ], "score": 0.91, "content": "h ( t ) = - h ( t ) + x ( t )", "type": "interline_equation", "image_path": "dba09118da2d3e0caed8ebffe36921e7ea2fdb63a46b6a1226a0bb0b7beae895.jpg" } ] } ], "index": 36, "virtual_lines": [ { "bbox": [ 265, 533, 345, 547 ], "spans": [], "index": 36 } ] }, { "type": "text", "bbox": [ 107, 549, 258, 560 ], "lines": [ { "bbox": [ 106, 547, 259, 562 ], "spans": [ { "bbox": [ 106, 547, 259, 562 ], "score": 1.0, "content": "which is also called a leaky integrator.", "type": "text" } ], "index": 37 } ], "index": 37 }, { "type": "text", "bbox": [ 107, 564, 223, 576 ], "lines": [ { "bbox": [ 105, 563, 224, 577 ], "spans": [ { "bbox": [ 105, 563, 224, 577 ], "score": 1.0, "content": "The discretization step size is", "type": "text" } ], "index": 38 } ], "index": 38 }, { "type": "interline_equation", "bbox": [ 225, 577, 387, 619 ], "lines": [ { "bbox": [ 225, 577, 387, 619 ], "spans": [ { "bbox": [ 225, 577, 387, 619 ], "score": 0.92, "content": "\\begin{array} { r l } & { \\Delta _ { k } = \\tau _ { \\Delta } ( \\mathsf { P a r a m e t e r } + s _ { \\Delta } ( x _ { k } ) ) } \\\\ & { \\quad = \\mathsf { s o f t p l u s } ( \\mathsf { P a r a m e t e r } + \\mathsf { L i n e a r } ( x _ { k } ) ) } \\\\ & { \\quad = \\mathsf { s o f t p l u s } ( \\mathsf { L i n e a r } ( x _ { k } ) ) } \\end{array}", "type": "interline_equation", "image_path": "51c7742c9bf92ac5d3e292d0572618641f2e1af7a0286f586aea5d126933eb1b.jpg" } ] } ], "index": 40, "virtual_lines": [ { "bbox": [ 225, 577, 387, 591.0 ], "spans": [], "index": 39 }, { "bbox": [ 225, 591.0, 387, 605.0 ], "spans": [], "index": 40 }, { "bbox": [ 225, 605.0, 387, 619.0 ], "spans": [], "index": 41 } ] }, { "type": "text", "bbox": [ 107, 621, 502, 643 ], "lines": [ { "bbox": [ 105, 619, 504, 633 ], "spans": [ { "bbox": [ 105, 619, 504, 633 ], "score": 1.0, "content": "where we observe that the parameter can be viewed as a learnable bias and folded into the linear", "type": "text" } ], "index": 42 }, { "bbox": [ 104, 631, 152, 644 ], "spans": [ { "bbox": [ 104, 631, 152, 644 ], "score": 1.0, "content": "projection.", "type": "text" } ], "index": 43 } ], "index": 42.5 }, { "type": "text", "bbox": [ 106, 646, 369, 659 ], "lines": [ { "bbox": [ 105, 646, 367, 660 ], "spans": [ { "bbox": [ 105, 646, 367, 660 ], "score": 1.0, "content": "Now applying the zero-order hold (ZOH) discretization formulas:", "type": "text" } ], "index": 44 } ], "index": 44 }, { "type": "interline_equation", "bbox": [ 175, 660, 437, 731 ], "lines": [ { "bbox": [ 175, 660, 437, 731 ], "spans": [ { "bbox": [ 175, 660, 437, 731 ], "score": 0.92, "content": "\\begin{array} { r l } & { \\overline { { \\pmb { A } } } _ { k } = \\mathrm { e x p } ( \\Delta \\pmb { A } ) = \\frac { 1 } { 1 + \\mathrm { e x p } \\left( \\mathsf { L i n e a r } \\left( x _ { k } \\right) \\right) } = \\sigma ( - \\mathsf { L i n e a r } \\left( x _ { k } \\right) ) } \\\\ & { \\qquad = 1 - \\sigma ( \\mathsf { L i n e a r } \\left( x _ { k } \\right) ) } \\\\ & { \\overline { { \\pmb { B } } } _ { k } = ( \\Delta \\pmb { A } ) ^ { - 1 } ( \\mathrm { e x p } ( \\Delta \\pmb { A } ) - \\pmb { I } ) \\cdot \\Delta \\pmb { B } = - ( \\mathrm { e x p } ( \\Delta \\pmb { A } ) - \\pmb { I } ) = 1 - \\overline { { \\pmb { A } } } } \\\\ & { \\qquad = \\sigma ( \\mathsf { L i n e a r } ( x _ { k } ) ) . } \\end{array}", "type": "interline_equation", "image_path": "9680cb2ac64f8ba75a14878d5016a367fe39ef89d7f7fb19e50ff993bc836edd.jpg" } ] } ], "index": 46, "virtual_lines": [ { "bbox": [ 175, 660, 437, 683.6666666666666 ], "spans": [], "index": 45 }, { "bbox": [ 175, 683.6666666666666, 437, 707.3333333333333 ], "spans": [], "index": 46 }, { "bbox": [ 175, 707.3333333333333, 437, 730.9999999999999 ], "spans": [], "index": 47 } ] } ], "page_idx": 16, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 304, 37 ], "lines": [ { "bbox": [ 106, 26, 304, 38 ], "spans": [ { "bbox": [ 106, 26, 304, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 310, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "", "type": "text", "height": 14, "width": 13 } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 148 ], "lines": [ { "bbox": [ 105, 81, 506, 95 ], "spans": [ { "bbox": [ 105, 81, 506, 95 ], "score": 1.0, "content": "Several older RNNs such as the strongly typed RNN (Balduzzi & Ghifary, 2016), quasi RNN (QRNN)", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 506, 105 ], "spans": [ { "bbox": [ 105, 93, 506, 105 ], "score": 1.0, "content": "(Bradbury et al., 2016), and simple recurrent unit (SRU) (Lei et al., 2017; Lei, 2021) involve forms", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 104, 505, 116 ], "spans": [ { "bbox": [ 105, 104, 505, 116 ], "score": 1.0, "content": "of gated RNNs without time-wise nonlinearities. Because of the connections of gating mechanisms", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 114, 505, 127 ], "spans": [ { "bbox": [ 105, 114, 505, 127 ], "score": 1.0, "content": "and selection mechanisms, these can be viewed as cases of selective SSMs, and are thus more powerful", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 126, 505, 137 ], "spans": [ { "bbox": [ 105, 126, 505, 137 ], "score": 1.0, "content": "in a sense than the family of LTI structured SSMs above. However, they do not use state expansion", "type": "text" } ], "index": 4 }, { "bbox": [ 109, 135, 489, 149 ], "spans": [ { "bbox": [ 109, 136, 136, 147 ], "score": 0.85, "content": "N { = } 1", "type": "inline_equation" }, { "bbox": [ 136, 135, 185, 149 ], "score": 1.0, "content": ") or selective", "type": "text" }, { "bbox": [ 186, 136, 208, 147 ], "score": 0.69, "content": "^ { _ { B , C } }", "type": "inline_equation" }, { "bbox": [ 208, 135, 489, 149 ], "score": 1.0, "content": "parameters, both of which are important for performance (Section 4.6).", "type": "text" } ], "index": 5 } ], "index": 2.5, "bbox_fs": [ 105, 81, 506, 149 ] }, { "type": "text", "bbox": [ 107, 151, 505, 217 ], "lines": [ { "bbox": [ 105, 151, 505, 164 ], "spans": [ { "bbox": [ 105, 151, 505, 164 ], "score": 1.0, "content": "Additionally, older RNNs famously suffered from efficiency issues and the vanishing gradients", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 163, 505, 175 ], "spans": [ { "bbox": [ 105, 163, 505, 175 ], "score": 1.0, "content": "problem (Pascanu et al., 2013), both caused by their sequential nature. The latter could be solved", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 173, 506, 186 ], "spans": [ { "bbox": [ 105, 173, 506, 186 ], "score": 1.0, "content": "for some of the above RNNs by leveraging the parallel scan (Martin & Cundy, 2018), but the former", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 184, 506, 196 ], "spans": [ { "bbox": [ 105, 184, 506, 196 ], "score": 1.0, "content": "was difficult without theory later developed for SSMs. For example, modern structured SSMs differ", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 193, 507, 209 ], "spans": [ { "bbox": [ 105, 193, 507, 209 ], "score": 1.0, "content": "in more careful parameterization of the recurrent dynamics inspired by classical SSM theory (e.g.", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 206, 451, 217 ], "spans": [ { "bbox": [ 106, 206, 451, 217 ], "score": 1.0, "content": "through discretization (Gu et al., 2021; 2023)), or direct analysis (Orvieto et al., 2023)).", "type": "text" } ], "index": 11 } ], "index": 8.5, "bbox_fs": [ 105, 151, 507, 217 ] }, { "type": "text", "bbox": [ 107, 221, 505, 298 ], "lines": [ { "bbox": [ 106, 221, 506, 233 ], "spans": [ { "bbox": [ 106, 221, 506, 233 ], "score": 1.0, "content": "We also note that there is a long line of work on orthogonal RNNs (Arjovsky et al., 2016; Henaff", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 231, 506, 244 ], "spans": [ { "bbox": [ 105, 231, 506, 244 ], "score": 1.0, "content": "et al., 2016; Mhammedi et al., 2017; Vorontsov et al., 2017; Lezcano-Casado & Mart´ınez-Rubio,", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 241, 506, 255 ], "spans": [ { "bbox": [ 105, 241, 299, 255 ], "score": 1.0, "content": "2019) which are motivated by constraining the", "type": "text" }, { "bbox": [ 299, 243, 309, 253 ], "score": 0.84, "content": "\\overline { { A } }", "type": "inline_equation" }, { "bbox": [ 309, 241, 506, 255 ], "score": 1.0, "content": "transition matrix to be orthogonal or unitary, in", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 253, 506, 266 ], "spans": [ { "bbox": [ 106, 253, 506, 266 ], "score": 1.0, "content": "order to control its eigenvalues and prevent the vanishing gradient problem. However, these had other", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 264, 505, 275 ], "spans": [ { "bbox": [ 106, 264, 505, 275 ], "score": 1.0, "content": "limitations; we believe that these stem from the fact that orthogonal/unitary RNNs are also LTI. For", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 276, 505, 287 ], "spans": [ { "bbox": [ 106, 276, 505, 287 ], "score": 1.0, "content": "example, they are almost always evaluated on the Copying task which they can solve perfectly, but", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 286, 379, 298 ], "spans": [ { "bbox": [ 106, 286, 379, 298 ], "score": 1.0, "content": "observed to struggle on the Selective Copying task (Jing et al., 2019).", "type": "text" } ], "index": 18 } ], "index": 15, "bbox_fs": [ 105, 221, 506, 298 ] }, { "type": "title", "bbox": [ 107, 307, 242, 318 ], "lines": [ { "bbox": [ 105, 306, 244, 320 ], "spans": [ { "bbox": [ 105, 306, 244, 320 ], "score": 1.0, "content": "B.4 LONG CONTEXT MODELS", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "text", "bbox": [ 107, 323, 505, 356 ], "lines": [ { "bbox": [ 105, 322, 505, 336 ], "spans": [ { "bbox": [ 105, 322, 505, 336 ], "score": 1.0, "content": "Long context has become a popular subject, and several recent models have claimed to scale to longer", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 334, 505, 346 ], "spans": [ { "bbox": [ 106, 334, 505, 346 ], "score": 1.0, "content": "and longer sequences. However, these are often from a computational standpoint and have not been", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 345, 255, 356 ], "spans": [ { "bbox": [ 105, 345, 255, 356 ], "score": 1.0, "content": "extensively validated. These include:", "type": "text" } ], "index": 22 } ], "index": 21, "bbox_fs": [ 105, 322, 505, 356 ] }, { "type": "list", "bbox": [ 106, 360, 505, 453 ], "lines": [ { "bbox": [ 105, 359, 505, 373 ], "spans": [ { "bbox": [ 105, 359, 505, 373 ], "score": 1.0, "content": "• Recurrent Memory Transformer (Bulatov et al., 2023), a lightweight wrapper around a Transformer", "type": "text" } ], "index": 23, "is_list_start_line": true }, { "bbox": [ 114, 370, 505, 383 ], "spans": [ { "bbox": [ 114, 370, 505, 383 ], "score": 1.0, "content": "backbone. It showed ability to generalize up to 1M sequences but only on synthetic memorization", "type": "text" } ], "index": 24 }, { "bbox": [ 114, 381, 456, 393 ], "spans": [ { "bbox": [ 114, 381, 456, 393 ], "score": 1.0, "content": "tasks; this result is similar to our Induction Heads extrapolation experiment (Figure 3).", "type": "text" } ], "index": 25, "is_list_end_line": true }, { "bbox": [ 105, 394, 505, 408 ], "spans": [ { "bbox": [ 105, 394, 505, 408 ], "score": 1.0, "content": "• LongNet (Ding et al., 2023), which claimed to scale to 1B length but only evaluated on length", "type": "text" } ], "index": 26, "is_list_start_line": true }, { "bbox": [ 114, 406, 214, 418 ], "spans": [ { "bbox": [ 114, 407, 149, 417 ], "score": 0.87, "content": "< 1 0 0 K", "type": "inline_equation" }, { "bbox": [ 150, 406, 214, 418 ], "score": 1.0, "content": "for actual tasks.", "type": "text" } ], "index": 27, "is_list_end_line": true }, { "bbox": [ 106, 420, 505, 433 ], "spans": [ { "bbox": [ 106, 420, 505, 433 ], "score": 1.0, "content": "• Hyena and HyenaDNA (Poli et al., 2023; Nguyen et al., 2023), which claimed to leverage up to 1M", "type": "text" } ], "index": 28, "is_list_start_line": true }, { "bbox": [ 114, 430, 505, 444 ], "spans": [ { "bbox": [ 114, 430, 505, 444 ], "score": 1.0, "content": "context, but did not control for computation time. In fact, its claims about efficiency and performance", "type": "text" } ], "index": 29 }, { "bbox": [ 114, 442, 361, 453 ], "spans": [ { "bbox": [ 114, 442, 361, 453 ], "score": 1.0, "content": "would be largely matched by any of the LTI S4 variants above.", "type": "text" } ], "index": 30, "is_list_end_line": true } ], "index": 26.5, "bbox_fs": [ 105, 359, 505, 453 ] }, { "type": "text", "bbox": [ 105, 457, 504, 479 ], "lines": [ { "bbox": [ 105, 457, 505, 470 ], "spans": [ { "bbox": [ 105, 457, 505, 470 ], "score": 1.0, "content": "In contrast, we believe this work presents one of the first approaches to meaningfully demonstrate", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 468, 283, 480 ], "spans": [ { "bbox": [ 106, 468, 283, 480 ], "score": 1.0, "content": "increasing performance with longer context.", "type": "text" } ], "index": 32 } ], "index": 31.5, "bbox_fs": [ 105, 457, 505, 480 ] }, { "type": "title", "bbox": [ 107, 491, 303, 504 ], "lines": [ { "bbox": [ 106, 490, 303, 505 ], "spans": [ { "bbox": [ 106, 490, 303, 505 ], "score": 1.0, "content": "C MECHANICS OF SELECTIVE SSMS", "type": "text" } ], "index": 33 } ], "index": 33 }, { "type": "text", "bbox": [ 108, 509, 504, 532 ], "lines": [ { "bbox": [ 105, 507, 505, 523 ], "spans": [ { "bbox": [ 105, 507, 379, 523 ], "score": 1.0, "content": "Proof of Theorem 1. Consider a selective SSM (Algorithm 2) with", "type": "text" }, { "bbox": [ 380, 509, 505, 521 ], "score": 0.91, "content": "N = 1 , A = - 1 , B = 1 , s _ { \\Delta } =", "type": "inline_equation" } ], "index": 34 }, { "bbox": [ 106, 520, 395, 533 ], "spans": [ { "bbox": [ 106, 520, 132, 533 ], "score": 1.0, "content": "Linear", "type": "text" }, { "bbox": [ 133, 521, 146, 532 ], "score": 0.58, "content": "( x )", "type": "inline_equation" }, { "bbox": [ 147, 520, 148, 533 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 148, 521, 170, 532 ], "score": 0.62, "content": "\\tau _ { \\Delta } =", "type": "inline_equation" }, { "bbox": [ 171, 520, 395, 533 ], "score": 1.0, "content": "softplus. The corresponding continuous-time SSM (1) is", "type": "text" } ], "index": 35 } ], "index": 34.5, "bbox_fs": [ 105, 507, 505, 533 ] }, { "type": "interline_equation", "bbox": [ 265, 533, 345, 547 ], "lines": [ { "bbox": [ 265, 533, 345, 547 ], "spans": [ { "bbox": [ 265, 533, 345, 547 ], "score": 0.91, "content": "h ( t ) = - h ( t ) + x ( t )", "type": "interline_equation", "image_path": "dba09118da2d3e0caed8ebffe36921e7ea2fdb63a46b6a1226a0bb0b7beae895.jpg" } ] } ], "index": 36, "virtual_lines": [ { "bbox": [ 265, 533, 345, 547 ], "spans": [], "index": 36 } ] }, { "type": "text", "bbox": [ 107, 549, 258, 560 ], "lines": [ { "bbox": [ 106, 547, 259, 562 ], "spans": [ { "bbox": [ 106, 547, 259, 562 ], "score": 1.0, "content": "which is also called a leaky integrator.", "type": "text" } ], "index": 37 } ], "index": 37, "bbox_fs": [ 106, 547, 259, 562 ] }, { "type": "text", "bbox": [ 107, 564, 223, 576 ], "lines": [ { "bbox": [ 105, 563, 224, 577 ], "spans": [ { "bbox": [ 105, 563, 224, 577 ], "score": 1.0, "content": "The discretization step size is", "type": "text" } ], "index": 38 } ], "index": 38, "bbox_fs": [ 105, 563, 224, 577 ] }, { "type": "interline_equation", "bbox": [ 225, 577, 387, 619 ], "lines": [ { "bbox": [ 225, 577, 387, 619 ], "spans": [ { "bbox": [ 225, 577, 387, 619 ], "score": 0.92, "content": "\\begin{array} { r l } & { \\Delta _ { k } = \\tau _ { \\Delta } ( \\mathsf { P a r a m e t e r } + s _ { \\Delta } ( x _ { k } ) ) } \\\\ & { \\quad = \\mathsf { s o f t p l u s } ( \\mathsf { P a r a m e t e r } + \\mathsf { L i n e a r } ( x _ { k } ) ) } \\\\ & { \\quad = \\mathsf { s o f t p l u s } ( \\mathsf { L i n e a r } ( x _ { k } ) ) } \\end{array}", "type": "interline_equation", "image_path": "51c7742c9bf92ac5d3e292d0572618641f2e1af7a0286f586aea5d126933eb1b.jpg" } ] } ], "index": 40, "virtual_lines": [ { "bbox": [ 225, 577, 387, 591.0 ], "spans": [], "index": 39 }, { "bbox": [ 225, 591.0, 387, 605.0 ], "spans": [], "index": 40 }, { "bbox": [ 225, 605.0, 387, 619.0 ], "spans": [], "index": 41 } ] }, { "type": "text", "bbox": [ 107, 621, 502, 643 ], "lines": [ { "bbox": [ 105, 619, 504, 633 ], "spans": [ { "bbox": [ 105, 619, 504, 633 ], "score": 1.0, "content": "where we observe that the parameter can be viewed as a learnable bias and folded into the linear", "type": "text" } ], "index": 42 }, { "bbox": [ 104, 631, 152, 644 ], "spans": [ { "bbox": [ 104, 631, 152, 644 ], "score": 1.0, "content": "projection.", "type": "text" } ], "index": 43 } ], "index": 42.5, "bbox_fs": [ 104, 619, 504, 644 ] }, { "type": "text", "bbox": [ 106, 646, 369, 659 ], "lines": [ { "bbox": [ 105, 646, 367, 660 ], "spans": [ { "bbox": [ 105, 646, 367, 660 ], "score": 1.0, "content": "Now applying the zero-order hold (ZOH) discretization formulas:", "type": "text" } ], "index": 44 } ], "index": 44, "bbox_fs": [ 105, 646, 367, 660 ] }, { "type": "interline_equation", "bbox": [ 175, 660, 437, 731 ], "lines": [ { "bbox": [ 175, 660, 437, 731 ], "spans": [ { "bbox": [ 175, 660, 437, 731 ], "score": 0.92, "content": "\\begin{array} { r l } & { \\overline { { \\pmb { A } } } _ { k } = \\mathrm { e x p } ( \\Delta \\pmb { A } ) = \\frac { 1 } { 1 + \\mathrm { e x p } \\left( \\mathsf { L i n e a r } \\left( x _ { k } \\right) \\right) } = \\sigma ( - \\mathsf { L i n e a r } \\left( x _ { k } \\right) ) } \\\\ & { \\qquad = 1 - \\sigma ( \\mathsf { L i n e a r } \\left( x _ { k } \\right) ) } \\\\ & { \\overline { { \\pmb { B } } } _ { k } = ( \\Delta \\pmb { A } ) ^ { - 1 } ( \\mathrm { e x p } ( \\Delta \\pmb { A } ) - \\pmb { I } ) \\cdot \\Delta \\pmb { B } = - ( \\mathrm { e x p } ( \\Delta \\pmb { A } ) - \\pmb { I } ) = 1 - \\overline { { \\pmb { A } } } } \\\\ & { \\qquad = \\sigma ( \\mathsf { L i n e a r } ( x _ { k } ) ) . } \\end{array}", "type": "interline_equation", "image_path": "9680cb2ac64f8ba75a14878d5016a367fe39ef89d7f7fb19e50ff993bc836edd.jpg" } ] } ], "index": 46, "virtual_lines": [ { "bbox": [ 175, 660, 437, 683.6666666666666 ], "spans": [], "index": 45 }, { "bbox": [ 175, 683.6666666666666, 437, 707.3333333333333 ], "spans": [], "index": 46 }, { "bbox": [ 175, 707.3333333333333, 437, 730.9999999999999 ], "spans": [], "index": 47 } ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 108, 82, 264, 93 ], "lines": [ { "bbox": [ 106, 80, 267, 95 ], "spans": [ { "bbox": [ 106, 80, 267, 95 ], "score": 1.0, "content": "Thus the final discrete recurrence (2a) is", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "interline_equation", "bbox": [ 252, 92, 358, 121 ], "lines": [ { "bbox": [ 252, 92, 358, 121 ], "spans": [ { "bbox": [ 252, 92, 358, 121 ], "score": 0.87, "content": "\\begin{array} { l } { g _ { k } = \\sigma ( \\mathsf { L i n e a r } ( x _ { k } ) ) } \\\\ { h _ { k } = ( 1 - g _ { k } ) h _ { k - 1 } + g _ { k } x _ { k } } \\end{array}", "type": "interline_equation", "image_path": "7bdb05ee7e5171720bf82b576796e8053a73c7dfcb369c4f0eb60ba57531675e.jpg" } ] } ], "index": 1.5, "virtual_lines": [ { "bbox": [ 252, 92, 358, 106.5 ], "spans": [], "index": 1 }, { "bbox": [ 252, 106.5, 358, 121.0 ], "spans": [], "index": 2 } ] }, { "type": "text", "bbox": [ 107, 119, 149, 130 ], "lines": [ { "bbox": [ 105, 118, 151, 131 ], "spans": [ { "bbox": [ 105, 118, 151, 131 ], "score": 1.0, "content": "as desired.", "type": "text" } ], "index": 3 } ], "index": 3 }, { "type": "title", "bbox": [ 108, 145, 415, 158 ], "lines": [ { "bbox": [ 106, 145, 415, 159 ], "spans": [ { "bbox": [ 106, 145, 415, 159 ], "score": 1.0, "content": "D HARDWARE-AWARE ALGORITHM FOR SELECTIVE SSMS", "type": "text" } ], "index": 4 } ], "index": 4 }, { "type": "text", "bbox": [ 106, 163, 505, 261 ], "lines": [ { "bbox": [ 106, 163, 506, 176 ], "spans": [ { "bbox": [ 106, 163, 506, 176 ], "score": 1.0, "content": "Without input-dependent selectivity, SSMs can be efficiently implemented as a convolution (Gu", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 174, 505, 187 ], "spans": [ { "bbox": [ 105, 174, 505, 187 ], "score": 1.0, "content": "et al., 2022a; Dao et al., 2023), which leverages the fast Fourier transform (FFT) as primitive. With", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 185, 505, 196 ], "spans": [ { "bbox": [ 106, 185, 505, 196 ], "score": 1.0, "content": "selectivity, SSMs are no-longer equivalent to convolution, but we leverage the parallel associative", "type": "text" } ], "index": 7 }, { "bbox": [ 104, 194, 506, 209 ], "spans": [ { "bbox": [ 104, 194, 306, 209 ], "score": 1.0, "content": "scan. While SSM scans are theoretically efficient", "type": "text" }, { "bbox": [ 306, 195, 357, 207 ], "score": 0.86, "content": "( O ( B L D N )", "type": "inline_equation" }, { "bbox": [ 357, 194, 456, 209 ], "score": 1.0, "content": "FLOPs, scaling linear in", "type": "text" }, { "bbox": [ 456, 196, 464, 205 ], "score": 0.69, "content": "L", "type": "inline_equation" }, { "bbox": [ 465, 194, 506, 209 ], "score": 1.0, "content": "), training", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 207, 506, 218 ], "spans": [ { "bbox": [ 106, 207, 506, 218 ], "score": 1.0, "content": "foundation models with selective SSMs requires them to be efficient on modern hardware (GPUs)", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 217, 506, 230 ], "spans": [ { "bbox": [ 105, 217, 506, 230 ], "score": 1.0, "content": "as well. We describe how we use kernel fusion and recomputation to make SSM scan fast and", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 228, 505, 240 ], "spans": [ { "bbox": [ 106, 228, 505, 240 ], "score": 1.0, "content": "memory-efficient. We evaluate the speed of our scan implementation compared to convolution and", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 239, 506, 252 ], "spans": [ { "bbox": [ 106, 239, 291, 252 ], "score": 1.0, "content": "attention in Section 4.5, showing that it is up to", "type": "text" }, { "bbox": [ 291, 239, 306, 249 ], "score": 0.87, "content": "7 \\times", "type": "inline_equation" }, { "bbox": [ 306, 239, 506, 252 ], "score": 1.0, "content": "times faster than attention at sequence length 32K,", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 250, 427, 261 ], "spans": [ { "bbox": [ 106, 250, 427, 261 ], "score": 1.0, "content": "and is as memory-efficient as the best attention implementation (FlashAttention).", "type": "text" } ], "index": 13 } ], "index": 9 }, { "type": "text", "bbox": [ 107, 270, 505, 313 ], "lines": [ { "bbox": [ 106, 270, 505, 282 ], "spans": [ { "bbox": [ 106, 270, 505, 282 ], "score": 1.0, "content": "Speed. On modern hardware accelerators (GPUs) most operations (except matrix multiply) are", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 280, 505, 292 ], "spans": [ { "bbox": [ 106, 280, 505, 292 ], "score": 1.0, "content": "bounded by memory-bandwidth (Williams et al., 2009; Ivanov et al., 2021; Dao et al., 2022). This the", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 290, 505, 305 ], "spans": [ { "bbox": [ 105, 290, 505, 305 ], "score": 1.0, "content": "case with our scan operation, and we use kernel fusion to reduce the amount of memory IOs, leading", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 302, 355, 314 ], "spans": [ { "bbox": [ 106, 302, 355, 314 ], "score": 1.0, "content": "to significant speedup compared to a standard implementation.", "type": "text" } ], "index": 17 } ], "index": 15.5 }, { "type": "text", "bbox": [ 107, 318, 505, 384 ], "lines": [ { "bbox": [ 105, 317, 505, 331 ], "spans": [ { "bbox": [ 105, 318, 482, 331 ], "score": 1.0, "content": "The standard way to implement the scan algorithm in Section 3.2 is to prepare the scan input", "type": "text" }, { "bbox": [ 482, 317, 505, 330 ], "score": 0.66, "content": "{ \\overline { { A } } } , { \\overline { { B } } }", "type": "inline_equation" } ], "index": 18 }, { "bbox": [ 105, 328, 507, 343 ], "spans": [ { "bbox": [ 105, 328, 135, 343 ], "score": 1.0, "content": "of size", "type": "text" }, { "bbox": [ 135, 329, 183, 342 ], "score": 0.88, "content": "( B , L , D , \\dot { N } )", "type": "inline_equation" }, { "bbox": [ 184, 328, 507, 343 ], "score": 1.0, "content": "in GPU HBM (high-bandwidth memory, commonly referred to as GPU memory),", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 340, 505, 353 ], "spans": [ { "bbox": [ 105, 340, 421, 353 ], "score": 1.0, "content": "call a parallel associative scan implementation to write the scan output of size", "type": "text" }, { "bbox": [ 422, 340, 471, 352 ], "score": 0.87, "content": "( B , L , D , N )", "type": "inline_equation" }, { "bbox": [ 471, 340, 505, 353 ], "score": 1.0, "content": "to GPU", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 349, 506, 364 ], "spans": [ { "bbox": [ 105, 349, 277, 364 ], "score": 1.0, "content": "HBM, then multiply that scan output with", "type": "text" }, { "bbox": [ 278, 352, 288, 361 ], "score": 0.81, "content": "C", "type": "inline_equation" }, { "bbox": [ 288, 349, 403, 364 ], "score": 1.0, "content": "to produce an output of size", "type": "text" }, { "bbox": [ 404, 351, 441, 363 ], "score": 0.9, "content": "( B , \\dot { L } , D )", "type": "inline_equation" }, { "bbox": [ 442, 349, 506, 364 ], "score": 1.0, "content": ". However, this", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 361, 506, 374 ], "spans": [ { "bbox": [ 105, 361, 352, 374 ], "score": 1.0, "content": "requires the number of memory reads/writes on the order of", "type": "text" }, { "bbox": [ 352, 362, 401, 374 ], "score": 0.92, "content": "\\bar { O ( } B L D N )", "type": "inline_equation" }, { "bbox": [ 401, 361, 506, 374 ], "score": 1.0, "content": ". We can instead fuse the", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 372, 403, 385 ], "spans": [ { "bbox": [ 105, 372, 327, 385 ], "score": 1.0, "content": "discretization step, the scan, and the multiplication with", "type": "text" }, { "bbox": [ 327, 373, 337, 383 ], "score": 0.83, "content": "C", "type": "inline_equation" }, { "bbox": [ 338, 372, 403, 385 ], "score": 1.0, "content": "into one kernel:", "type": "text" } ], "index": 23 } ], "index": 20.5 }, { "type": "text", "bbox": [ 125, 392, 505, 462 ], "lines": [ { "bbox": [ 130, 392, 499, 405 ], "spans": [ { "bbox": [ 130, 392, 186, 405 ], "score": 1.0, "content": "1. We read in", "type": "text" }, { "bbox": [ 186, 392, 252, 405 ], "score": 0.9, "content": "O ( B L D + D N )", "type": "inline_equation" }, { "bbox": [ 252, 392, 323, 405 ], "score": 1.0, "content": "bytes of memory", "type": "text" }, { "bbox": [ 323, 392, 372, 404 ], "score": 0.63, "content": "( \\Delta , A , B , C )", "type": "inline_equation" }, { "bbox": [ 373, 392, 499, 405 ], "score": 1.0, "content": "from slow HBM to fast SRAM.", "type": "text" } ], "index": 24 }, { "bbox": [ 129, 408, 384, 421 ], "spans": [ { "bbox": [ 129, 408, 241, 421 ], "score": 1.0, "content": "2. We discretize to produce", "type": "text" }, { "bbox": [ 241, 408, 263, 420 ], "score": 0.47, "content": "\\overline { { A } } , \\overline { { B } }", "type": "inline_equation" }, { "bbox": [ 263, 408, 292, 421 ], "score": 1.0, "content": "of size", "type": "text" }, { "bbox": [ 292, 409, 340, 421 ], "score": 0.63, "content": "( B , L , D , N )", "type": "inline_equation" }, { "bbox": [ 341, 408, 384, 421 ], "score": 1.0, "content": "in SRAM.", "type": "text" } ], "index": 25 }, { "bbox": [ 129, 424, 505, 437 ], "spans": [ { "bbox": [ 129, 424, 444, 437 ], "score": 1.0, "content": "3. We perform a parallel associative scan, yielding intermediate states of size", "type": "text" }, { "bbox": [ 444, 424, 493, 436 ], "score": 0.74, "content": "( B , L , D , N )", "type": "inline_equation" }, { "bbox": [ 494, 424, 505, 437 ], "score": 1.0, "content": "in", "type": "text" } ], "index": 26 }, { "bbox": [ 141, 435, 176, 447 ], "spans": [ { "bbox": [ 141, 435, 176, 447 ], "score": 1.0, "content": "SRAM.", "type": "text" } ], "index": 27 }, { "bbox": [ 129, 450, 484, 463 ], "spans": [ { "bbox": [ 129, 450, 247, 463 ], "score": 1.0, "content": "4. We multiply and sum with", "type": "text" }, { "bbox": [ 248, 451, 257, 460 ], "score": 0.82, "content": "C", "type": "inline_equation" }, { "bbox": [ 258, 450, 363, 463 ], "score": 1.0, "content": ", producing outputs of size", "type": "text" }, { "bbox": [ 363, 450, 399, 462 ], "score": 0.45, "content": "( \\boldsymbol { B } , \\boldsymbol { L } , \\boldsymbol { D } )", "type": "inline_equation" }, { "bbox": [ 399, 450, 484, 463 ], "score": 1.0, "content": "and write it to HBM.", "type": "text" } ], "index": 28 } ], "index": 26 }, { "type": "text", "bbox": [ 108, 470, 504, 493 ], "lines": [ { "bbox": [ 106, 469, 505, 484 ], "spans": [ { "bbox": [ 106, 469, 262, 484 ], "score": 1.0, "content": "This way, we reduce IOs by a factor of", "type": "text" }, { "bbox": [ 262, 470, 288, 483 ], "score": 0.92, "content": "O ( N )", "type": "inline_equation" }, { "bbox": [ 288, 469, 505, 484 ], "score": 1.0, "content": "(the state dimension), which in practice speeds up the", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 480, 263, 494 ], "spans": [ { "bbox": [ 106, 480, 263, 494 ], "score": 1.0, "content": "operation by 20-40 times (Section 4.5).", "type": "text" } ], "index": 30 } ], "index": 29.5 }, { "type": "text", "bbox": [ 108, 497, 503, 530 ], "lines": [ { "bbox": [ 106, 496, 505, 510 ], "spans": [ { "bbox": [ 106, 496, 189, 510 ], "score": 1.0, "content": "For sequence length", "type": "text" }, { "bbox": [ 189, 497, 198, 507 ], "score": 0.71, "content": "L", "type": "inline_equation" }, { "bbox": [ 198, 496, 505, 510 ], "score": 1.0, "content": "too long where we cannot fit the sequence in SRAM (which is much smaller", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 507, 505, 521 ], "spans": [ { "bbox": [ 105, 507, 505, 521 ], "score": 1.0, "content": "than HBM), we split the sequences into chunks and perform the fused scan on each chunk. As long", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 519, 444, 530 ], "spans": [ { "bbox": [ 105, 519, 444, 530 ], "score": 1.0, "content": "as we have the intermediate scan states, we can continue the scan with the next chunk.", "type": "text" } ], "index": 33 } ], "index": 32 }, { "type": "text", "bbox": [ 106, 538, 504, 561 ], "lines": [ { "bbox": [ 106, 539, 504, 550 ], "spans": [ { "bbox": [ 106, 539, 504, 550 ], "score": 1.0, "content": "Memory. We describe how we use the classical technique of recomputation to reduce the total", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 550, 335, 561 ], "spans": [ { "bbox": [ 106, 550, 335, 561 ], "score": 1.0, "content": "amount of memory required to train selective SSM layers.", "type": "text" } ], "index": 35 } ], "index": 34.5 }, { "type": "text", "bbox": [ 107, 565, 505, 641 ], "lines": [ { "bbox": [ 105, 565, 505, 577 ], "spans": [ { "bbox": [ 105, 565, 454, 577 ], "score": 1.0, "content": "From the way we fuse the forward pass, we do not save the intermediate states of size", "type": "text" }, { "bbox": [ 455, 565, 505, 577 ], "score": 0.85, "content": "( B , L , D , N )", "type": "inline_equation" } ], "index": 36 }, { "bbox": [ 105, 577, 506, 589 ], "spans": [ { "bbox": [ 105, 577, 506, 589 ], "score": 1.0, "content": "to avoid memory blowup. However, these intermediate states are necessary for the backward pass", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 587, 505, 599 ], "spans": [ { "bbox": [ 105, 587, 505, 599 ], "score": 1.0, "content": "to compute gradients. We instead recompute those intermediate states in the backward pass. Since the", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 597, 505, 609 ], "spans": [ { "bbox": [ 105, 597, 133, 609 ], "score": 1.0, "content": "inputs", "type": "text" }, { "bbox": [ 133, 597, 177, 608 ], "score": 0.87, "content": "\\bar { \\Delta } , \\bar { A , } \\bar { B , } C", "type": "inline_equation" }, { "bbox": [ 178, 597, 403, 609 ], "score": 1.0, "content": "and output gradient read from HBM to SRAM are of size", "type": "text" }, { "bbox": [ 404, 597, 470, 609 ], "score": 0.89, "content": "O ( B L N { + } D N )", "type": "inline_equation" }, { "bbox": [ 471, 597, 505, 609 ], "score": 1.0, "content": ", and the", "type": "text" } ], "index": 39 }, { "bbox": [ 104, 608, 505, 621 ], "spans": [ { "bbox": [ 104, 608, 226, 621 ], "score": 1.0, "content": "input gradients are also of size", "type": "text" }, { "bbox": [ 226, 608, 293, 620 ], "score": 0.93, "content": "O ( B L N + D N )", "type": "inline_equation" }, { "bbox": [ 293, 608, 456, 621 ], "score": 1.0, "content": ", recomputation avoids the cost of reading", "type": "text" }, { "bbox": [ 456, 608, 505, 620 ], "score": 0.91, "content": "O ( B L N D )", "type": "inline_equation" } ], "index": 40 }, { "bbox": [ 105, 619, 506, 632 ], "spans": [ { "bbox": [ 105, 619, 506, 632 ], "score": 1.0, "content": "elements from HBM. This means that recomputation of the SSM states in the backward pass speeds", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 630, 406, 642 ], "spans": [ { "bbox": [ 105, 630, 406, 642 ], "score": 1.0, "content": "up the computation compared to storing them and reading them from HBM.", "type": "text" } ], "index": 42 } ], "index": 39 }, { "type": "text", "bbox": [ 107, 645, 505, 732 ], "lines": [ { "bbox": [ 106, 646, 505, 658 ], "spans": [ { "bbox": [ 106, 646, 505, 658 ], "score": 1.0, "content": "Beyond optimizing for the memory requirement of just the scan operation, we also use recomputation", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 656, 506, 669 ], "spans": [ { "bbox": [ 106, 656, 506, 669 ], "score": 1.0, "content": "to optimize the memory requirement of the entire selective SSM block (input projection, convolution,", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 667, 505, 679 ], "spans": [ { "bbox": [ 105, 667, 505, 679 ], "score": 1.0, "content": "activation, scan, output projection). In particular, we do not save intermediate activations that take", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 677, 506, 691 ], "spans": [ { "bbox": [ 105, 677, 506, 691 ], "score": 1.0, "content": "a lot of memory but are fast to recompute (e.g. output of activation function or short convolution).", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 687, 506, 701 ], "spans": [ { "bbox": [ 105, 687, 506, 701 ], "score": 1.0, "content": "As a result, the selective SSM layer has the same memory requirement as an optimized Transformer", "type": "text" } ], "index": 47 }, { "bbox": [ 106, 699, 505, 711 ], "spans": [ { "bbox": [ 106, 699, 505, 711 ], "score": 1.0, "content": "implementation with FlashAttention. In particular, each attention layer (FlashAttention) stores around", "type": "text" } ], "index": 48 }, { "bbox": [ 105, 709, 506, 723 ], "spans": [ { "bbox": [ 105, 709, 506, 723 ], "score": 1.0, "content": "12 bytes of activations per token, an each MLP layer stores around 20 bytes of activations per token,", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 720, 506, 733 ], "spans": [ { "bbox": [ 105, 720, 506, 733 ], "score": 1.0, "content": "for a total of 32 bytes ((assuming mixed-precision training in FP16 or BF16)). Each selective SSM", "type": "text" } ], "index": 50 } ], "index": 46.5 } ], "page_idx": 17, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 304, 37 ], "lines": [ { "bbox": [ 106, 25, 305, 39 ], "spans": [ { "bbox": [ 106, 25, 305, 39 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 310, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 763 ], "spans": [ { "bbox": [ 299, 750, 312, 763 ], "score": 1.0, "content": "18", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 495, 119, 505, 130 ], "lines": [ { "bbox": [ 496, 121, 504, 129 ], "spans": [ { "bbox": [ 496, 121, 504, 129 ], "score": 0.994, "content": "□", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 108, 82, 264, 93 ], "lines": [ { "bbox": [ 106, 80, 267, 95 ], "spans": [ { "bbox": [ 106, 80, 267, 95 ], "score": 1.0, "content": "Thus the final discrete recurrence (2a) is", "type": "text" } ], "index": 0 } ], "index": 0, "bbox_fs": [ 106, 80, 267, 95 ] }, { "type": "interline_equation", "bbox": [ 252, 92, 358, 121 ], "lines": [ { "bbox": [ 252, 92, 358, 121 ], "spans": [ { "bbox": [ 252, 92, 358, 121 ], "score": 0.87, "content": "\\begin{array} { l } { g _ { k } = \\sigma ( \\mathsf { L i n e a r } ( x _ { k } ) ) } \\\\ { h _ { k } = ( 1 - g _ { k } ) h _ { k - 1 } + g _ { k } x _ { k } } \\end{array}", "type": "interline_equation", "image_path": "7bdb05ee7e5171720bf82b576796e8053a73c7dfcb369c4f0eb60ba57531675e.jpg" } ] } ], "index": 1.5, "virtual_lines": [ { "bbox": [ 252, 92, 358, 106.5 ], "spans": [], "index": 1 }, { "bbox": [ 252, 106.5, 358, 121.0 ], "spans": [], "index": 2 } ] }, { "type": "text", "bbox": [ 107, 119, 149, 130 ], "lines": [ { "bbox": [ 105, 118, 151, 131 ], "spans": [ { "bbox": [ 105, 118, 151, 131 ], "score": 1.0, "content": "as desired.", "type": "text" } ], "index": 3 } ], "index": 3, "bbox_fs": [ 105, 118, 151, 131 ] }, { "type": "title", "bbox": [ 108, 145, 415, 158 ], "lines": [ { "bbox": [ 106, 145, 415, 159 ], "spans": [ { "bbox": [ 106, 145, 415, 159 ], "score": 1.0, "content": "D HARDWARE-AWARE ALGORITHM FOR SELECTIVE SSMS", "type": "text" } ], "index": 4 } ], "index": 4 }, { "type": "text", "bbox": [ 106, 163, 505, 261 ], "lines": [ { "bbox": [ 106, 163, 506, 176 ], "spans": [ { "bbox": [ 106, 163, 506, 176 ], "score": 1.0, "content": "Without input-dependent selectivity, SSMs can be efficiently implemented as a convolution (Gu", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 174, 505, 187 ], "spans": [ { "bbox": [ 105, 174, 505, 187 ], "score": 1.0, "content": "et al., 2022a; Dao et al., 2023), which leverages the fast Fourier transform (FFT) as primitive. With", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 185, 505, 196 ], "spans": [ { "bbox": [ 106, 185, 505, 196 ], "score": 1.0, "content": "selectivity, SSMs are no-longer equivalent to convolution, but we leverage the parallel associative", "type": "text" } ], "index": 7 }, { "bbox": [ 104, 194, 506, 209 ], "spans": [ { "bbox": [ 104, 194, 306, 209 ], "score": 1.0, "content": "scan. While SSM scans are theoretically efficient", "type": "text" }, { "bbox": [ 306, 195, 357, 207 ], "score": 0.86, "content": "( O ( B L D N )", "type": "inline_equation" }, { "bbox": [ 357, 194, 456, 209 ], "score": 1.0, "content": "FLOPs, scaling linear in", "type": "text" }, { "bbox": [ 456, 196, 464, 205 ], "score": 0.69, "content": "L", "type": "inline_equation" }, { "bbox": [ 465, 194, 506, 209 ], "score": 1.0, "content": "), training", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 207, 506, 218 ], "spans": [ { "bbox": [ 106, 207, 506, 218 ], "score": 1.0, "content": "foundation models with selective SSMs requires them to be efficient on modern hardware (GPUs)", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 217, 506, 230 ], "spans": [ { "bbox": [ 105, 217, 506, 230 ], "score": 1.0, "content": "as well. We describe how we use kernel fusion and recomputation to make SSM scan fast and", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 228, 505, 240 ], "spans": [ { "bbox": [ 106, 228, 505, 240 ], "score": 1.0, "content": "memory-efficient. We evaluate the speed of our scan implementation compared to convolution and", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 239, 506, 252 ], "spans": [ { "bbox": [ 106, 239, 291, 252 ], "score": 1.0, "content": "attention in Section 4.5, showing that it is up to", "type": "text" }, { "bbox": [ 291, 239, 306, 249 ], "score": 0.87, "content": "7 \\times", "type": "inline_equation" }, { "bbox": [ 306, 239, 506, 252 ], "score": 1.0, "content": "times faster than attention at sequence length 32K,", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 250, 427, 261 ], "spans": [ { "bbox": [ 106, 250, 427, 261 ], "score": 1.0, "content": "and is as memory-efficient as the best attention implementation (FlashAttention).", "type": "text" } ], "index": 13 } ], "index": 9, "bbox_fs": [ 104, 163, 506, 261 ] }, { "type": "text", "bbox": [ 107, 270, 505, 313 ], "lines": [ { "bbox": [ 106, 270, 505, 282 ], "spans": [ { "bbox": [ 106, 270, 505, 282 ], "score": 1.0, "content": "Speed. On modern hardware accelerators (GPUs) most operations (except matrix multiply) are", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 280, 505, 292 ], "spans": [ { "bbox": [ 106, 280, 505, 292 ], "score": 1.0, "content": "bounded by memory-bandwidth (Williams et al., 2009; Ivanov et al., 2021; Dao et al., 2022). This the", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 290, 505, 305 ], "spans": [ { "bbox": [ 105, 290, 505, 305 ], "score": 1.0, "content": "case with our scan operation, and we use kernel fusion to reduce the amount of memory IOs, leading", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 302, 355, 314 ], "spans": [ { "bbox": [ 106, 302, 355, 314 ], "score": 1.0, "content": "to significant speedup compared to a standard implementation.", "type": "text" } ], "index": 17 } ], "index": 15.5, "bbox_fs": [ 105, 270, 505, 314 ] }, { "type": "text", "bbox": [ 107, 318, 505, 384 ], "lines": [ { "bbox": [ 105, 317, 505, 331 ], "spans": [ { "bbox": [ 105, 318, 482, 331 ], "score": 1.0, "content": "The standard way to implement the scan algorithm in Section 3.2 is to prepare the scan input", "type": "text" }, { "bbox": [ 482, 317, 505, 330 ], "score": 0.66, "content": "{ \\overline { { A } } } , { \\overline { { B } } }", "type": "inline_equation" } ], "index": 18 }, { "bbox": [ 105, 328, 507, 343 ], "spans": [ { "bbox": [ 105, 328, 135, 343 ], "score": 1.0, "content": "of size", "type": "text" }, { "bbox": [ 135, 329, 183, 342 ], "score": 0.88, "content": "( B , L , D , \\dot { N } )", "type": "inline_equation" }, { "bbox": [ 184, 328, 507, 343 ], "score": 1.0, "content": "in GPU HBM (high-bandwidth memory, commonly referred to as GPU memory),", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 340, 505, 353 ], "spans": [ { "bbox": [ 105, 340, 421, 353 ], "score": 1.0, "content": "call a parallel associative scan implementation to write the scan output of size", "type": "text" }, { "bbox": [ 422, 340, 471, 352 ], "score": 0.87, "content": "( B , L , D , N )", "type": "inline_equation" }, { "bbox": [ 471, 340, 505, 353 ], "score": 1.0, "content": "to GPU", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 349, 506, 364 ], "spans": [ { "bbox": [ 105, 349, 277, 364 ], "score": 1.0, "content": "HBM, then multiply that scan output with", "type": "text" }, { "bbox": [ 278, 352, 288, 361 ], "score": 0.81, "content": "C", "type": "inline_equation" }, { "bbox": [ 288, 349, 403, 364 ], "score": 1.0, "content": "to produce an output of size", "type": "text" }, { "bbox": [ 404, 351, 441, 363 ], "score": 0.9, "content": "( B , \\dot { L } , D )", "type": "inline_equation" }, { "bbox": [ 442, 349, 506, 364 ], "score": 1.0, "content": ". However, this", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 361, 506, 374 ], "spans": [ { "bbox": [ 105, 361, 352, 374 ], "score": 1.0, "content": "requires the number of memory reads/writes on the order of", "type": "text" }, { "bbox": [ 352, 362, 401, 374 ], "score": 0.92, "content": "\\bar { O ( } B L D N )", "type": "inline_equation" }, { "bbox": [ 401, 361, 506, 374 ], "score": 1.0, "content": ". We can instead fuse the", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 372, 403, 385 ], "spans": [ { "bbox": [ 105, 372, 327, 385 ], "score": 1.0, "content": "discretization step, the scan, and the multiplication with", "type": "text" }, { "bbox": [ 327, 373, 337, 383 ], "score": 0.83, "content": "C", "type": "inline_equation" }, { "bbox": [ 338, 372, 403, 385 ], "score": 1.0, "content": "into one kernel:", "type": "text" } ], "index": 23 } ], "index": 20.5, "bbox_fs": [ 105, 317, 507, 385 ] }, { "type": "index", "bbox": [ 125, 392, 505, 462 ], "lines": [ { "bbox": [ 130, 392, 499, 405 ], "spans": [ { "bbox": [ 130, 392, 186, 405 ], "score": 1.0, "content": "1. We read in", "type": "text" }, { "bbox": [ 186, 392, 252, 405 ], "score": 0.9, "content": "O ( B L D + D N )", "type": "inline_equation" }, { "bbox": [ 252, 392, 323, 405 ], "score": 1.0, "content": "bytes of memory", "type": "text" }, { "bbox": [ 323, 392, 372, 404 ], "score": 0.63, "content": "( \\Delta , A , B , C )", "type": "inline_equation" }, { "bbox": [ 373, 392, 499, 405 ], "score": 1.0, "content": "from slow HBM to fast SRAM.", "type": "text" } ], "index": 24, "is_list_start_line": true }, { "bbox": [ 129, 408, 384, 421 ], "spans": [ { "bbox": [ 129, 408, 241, 421 ], "score": 1.0, "content": "2. We discretize to produce", "type": "text" }, { "bbox": [ 241, 408, 263, 420 ], "score": 0.47, "content": "\\overline { { A } } , \\overline { { B } }", "type": "inline_equation" }, { "bbox": [ 263, 408, 292, 421 ], "score": 1.0, "content": "of size", "type": "text" }, { "bbox": [ 292, 409, 340, 421 ], "score": 0.63, "content": "( B , L , D , N )", "type": "inline_equation" }, { "bbox": [ 341, 408, 384, 421 ], "score": 1.0, "content": "in SRAM.", "type": "text" } ], "index": 25, "is_list_start_line": true }, { "bbox": [ 129, 424, 505, 437 ], "spans": [ { "bbox": [ 129, 424, 444, 437 ], "score": 1.0, "content": "3. We perform a parallel associative scan, yielding intermediate states of size", "type": "text" }, { "bbox": [ 444, 424, 493, 436 ], "score": 0.74, "content": "( B , L , D , N )", "type": "inline_equation" }, { "bbox": [ 494, 424, 505, 437 ], "score": 1.0, "content": "in", "type": "text" } ], "index": 26, "is_list_start_line": true }, { "bbox": [ 141, 435, 176, 447 ], "spans": [ { "bbox": [ 141, 435, 176, 447 ], "score": 1.0, "content": "SRAM.", "type": "text" } ], "index": 27, "is_list_start_line": true }, { "bbox": [ 129, 450, 484, 463 ], "spans": [ { "bbox": [ 129, 450, 247, 463 ], "score": 1.0, "content": "4. We multiply and sum with", "type": "text" }, { "bbox": [ 248, 451, 257, 460 ], "score": 0.82, "content": "C", "type": "inline_equation" }, { "bbox": [ 258, 450, 363, 463 ], "score": 1.0, "content": ", producing outputs of size", "type": "text" }, { "bbox": [ 363, 450, 399, 462 ], "score": 0.45, "content": "( \\boldsymbol { B } , \\boldsymbol { L } , \\boldsymbol { D } )", "type": "inline_equation" }, { "bbox": [ 399, 450, 484, 463 ], "score": 1.0, "content": "and write it to HBM.", "type": "text" } ], "index": 28, "is_list_start_line": true } ], "index": 26, "bbox_fs": [ 129, 392, 505, 463 ] }, { "type": "text", "bbox": [ 108, 470, 504, 493 ], "lines": [ { "bbox": [ 106, 469, 505, 484 ], "spans": [ { "bbox": [ 106, 469, 262, 484 ], "score": 1.0, "content": "This way, we reduce IOs by a factor of", "type": "text" }, { "bbox": [ 262, 470, 288, 483 ], "score": 0.92, "content": "O ( N )", "type": "inline_equation" }, { "bbox": [ 288, 469, 505, 484 ], "score": 1.0, "content": "(the state dimension), which in practice speeds up the", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 480, 263, 494 ], "spans": [ { "bbox": [ 106, 480, 263, 494 ], "score": 1.0, "content": "operation by 20-40 times (Section 4.5).", "type": "text" } ], "index": 30 } ], "index": 29.5, "bbox_fs": [ 106, 469, 505, 494 ] }, { "type": "text", "bbox": [ 108, 497, 503, 530 ], "lines": [ { "bbox": [ 106, 496, 505, 510 ], "spans": [ { "bbox": [ 106, 496, 189, 510 ], "score": 1.0, "content": "For sequence length", "type": "text" }, { "bbox": [ 189, 497, 198, 507 ], "score": 0.71, "content": "L", "type": "inline_equation" }, { "bbox": [ 198, 496, 505, 510 ], "score": 1.0, "content": "too long where we cannot fit the sequence in SRAM (which is much smaller", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 507, 505, 521 ], "spans": [ { "bbox": [ 105, 507, 505, 521 ], "score": 1.0, "content": "than HBM), we split the sequences into chunks and perform the fused scan on each chunk. As long", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 519, 444, 530 ], "spans": [ { "bbox": [ 105, 519, 444, 530 ], "score": 1.0, "content": "as we have the intermediate scan states, we can continue the scan with the next chunk.", "type": "text" } ], "index": 33 } ], "index": 32, "bbox_fs": [ 105, 496, 505, 530 ] }, { "type": "text", "bbox": [ 106, 538, 504, 561 ], "lines": [ { "bbox": [ 106, 539, 504, 550 ], "spans": [ { "bbox": [ 106, 539, 504, 550 ], "score": 1.0, "content": "Memory. We describe how we use the classical technique of recomputation to reduce the total", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 550, 335, 561 ], "spans": [ { "bbox": [ 106, 550, 335, 561 ], "score": 1.0, "content": "amount of memory required to train selective SSM layers.", "type": "text" } ], "index": 35 } ], "index": 34.5, "bbox_fs": [ 106, 539, 504, 561 ] }, { "type": "text", "bbox": [ 107, 565, 505, 641 ], "lines": [ { "bbox": [ 105, 565, 505, 577 ], "spans": [ { "bbox": [ 105, 565, 454, 577 ], "score": 1.0, "content": "From the way we fuse the forward pass, we do not save the intermediate states of size", "type": "text" }, { "bbox": [ 455, 565, 505, 577 ], "score": 0.85, "content": "( B , L , D , N )", "type": "inline_equation" } ], "index": 36 }, { "bbox": [ 105, 577, 506, 589 ], "spans": [ { "bbox": [ 105, 577, 506, 589 ], "score": 1.0, "content": "to avoid memory blowup. However, these intermediate states are necessary for the backward pass", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 587, 505, 599 ], "spans": [ { "bbox": [ 105, 587, 505, 599 ], "score": 1.0, "content": "to compute gradients. We instead recompute those intermediate states in the backward pass. Since the", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 597, 505, 609 ], "spans": [ { "bbox": [ 105, 597, 133, 609 ], "score": 1.0, "content": "inputs", "type": "text" }, { "bbox": [ 133, 597, 177, 608 ], "score": 0.87, "content": "\\bar { \\Delta } , \\bar { A , } \\bar { B , } C", "type": "inline_equation" }, { "bbox": [ 178, 597, 403, 609 ], "score": 1.0, "content": "and output gradient read from HBM to SRAM are of size", "type": "text" }, { "bbox": [ 404, 597, 470, 609 ], "score": 0.89, "content": "O ( B L N { + } D N )", "type": "inline_equation" }, { "bbox": [ 471, 597, 505, 609 ], "score": 1.0, "content": ", and the", "type": "text" } ], "index": 39 }, { "bbox": [ 104, 608, 505, 621 ], "spans": [ { "bbox": [ 104, 608, 226, 621 ], "score": 1.0, "content": "input gradients are also of size", "type": "text" }, { "bbox": [ 226, 608, 293, 620 ], "score": 0.93, "content": "O ( B L N + D N )", "type": "inline_equation" }, { "bbox": [ 293, 608, 456, 621 ], "score": 1.0, "content": ", recomputation avoids the cost of reading", "type": "text" }, { "bbox": [ 456, 608, 505, 620 ], "score": 0.91, "content": "O ( B L N D )", "type": "inline_equation" } ], "index": 40 }, { "bbox": [ 105, 619, 506, 632 ], "spans": [ { "bbox": [ 105, 619, 506, 632 ], "score": 1.0, "content": "elements from HBM. This means that recomputation of the SSM states in the backward pass speeds", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 630, 406, 642 ], "spans": [ { "bbox": [ 105, 630, 406, 642 ], "score": 1.0, "content": "up the computation compared to storing them and reading them from HBM.", "type": "text" } ], "index": 42 } ], "index": 39, "bbox_fs": [ 104, 565, 506, 642 ] }, { "type": "text", "bbox": [ 107, 645, 505, 732 ], "lines": [ { "bbox": [ 106, 646, 505, 658 ], "spans": [ { "bbox": [ 106, 646, 505, 658 ], "score": 1.0, "content": "Beyond optimizing for the memory requirement of just the scan operation, we also use recomputation", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 656, 506, 669 ], "spans": [ { "bbox": [ 106, 656, 506, 669 ], "score": 1.0, "content": "to optimize the memory requirement of the entire selective SSM block (input projection, convolution,", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 667, 505, 679 ], "spans": [ { "bbox": [ 105, 667, 505, 679 ], "score": 1.0, "content": "activation, scan, output projection). In particular, we do not save intermediate activations that take", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 677, 506, 691 ], "spans": [ { "bbox": [ 105, 677, 506, 691 ], "score": 1.0, "content": "a lot of memory but are fast to recompute (e.g. output of activation function or short convolution).", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 687, 506, 701 ], "spans": [ { "bbox": [ 105, 687, 506, 701 ], "score": 1.0, "content": "As a result, the selective SSM layer has the same memory requirement as an optimized Transformer", "type": "text" } ], "index": 47 }, { "bbox": [ 106, 699, 505, 711 ], "spans": [ { "bbox": [ 106, 699, 505, 711 ], "score": 1.0, "content": "implementation with FlashAttention. In particular, each attention layer (FlashAttention) stores around", "type": "text" } ], "index": 48 }, { "bbox": [ 105, 709, 506, 723 ], "spans": [ { "bbox": [ 105, 709, 506, 723 ], "score": 1.0, "content": "12 bytes of activations per token, an each MLP layer stores around 20 bytes of activations per token,", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 720, 506, 733 ], "spans": [ { "bbox": [ 105, 720, 506, 733 ], "score": 1.0, "content": "for a total of 32 bytes ((assuming mixed-precision training in FP16 or BF16)). Each selective SSM", "type": "text" } ], "index": 50 }, { "bbox": [ 105, 82, 505, 95 ], "spans": [ { "bbox": [ 105, 82, 505, 95 ], "score": 1.0, "content": "stores around 16 bytes of activations per token. Hence two layers of selective SSMs have around the", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 105, 93, 360, 106 ], "spans": [ { "bbox": [ 105, 93, 360, 106 ], "score": 1.0, "content": "same activation memory as an attention layer and an MLP layer.", "type": "text", "cross_page": true } ], "index": 1 } ], "index": 46.5, "bbox_fs": [ 105, 646, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 82, 504, 105 ], "lines": [ { "bbox": [ 105, 82, 505, 95 ], "spans": [ { "bbox": [ 105, 82, 505, 95 ], "score": 1.0, "content": "stores around 16 bytes of activations per token. Hence two layers of selective SSMs have around the", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 360, 106 ], "spans": [ { "bbox": [ 105, 93, 360, 106 ], "score": 1.0, "content": "same activation memory as an attention layer and an MLP layer.", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "title", "bbox": [ 107, 117, 232, 129 ], "lines": [ { "bbox": [ 105, 116, 232, 131 ], "spans": [ { "bbox": [ 105, 116, 232, 131 ], "score": 1.0, "content": "E FULL EXPERIMENTS", "type": "text" } ], "index": 2 } ], "index": 2 }, { "type": "title", "bbox": [ 107, 136, 213, 147 ], "lines": [ { "bbox": [ 105, 135, 214, 148 ], "spans": [ { "bbox": [ 105, 135, 214, 148 ], "score": 1.0, "content": "E.1 SYNTHETIC TASKS", "type": "text" } ], "index": 3 } ], "index": 3 }, { "type": "text", "bbox": [ 106, 151, 504, 163 ], "lines": [ { "bbox": [ 106, 151, 505, 165 ], "spans": [ { "bbox": [ 106, 151, 505, 165 ], "score": 1.0, "content": "Full experiment details for these tasks including task details and training protocol are in Appendix F.1.", "type": "text" } ], "index": 4 } ], "index": 4 }, { "type": "title", "bbox": [ 107, 173, 232, 185 ], "lines": [ { "bbox": [ 106, 173, 232, 185 ], "spans": [ { "bbox": [ 106, 173, 232, 185 ], "score": 1.0, "content": "E.1.1 SELECTIVE COPYING", "type": "text" } ], "index": 5 } ], "index": 5 }, { "type": "text", "bbox": [ 106, 194, 505, 270 ], "lines": [ { "bbox": [ 106, 194, 505, 206 ], "spans": [ { "bbox": [ 106, 194, 505, 206 ], "score": 1.0, "content": "The Copying task is one of the most well-studied synthetic tasks for sequence modeling, originally", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 204, 505, 216 ], "spans": [ { "bbox": [ 105, 204, 505, 216 ], "score": 1.0, "content": "designed to test the memorization abilities of recurrent models. As discussed in Section 3.1, LTI SSMs", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 216, 504, 227 ], "spans": [ { "bbox": [ 106, 216, 504, 227 ], "score": 1.0, "content": "(linear recurrences and global convolutions) can easily solve this task by only keeping track of time", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 227, 504, 238 ], "spans": [ { "bbox": [ 106, 227, 504, 238 ], "score": 1.0, "content": "instead of reasoning about the data; for example, by constructing a convolution kernel of exactly the", "type": "text" } ], "index": 9 }, { "bbox": [ 104, 236, 506, 249 ], "spans": [ { "bbox": [ 104, 236, 506, 249 ], "score": 1.0, "content": "right length (Figure 1). This was explicitly validated in earlier work on global convolutions (Romero", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 247, 505, 260 ], "spans": [ { "bbox": [ 105, 247, 505, 260 ], "score": 1.0, "content": "et al., 2021). The Selective Copying task prevents this shortcut by randomizing the spacing between", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 258, 476, 270 ], "spans": [ { "bbox": [ 106, 258, 476, 270 ], "score": 1.0, "content": "tokens. Note that this task has been introduced before as the Denoising task (Jing et al., 2019).", "type": "text" } ], "index": 12 } ], "index": 9 }, { "type": "text", "bbox": [ 107, 274, 505, 329 ], "lines": [ { "bbox": [ 106, 275, 505, 286 ], "spans": [ { "bbox": [ 106, 275, 505, 286 ], "score": 1.0, "content": "Note that many previous works argue that adding architecture gating (multiplicative interactions) can", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 284, 505, 296 ], "spans": [ { "bbox": [ 106, 284, 505, 296 ], "score": 1.0, "content": "endow models with “data-dependence” and solve related tasks (Dao et al., 2023; Poli et al., 2023).", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 294, 506, 309 ], "spans": [ { "bbox": [ 105, 294, 506, 309 ], "score": 1.0, "content": "However, we find this explanation insufficient intuitively because such gating does not interact along", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 306, 506, 319 ], "spans": [ { "bbox": [ 105, 306, 506, 319 ], "score": 1.0, "content": "the sequence axis, and cannot affect the spacing between tokens. In particular architecture gating is", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 317, 336, 329 ], "spans": [ { "bbox": [ 105, 317, 336, 329 ], "score": 1.0, "content": "not an instance of a selection mechanism (Appendix A.1).", "type": "text" } ], "index": 17 } ], "index": 15 }, { "type": "text", "bbox": [ 107, 333, 505, 366 ], "lines": [ { "bbox": [ 105, 331, 506, 347 ], "spans": [ { "bbox": [ 105, 331, 506, 347 ], "score": 1.0, "content": "Table 1 confirms that gated architectures such as H3 and Mamba only partially improve performance,", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 344, 505, 356 ], "spans": [ { "bbox": [ 106, 344, 505, 356 ], "score": 1.0, "content": "while the selection mechanism (modifying S4 to S6) easily solves this task, particularly when combined", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 355, 264, 366 ], "spans": [ { "bbox": [ 106, 355, 264, 366 ], "score": 1.0, "content": "with these more powerful architectures.", "type": "text" } ], "index": 20 } ], "index": 19 }, { "type": "title", "bbox": [ 107, 376, 223, 388 ], "lines": [ { "bbox": [ 106, 376, 224, 389 ], "spans": [ { "bbox": [ 106, 376, 224, 389 ], "score": 1.0, "content": "E.1.2 INDUCTION HEADS", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "text", "bbox": [ 107, 397, 505, 451 ], "lines": [ { "bbox": [ 105, 395, 505, 410 ], "spans": [ { "bbox": [ 105, 395, 505, 410 ], "score": 1.0, "content": "Induction heads (Olsson et al., 2022) is a simple task from the mechanistic interpretability lens (Elhage", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 408, 505, 420 ], "spans": [ { "bbox": [ 105, 408, 505, 420 ], "score": 1.0, "content": "et al., 2021) that is surprisingly predictive of the in-context learning ability of LLMs. It requires models", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 418, 505, 432 ], "spans": [ { "bbox": [ 105, 418, 505, 432 ], "score": 1.0, "content": "to perform associative recall and copy: for example, if the model has seen a bigram such as “Harry", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 429, 505, 442 ], "spans": [ { "bbox": [ 105, 429, 505, 442 ], "score": 1.0, "content": "Potter” in the sequence, then the next time “Harry” appears in the same sequence, the model should", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 439, 310, 453 ], "spans": [ { "bbox": [ 105, 439, 310, 453 ], "score": 1.0, "content": "be able to predict “Potter” by copying from history.", "type": "text" } ], "index": 26 } ], "index": 24 }, { "type": "text", "bbox": [ 107, 462, 505, 505 ], "lines": [ { "bbox": [ 106, 461, 505, 473 ], "spans": [ { "bbox": [ 106, 461, 505, 473 ], "score": 1.0, "content": "Dataset We train a 2-layer model on the induction heads task at sequence length 256, with a vocab", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 472, 507, 485 ], "spans": [ { "bbox": [ 105, 472, 507, 485 ], "score": 1.0, "content": "size of 16, which is comparable to prior work on this task (Dao et al., 2023) but with longer sequences.", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 483, 506, 496 ], "spans": [ { "bbox": [ 105, 483, 506, 496 ], "score": 1.0, "content": "We additionally investigate generalization and extrapolation abilities by evaluating on a range of", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 493, 359, 507 ], "spans": [ { "bbox": [ 105, 493, 197, 507 ], "score": 1.0, "content": "sequence lengths from", "type": "text" }, { "bbox": [ 198, 494, 228, 505 ], "score": 0.9, "content": "2 ^ { \\overline { { 6 } } } = 6 \\overline { { 4 } }", "type": "inline_equation" }, { "bbox": [ 228, 493, 250, 507 ], "score": 1.0, "content": "up to", "type": "text" }, { "bbox": [ 250, 494, 310, 505 ], "score": 0.89, "content": "2 ^ { 2 0 } = 1 0 4 8 5 7 6", "type": "inline_equation" }, { "bbox": [ 310, 493, 359, 507 ], "score": 1.0, "content": "at test time.", "type": "text" } ], "index": 30 } ], "index": 28.5 }, { "type": "text", "bbox": [ 107, 516, 505, 559 ], "lines": [ { "bbox": [ 105, 515, 505, 528 ], "spans": [ { "bbox": [ 105, 515, 505, 528 ], "score": 1.0, "content": "Models Following established work on induction heads, we use 2 layer models, which allows", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 527, 505, 538 ], "spans": [ { "bbox": [ 105, 527, 505, 538 ], "score": 1.0, "content": "attention to mechanistically solve the induction heads task (Olsson et al., 2022). We test both", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 538, 505, 550 ], "spans": [ { "bbox": [ 105, 538, 505, 550 ], "score": 1.0, "content": "multi-head attention (8 heads, with various positional encodings) and SSM variants. We use a model", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 549, 346, 559 ], "spans": [ { "bbox": [ 106, 549, 150, 559 ], "score": 1.0, "content": "dimension", "type": "text" }, { "bbox": [ 150, 549, 160, 558 ], "score": 0.78, "content": "D", "type": "inline_equation" }, { "bbox": [ 160, 549, 346, 559 ], "score": 1.0, "content": "of 64 for Mamba and 128 for the other models.", "type": "text" } ], "index": 34 } ], "index": 32.5 }, { "type": "text", "bbox": [ 107, 570, 505, 614 ], "lines": [ { "bbox": [ 105, 569, 505, 583 ], "spans": [ { "bbox": [ 105, 569, 505, 583 ], "score": 1.0, "content": "Results Figure 3 shows that Mamba—or more precisely, its selective SSM layer—has the ability to", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 580, 506, 594 ], "spans": [ { "bbox": [ 105, 580, 506, 594 ], "score": 1.0, "content": "solve the task perfectly because of its ability to selectively remember the relevant token while ignoring", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 592, 506, 605 ], "spans": [ { "bbox": [ 105, 592, 446, 605 ], "score": 1.0, "content": "everything else in between. It generalizes perfectly to million-length sequences, or", "type": "text" }, { "bbox": [ 446, 592, 475, 603 ], "score": 0.87, "content": "4 0 0 0 \\times", "type": "inline_equation" }, { "bbox": [ 476, 592, 506, 605 ], "score": 1.0, "content": "longer", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 603, 374, 615 ], "spans": [ { "bbox": [ 106, 603, 356, 615 ], "score": 1.0, "content": "than it saw during training, while no other method goes beyond", "type": "text" }, { "bbox": [ 356, 603, 370, 613 ], "score": 0.86, "content": "2 \\times", "type": "inline_equation" }, { "bbox": [ 370, 603, 374, 615 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 38 } ], "index": 36.5 }, { "type": "text", "bbox": [ 107, 618, 505, 662 ], "lines": [ { "bbox": [ 106, 619, 505, 630 ], "spans": [ { "bbox": [ 106, 619, 505, 630 ], "score": 1.0, "content": "Out of positional encoding variants for attention models, xPos (which was designed for length", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 630, 505, 641 ], "spans": [ { "bbox": [ 105, 630, 505, 641 ], "score": 1.0, "content": "extrapolation) is slightly better than the others; also note that all attention models were only tested", "type": "text" } ], "index": 40 }, { "bbox": [ 104, 638, 506, 653 ], "spans": [ { "bbox": [ 104, 638, 197, 653 ], "score": 1.0, "content": "up to sequence length", "type": "text" }, { "bbox": [ 197, 639, 248, 650 ], "score": 0.9, "content": "\\dot { 2 } ^ { 1 4 } = 1 6 3 8 4", "type": "inline_equation" }, { "bbox": [ 249, 638, 506, 653 ], "score": 1.0, "content": "due to memory limitations. Out of other SSMs, H3 and Hyena", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 650, 325, 663 ], "spans": [ { "bbox": [ 105, 650, 325, 663 ], "score": 1.0, "content": "are similar, contrary to the findings in Poli et al. (2023).", "type": "text" } ], "index": 42 } ], "index": 40.5 }, { "type": "title", "bbox": [ 107, 672, 206, 684 ], "lines": [ { "bbox": [ 105, 671, 207, 685 ], "spans": [ { "bbox": [ 105, 671, 207, 685 ], "score": 1.0, "content": "E.2 DNA MODELING", "type": "text" } ], "index": 43 } ], "index": 43 }, { "type": "text", "bbox": [ 108, 689, 504, 732 ], "lines": [ { "bbox": [ 105, 688, 506, 702 ], "spans": [ { "bbox": [ 105, 688, 506, 702 ], "score": 1.0, "content": "For pretraining, we largely follow a standard causal language modeling (next token prediction) setup", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 698, 505, 712 ], "spans": [ { "bbox": [ 105, 698, 505, 712 ], "score": 1.0, "content": "for the training and model details (see also Appendix F.2). For the dataset, we largely follow the", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 708, 506, 724 ], "spans": [ { "bbox": [ 105, 708, 506, 724 ], "score": 1.0, "content": "setup of HyenaDNA (Nguyen et al., 2023), which uses the HG38 dataset for pretraining consisting", "type": "text" } ], "index": 46 }, { "bbox": [ 106, 721, 475, 733 ], "spans": [ { "bbox": [ 106, 721, 475, 733 ], "score": 1.0, "content": "of a single human genome with about 4.5 billion tokens (DNA base pairs) in the training split.", "type": "text" } ], "index": 47 } ], "index": 45.5 } ], "page_idx": 18, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 304, 37 ], "lines": [ { "bbox": [ 106, 26, 305, 38 ], "spans": [ { "bbox": [ 106, 26, 305, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 310, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "", "type": "text", "height": 14, "width": 13 } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 82, 504, 105 ], "lines": [], "index": 0.5, "bbox_fs": [ 105, 82, 505, 106 ], "lines_deleted": true }, { "type": "title", "bbox": [ 107, 117, 232, 129 ], "lines": [ { "bbox": [ 105, 116, 232, 131 ], "spans": [ { "bbox": [ 105, 116, 232, 131 ], "score": 1.0, "content": "E FULL EXPERIMENTS", "type": "text" } ], "index": 2 } ], "index": 2 }, { "type": "title", "bbox": [ 107, 136, 213, 147 ], "lines": [ { "bbox": [ 105, 135, 214, 148 ], "spans": [ { "bbox": [ 105, 135, 214, 148 ], "score": 1.0, "content": "E.1 SYNTHETIC TASKS", "type": "text" } ], "index": 3 } ], "index": 3 }, { "type": "text", "bbox": [ 106, 151, 504, 163 ], "lines": [ { "bbox": [ 106, 151, 505, 165 ], "spans": [ { "bbox": [ 106, 151, 505, 165 ], "score": 1.0, "content": "Full experiment details for these tasks including task details and training protocol are in Appendix F.1.", "type": "text" } ], "index": 4 } ], "index": 4, "bbox_fs": [ 106, 151, 505, 165 ] }, { "type": "title", "bbox": [ 107, 173, 232, 185 ], "lines": [ { "bbox": [ 106, 173, 232, 185 ], "spans": [ { "bbox": [ 106, 173, 232, 185 ], "score": 1.0, "content": "E.1.1 SELECTIVE COPYING", "type": "text" } ], "index": 5 } ], "index": 5 }, { "type": "text", "bbox": [ 106, 194, 505, 270 ], "lines": [ { "bbox": [ 106, 194, 505, 206 ], "spans": [ { "bbox": [ 106, 194, 505, 206 ], "score": 1.0, "content": "The Copying task is one of the most well-studied synthetic tasks for sequence modeling, originally", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 204, 505, 216 ], "spans": [ { "bbox": [ 105, 204, 505, 216 ], "score": 1.0, "content": "designed to test the memorization abilities of recurrent models. As discussed in Section 3.1, LTI SSMs", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 216, 504, 227 ], "spans": [ { "bbox": [ 106, 216, 504, 227 ], "score": 1.0, "content": "(linear recurrences and global convolutions) can easily solve this task by only keeping track of time", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 227, 504, 238 ], "spans": [ { "bbox": [ 106, 227, 504, 238 ], "score": 1.0, "content": "instead of reasoning about the data; for example, by constructing a convolution kernel of exactly the", "type": "text" } ], "index": 9 }, { "bbox": [ 104, 236, 506, 249 ], "spans": [ { "bbox": [ 104, 236, 506, 249 ], "score": 1.0, "content": "right length (Figure 1). This was explicitly validated in earlier work on global convolutions (Romero", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 247, 505, 260 ], "spans": [ { "bbox": [ 105, 247, 505, 260 ], "score": 1.0, "content": "et al., 2021). The Selective Copying task prevents this shortcut by randomizing the spacing between", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 258, 476, 270 ], "spans": [ { "bbox": [ 106, 258, 476, 270 ], "score": 1.0, "content": "tokens. Note that this task has been introduced before as the Denoising task (Jing et al., 2019).", "type": "text" } ], "index": 12 } ], "index": 9, "bbox_fs": [ 104, 194, 506, 270 ] }, { "type": "text", "bbox": [ 107, 274, 505, 329 ], "lines": [ { "bbox": [ 106, 275, 505, 286 ], "spans": [ { "bbox": [ 106, 275, 505, 286 ], "score": 1.0, "content": "Note that many previous works argue that adding architecture gating (multiplicative interactions) can", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 284, 505, 296 ], "spans": [ { "bbox": [ 106, 284, 505, 296 ], "score": 1.0, "content": "endow models with “data-dependence” and solve related tasks (Dao et al., 2023; Poli et al., 2023).", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 294, 506, 309 ], "spans": [ { "bbox": [ 105, 294, 506, 309 ], "score": 1.0, "content": "However, we find this explanation insufficient intuitively because such gating does not interact along", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 306, 506, 319 ], "spans": [ { "bbox": [ 105, 306, 506, 319 ], "score": 1.0, "content": "the sequence axis, and cannot affect the spacing between tokens. In particular architecture gating is", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 317, 336, 329 ], "spans": [ { "bbox": [ 105, 317, 336, 329 ], "score": 1.0, "content": "not an instance of a selection mechanism (Appendix A.1).", "type": "text" } ], "index": 17 } ], "index": 15, "bbox_fs": [ 105, 275, 506, 329 ] }, { "type": "text", "bbox": [ 107, 333, 505, 366 ], "lines": [ { "bbox": [ 105, 331, 506, 347 ], "spans": [ { "bbox": [ 105, 331, 506, 347 ], "score": 1.0, "content": "Table 1 confirms that gated architectures such as H3 and Mamba only partially improve performance,", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 344, 505, 356 ], "spans": [ { "bbox": [ 106, 344, 505, 356 ], "score": 1.0, "content": "while the selection mechanism (modifying S4 to S6) easily solves this task, particularly when combined", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 355, 264, 366 ], "spans": [ { "bbox": [ 106, 355, 264, 366 ], "score": 1.0, "content": "with these more powerful architectures.", "type": "text" } ], "index": 20 } ], "index": 19, "bbox_fs": [ 105, 331, 506, 366 ] }, { "type": "title", "bbox": [ 107, 376, 223, 388 ], "lines": [ { "bbox": [ 106, 376, 224, 389 ], "spans": [ { "bbox": [ 106, 376, 224, 389 ], "score": 1.0, "content": "E.1.2 INDUCTION HEADS", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "text", "bbox": [ 107, 397, 505, 451 ], "lines": [ { "bbox": [ 105, 395, 505, 410 ], "spans": [ { "bbox": [ 105, 395, 505, 410 ], "score": 1.0, "content": "Induction heads (Olsson et al., 2022) is a simple task from the mechanistic interpretability lens (Elhage", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 408, 505, 420 ], "spans": [ { "bbox": [ 105, 408, 505, 420 ], "score": 1.0, "content": "et al., 2021) that is surprisingly predictive of the in-context learning ability of LLMs. It requires models", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 418, 505, 432 ], "spans": [ { "bbox": [ 105, 418, 505, 432 ], "score": 1.0, "content": "to perform associative recall and copy: for example, if the model has seen a bigram such as “Harry", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 429, 505, 442 ], "spans": [ { "bbox": [ 105, 429, 505, 442 ], "score": 1.0, "content": "Potter” in the sequence, then the next time “Harry” appears in the same sequence, the model should", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 439, 310, 453 ], "spans": [ { "bbox": [ 105, 439, 310, 453 ], "score": 1.0, "content": "be able to predict “Potter” by copying from history.", "type": "text" } ], "index": 26 } ], "index": 24, "bbox_fs": [ 105, 395, 505, 453 ] }, { "type": "text", "bbox": [ 107, 462, 505, 505 ], "lines": [ { "bbox": [ 106, 461, 505, 473 ], "spans": [ { "bbox": [ 106, 461, 505, 473 ], "score": 1.0, "content": "Dataset We train a 2-layer model on the induction heads task at sequence length 256, with a vocab", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 472, 507, 485 ], "spans": [ { "bbox": [ 105, 472, 507, 485 ], "score": 1.0, "content": "size of 16, which is comparable to prior work on this task (Dao et al., 2023) but with longer sequences.", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 483, 506, 496 ], "spans": [ { "bbox": [ 105, 483, 506, 496 ], "score": 1.0, "content": "We additionally investigate generalization and extrapolation abilities by evaluating on a range of", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 493, 359, 507 ], "spans": [ { "bbox": [ 105, 493, 197, 507 ], "score": 1.0, "content": "sequence lengths from", "type": "text" }, { "bbox": [ 198, 494, 228, 505 ], "score": 0.9, "content": "2 ^ { \\overline { { 6 } } } = 6 \\overline { { 4 } }", "type": "inline_equation" }, { "bbox": [ 228, 493, 250, 507 ], "score": 1.0, "content": "up to", "type": "text" }, { "bbox": [ 250, 494, 310, 505 ], "score": 0.89, "content": "2 ^ { 2 0 } = 1 0 4 8 5 7 6", "type": "inline_equation" }, { "bbox": [ 310, 493, 359, 507 ], "score": 1.0, "content": "at test time.", "type": "text" } ], "index": 30 } ], "index": 28.5, "bbox_fs": [ 105, 461, 507, 507 ] }, { "type": "text", "bbox": [ 107, 516, 505, 559 ], "lines": [ { "bbox": [ 105, 515, 505, 528 ], "spans": [ { "bbox": [ 105, 515, 505, 528 ], "score": 1.0, "content": "Models Following established work on induction heads, we use 2 layer models, which allows", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 527, 505, 538 ], "spans": [ { "bbox": [ 105, 527, 505, 538 ], "score": 1.0, "content": "attention to mechanistically solve the induction heads task (Olsson et al., 2022). We test both", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 538, 505, 550 ], "spans": [ { "bbox": [ 105, 538, 505, 550 ], "score": 1.0, "content": "multi-head attention (8 heads, with various positional encodings) and SSM variants. We use a model", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 549, 346, 559 ], "spans": [ { "bbox": [ 106, 549, 150, 559 ], "score": 1.0, "content": "dimension", "type": "text" }, { "bbox": [ 150, 549, 160, 558 ], "score": 0.78, "content": "D", "type": "inline_equation" }, { "bbox": [ 160, 549, 346, 559 ], "score": 1.0, "content": "of 64 for Mamba and 128 for the other models.", "type": "text" } ], "index": 34 } ], "index": 32.5, "bbox_fs": [ 105, 515, 505, 559 ] }, { "type": "text", "bbox": [ 107, 570, 505, 614 ], "lines": [ { "bbox": [ 105, 569, 505, 583 ], "spans": [ { "bbox": [ 105, 569, 505, 583 ], "score": 1.0, "content": "Results Figure 3 shows that Mamba—or more precisely, its selective SSM layer—has the ability to", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 580, 506, 594 ], "spans": [ { "bbox": [ 105, 580, 506, 594 ], "score": 1.0, "content": "solve the task perfectly because of its ability to selectively remember the relevant token while ignoring", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 592, 506, 605 ], "spans": [ { "bbox": [ 105, 592, 446, 605 ], "score": 1.0, "content": "everything else in between. It generalizes perfectly to million-length sequences, or", "type": "text" }, { "bbox": [ 446, 592, 475, 603 ], "score": 0.87, "content": "4 0 0 0 \\times", "type": "inline_equation" }, { "bbox": [ 476, 592, 506, 605 ], "score": 1.0, "content": "longer", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 603, 374, 615 ], "spans": [ { "bbox": [ 106, 603, 356, 615 ], "score": 1.0, "content": "than it saw during training, while no other method goes beyond", "type": "text" }, { "bbox": [ 356, 603, 370, 613 ], "score": 0.86, "content": "2 \\times", "type": "inline_equation" }, { "bbox": [ 370, 603, 374, 615 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 38 } ], "index": 36.5, "bbox_fs": [ 105, 569, 506, 615 ] }, { "type": "text", "bbox": [ 107, 618, 505, 662 ], "lines": [ { "bbox": [ 106, 619, 505, 630 ], "spans": [ { "bbox": [ 106, 619, 505, 630 ], "score": 1.0, "content": "Out of positional encoding variants for attention models, xPos (which was designed for length", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 630, 505, 641 ], "spans": [ { "bbox": [ 105, 630, 505, 641 ], "score": 1.0, "content": "extrapolation) is slightly better than the others; also note that all attention models were only tested", "type": "text" } ], "index": 40 }, { "bbox": [ 104, 638, 506, 653 ], "spans": [ { "bbox": [ 104, 638, 197, 653 ], "score": 1.0, "content": "up to sequence length", "type": "text" }, { "bbox": [ 197, 639, 248, 650 ], "score": 0.9, "content": "\\dot { 2 } ^ { 1 4 } = 1 6 3 8 4", "type": "inline_equation" }, { "bbox": [ 249, 638, 506, 653 ], "score": 1.0, "content": "due to memory limitations. Out of other SSMs, H3 and Hyena", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 650, 325, 663 ], "spans": [ { "bbox": [ 105, 650, 325, 663 ], "score": 1.0, "content": "are similar, contrary to the findings in Poli et al. (2023).", "type": "text" } ], "index": 42 } ], "index": 40.5, "bbox_fs": [ 104, 619, 506, 663 ] }, { "type": "title", "bbox": [ 107, 672, 206, 684 ], "lines": [ { "bbox": [ 105, 671, 207, 685 ], "spans": [ { "bbox": [ 105, 671, 207, 685 ], "score": 1.0, "content": "E.2 DNA MODELING", "type": "text" } ], "index": 43 } ], "index": 43 }, { "type": "text", "bbox": [ 108, 689, 504, 732 ], "lines": [ { "bbox": [ 105, 688, 506, 702 ], "spans": [ { "bbox": [ 105, 688, 506, 702 ], "score": 1.0, "content": "For pretraining, we largely follow a standard causal language modeling (next token prediction) setup", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 698, 505, 712 ], "spans": [ { "bbox": [ 105, 698, 505, 712 ], "score": 1.0, "content": "for the training and model details (see also Appendix F.2). For the dataset, we largely follow the", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 708, 506, 724 ], "spans": [ { "bbox": [ 105, 708, 506, 724 ], "score": 1.0, "content": "setup of HyenaDNA (Nguyen et al., 2023), which uses the HG38 dataset for pretraining consisting", "type": "text" } ], "index": 46 }, { "bbox": [ 106, 721, 475, 733 ], "spans": [ { "bbox": [ 106, 721, 475, 733 ], "score": 1.0, "content": "of a single human genome with about 4.5 billion tokens (DNA base pairs) in the training split.", "type": "text" } ], "index": 47 } ], "index": 45.5, "bbox_fs": [ 105, 688, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 105, 103, 504, 125 ], "lines": [ { "bbox": [ 105, 102, 505, 115 ], "spans": [ { "bbox": [ 105, 102, 505, 115 ], "score": 1.0, "content": "In this experiment, we investigate the scaling properties of genomics foundation models with various", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 114, 240, 126 ], "spans": [ { "bbox": [ 106, 114, 240, 126 ], "score": 1.0, "content": "model backbones (Figure 5 Left).", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "text", "bbox": [ 107, 129, 505, 172 ], "lines": [ { "bbox": [ 105, 128, 505, 142 ], "spans": [ { "bbox": [ 105, 128, 505, 142 ], "score": 1.0, "content": "Training To advantage the baselines, we train on a short sequence length of 1024; as shown in", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 139, 506, 153 ], "spans": [ { "bbox": [ 105, 139, 506, 153 ], "score": 1.0, "content": "Appendix E.2.2, we expect results to favor Mamba even more at longer sequence lengths. We fix", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 149, 504, 164 ], "spans": [ { "bbox": [ 105, 149, 270, 164 ], "score": 1.0, "content": "a global batch size of 1024, for a total of", "type": "text" }, { "bbox": [ 271, 150, 312, 161 ], "score": 0.92, "content": "2 ^ { 2 0 } \\approx 1 M", "type": "inline_equation" }, { "bbox": [ 312, 149, 483, 164 ], "score": 1.0, "content": "tokens per batch. Models were trained for", "type": "text" }, { "bbox": [ 484, 151, 504, 161 ], "score": 0.78, "content": "1 0 K", "type": "inline_equation" } ], "index": 4 }, { "bbox": [ 105, 161, 264, 174 ], "spans": [ { "bbox": [ 105, 161, 213, 174 ], "score": 1.0, "content": "gradient steps for a total of", "type": "text" }, { "bbox": [ 213, 162, 232, 172 ], "score": 0.82, "content": "1 0 B", "type": "inline_equation" }, { "bbox": [ 233, 161, 264, 174 ], "score": 1.0, "content": "tokens.", "type": "text" } ], "index": 5 } ], "index": 3.5 }, { "type": "text", "bbox": [ 107, 177, 505, 221 ], "lines": [ { "bbox": [ 105, 177, 505, 190 ], "spans": [ { "bbox": [ 105, 177, 505, 190 ], "score": 1.0, "content": "Results Figure 5 (Left) shows that Mamba’s pretraining perplexity improves smoothly with model", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 188, 505, 200 ], "spans": [ { "bbox": [ 105, 188, 407, 200 ], "score": 1.0, "content": "size, and that Mamba scales better than both HyenaDNA and Transformer", "type": "text" }, { "bbox": [ 407, 190, 419, 198 ], "score": 0.56, "content": "^ { + + }", "type": "inline_equation" }, { "bbox": [ 420, 188, 505, 200 ], "score": 1.0, "content": ". For example, at the", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 199, 507, 211 ], "spans": [ { "bbox": [ 105, 199, 190, 211 ], "score": 1.0, "content": "largest model size of", "type": "text" }, { "bbox": [ 190, 199, 221, 209 ], "score": 0.89, "content": "{ \\approx } 4 0 M", "type": "inline_equation" }, { "bbox": [ 222, 199, 507, 211 ], "score": 1.0, "content": "parameters, the curve shows that Mamba can match the Transformer++", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 210, 367, 222 ], "spans": [ { "bbox": [ 106, 210, 255, 222 ], "score": 1.0, "content": "and HyenaDNA models with roughly", "type": "text" }, { "bbox": [ 255, 210, 270, 220 ], "score": 0.87, "content": "3 \\times", "type": "inline_equation" }, { "bbox": [ 270, 210, 280, 222 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 280, 210, 294, 220 ], "score": 0.87, "content": "4 \\times", "type": "inline_equation" }, { "bbox": [ 294, 210, 367, 222 ], "score": 1.0, "content": "fewer parameters.", "type": "text" } ], "index": 9 } ], "index": 7.5 }, { "type": "title", "bbox": [ 108, 230, 266, 241 ], "lines": [ { "bbox": [ 105, 228, 268, 243 ], "spans": [ { "bbox": [ 105, 228, 268, 243 ], "score": 1.0, "content": "E.2.2 SCALING: CONTEXT LENGTH", "type": "text" } ], "index": 10 } ], "index": 10 }, { "type": "text", "bbox": [ 107, 250, 505, 326 ], "lines": [ { "bbox": [ 105, 249, 506, 264 ], "spans": [ { "bbox": [ 105, 249, 506, 264 ], "score": 1.0, "content": "In the next DNA experiment, we investigate the scaling properties of models with respect to sequence", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 262, 505, 274 ], "spans": [ { "bbox": [ 105, 262, 505, 274 ], "score": 1.0, "content": "length. We only compare the HyenaDNA and Mamba models, as quadratic attention becomes", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 272, 505, 284 ], "spans": [ { "bbox": [ 105, 272, 505, 284 ], "score": 1.0, "content": "prohibitively expensive at longer sequence lengths. We pretrain models on sequence lengths", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 280, 506, 295 ], "spans": [ { "bbox": [ 106, 282, 152, 293 ], "score": 0.88, "content": "\\dot { 2 } ^ { 1 0 } = 1 0 2 4", "type": "inline_equation" }, { "bbox": [ 153, 280, 156, 295 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 156, 282, 202, 294 ], "score": 0.86, "content": "2 ^ { 1 2 } = 4 0 9 6", "type": "inline_equation" }, { "bbox": [ 203, 280, 206, 295 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 206, 282, 257, 293 ], "score": 0.86, "content": "2 ^ { 1 4 } = \\bar { 1 } 6 3 8 4", "type": "inline_equation" }, { "bbox": [ 257, 280, 261, 295 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 261, 282, 312, 293 ], "score": 0.87, "content": "2 ^ { 1 6 } = 6 5 5 \\bar { 3 6 }", "type": "inline_equation" }, { "bbox": [ 312, 280, 315, 295 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 316, 282, 371, 293 ], "score": 0.84, "content": "2 ^ { 1 8 } = 2 \\dot { 6 } 2 1 4 4", "type": "inline_equation" }, { "bbox": [ 372, 280, 375, 295 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 376, 282, 436, 293 ], "score": 0.88, "content": "2 ^ { 2 0 } = 1 0 4 8 5 7 6", "type": "inline_equation" }, { "bbox": [ 437, 280, 506, 295 ], "score": 1.0, "content": ". We fix a model", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 293, 505, 306 ], "spans": [ { "bbox": [ 105, 293, 448, 306 ], "score": 1.0, "content": "size of 6 layers by width 128 (about 1.3M-1.4M parameters). Models were trained for", "type": "text" }, { "bbox": [ 448, 294, 469, 304 ], "score": 0.83, "content": "2 0 K", "type": "inline_equation" }, { "bbox": [ 469, 293, 505, 306 ], "score": 1.0, "content": "gradient", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 303, 506, 317 ], "spans": [ { "bbox": [ 105, 303, 178, 317 ], "score": 1.0, "content": "steps for a total of", "type": "text" }, { "bbox": [ 178, 304, 211, 315 ], "score": 0.91, "content": "{ \\approx } 3 3 0 B", "type": "inline_equation" }, { "bbox": [ 211, 303, 506, 317 ], "score": 1.0, "content": "tokens. The longer sequence lengths used sequence length warmup similar", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 315, 204, 327 ], "spans": [ { "bbox": [ 105, 315, 204, 327 ], "score": 1.0, "content": "to (Nguyen et al., 2023).", "type": "text" } ], "index": 17 } ], "index": 14 }, { "type": "text", "bbox": [ 107, 331, 505, 407 ], "lines": [ { "bbox": [ 106, 330, 505, 343 ], "spans": [ { "bbox": [ 106, 330, 505, 343 ], "score": 1.0, "content": "Results Figure 5 (Right) shows that Mamba is able to make use of longer context even up to extremely", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 341, 506, 355 ], "spans": [ { "bbox": [ 105, 341, 506, 355 ], "score": 1.0, "content": "long sequences of length 1M, and its pretraining perplexity improves as the context increases. On the", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 353, 505, 365 ], "spans": [ { "bbox": [ 105, 353, 505, 365 ], "score": 1.0, "content": "other hand, the HyenaDNA model gets worse with sequence length. This is intuitive from the discussion", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 362, 505, 375 ], "spans": [ { "bbox": [ 105, 362, 505, 375 ], "score": 1.0, "content": "in Section 3.5 that motivated the selection mechanism. In particular, LTI models cannot selectively", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 373, 505, 387 ], "spans": [ { "bbox": [ 105, 373, 505, 387 ], "score": 1.0, "content": "ignore information; from a convolutional perspective, a very long convolution kernel is aggregating", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 384, 506, 398 ], "spans": [ { "bbox": [ 105, 384, 506, 398 ], "score": 1.0, "content": "all information across a long sequence which may be very noisy. Note that while Nguyen et al. (2023)", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 396, 492, 408 ], "spans": [ { "bbox": [ 105, 396, 492, 408 ], "score": 1.0, "content": "claims that Hyena improves with longer context, their results do not control for computation time.", "type": "text" } ], "index": 24 } ], "index": 21 }, { "type": "title", "bbox": [ 108, 416, 302, 427 ], "lines": [ { "bbox": [ 105, 415, 304, 429 ], "spans": [ { "bbox": [ 105, 415, 304, 429 ], "score": 1.0, "content": "E.2.3 SYNTHETIC SPECIES CLASSIFICATION", "type": "text" } ], "index": 25 } ], "index": 25 }, { "type": "text", "bbox": [ 107, 437, 505, 480 ], "lines": [ { "bbox": [ 106, 437, 505, 448 ], "spans": [ { "bbox": [ 106, 437, 505, 448 ], "score": 1.0, "content": "We evaluate models on a downstream task of classifying between 5 different species by randomly", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 447, 505, 459 ], "spans": [ { "bbox": [ 105, 447, 505, 459 ], "score": 1.0, "content": "sampling a contiguous segment of their DNA. This task is adapted from HyenaDNA, which used the", "type": "text" } ], "index": 27 }, { "bbox": [ 104, 457, 506, 472 ], "spans": [ { "bbox": [ 104, 457, 137, 472 ], "score": 1.0, "content": "species", "type": "text" }, { "bbox": [ 137, 458, 143, 470 ], "score": 0.55, "content": "\\{", "type": "inline_equation" }, { "bbox": [ 144, 457, 275, 472 ], "score": 1.0, "content": "human, lemur, mouse, pig, hippo", "type": "text" }, { "bbox": [ 275, 458, 281, 470 ], "score": 0.51, "content": "\\}", "type": "inline_equation" }, { "bbox": [ 281, 457, 506, 472 ], "score": 1.0, "content": ". We modify the task to be significantly more challenging", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 469, 503, 482 ], "spans": [ { "bbox": [ 105, 469, 299, 482 ], "score": 1.0, "content": "by classifying between the five great apes species", "type": "text" }, { "bbox": [ 299, 469, 306, 481 ], "score": 0.26, "content": "\\{", "type": "inline_equation" }, { "bbox": [ 306, 469, 492, 482 ], "score": 1.0, "content": "human, chimpanzee, gorilla, orangutan, bonobo", "type": "text" }, { "bbox": [ 493, 469, 498, 481 ], "score": 0.41, "content": "\\}", "type": "inline_equation" }, { "bbox": [ 499, 469, 503, 482 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 29 } ], "index": 27.5 }, { "type": "title", "bbox": [ 107, 489, 217, 501 ], "lines": [ { "bbox": [ 105, 488, 219, 502 ], "spans": [ { "bbox": [ 105, 488, 219, 502 ], "score": 1.0, "content": "E.3 MODEL ABLATIONS", "type": "text" } ], "index": 30 } ], "index": 30 }, { "type": "text", "bbox": [ 105, 505, 504, 517 ], "lines": [ { "bbox": [ 106, 505, 506, 519 ], "spans": [ { "bbox": [ 106, 505, 506, 519 ], "score": 1.0, "content": "Table 5 investigates the effects of the architecture (block) and its inner SSM layer (Figure 2). We find that", "type": "text" } ], "index": 31 } ], "index": 31 }, { "type": "text", "bbox": [ 106, 525, 505, 631 ], "lines": [ { "bbox": [ 104, 525, 505, 540 ], "spans": [ { "bbox": [ 104, 525, 505, 540 ], "score": 1.0, "content": "• Among previous non-selective (LTI) SSMs, which are equivalent to global convolutions, performance", "type": "text" } ], "index": 32 }, { "bbox": [ 112, 536, 175, 550 ], "spans": [ { "bbox": [ 112, 536, 175, 550 ], "score": 1.0, "content": "is very similar.", "type": "text" } ], "index": 33 }, { "bbox": [ 104, 549, 505, 563 ], "spans": [ { "bbox": [ 104, 549, 505, 563 ], "score": 1.0, "content": "• Replacing the complex-valued S4 variant from previous work with a real-valued one does not affect", "type": "text" } ], "index": 34 }, { "bbox": [ 113, 561, 505, 573 ], "spans": [ { "bbox": [ 113, 561, 505, 573 ], "score": 1.0, "content": "performance much, suggesting that (at least for LM) real-valued SSMs may be a better choice when", "type": "text" } ], "index": 35 }, { "bbox": [ 113, 572, 256, 585 ], "spans": [ { "bbox": [ 113, 572, 256, 585 ], "score": 1.0, "content": "accounting for hardware efficiency.", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 584, 505, 599 ], "spans": [ { "bbox": [ 105, 584, 505, 599 ], "score": 1.0, "content": "• Replacing any of these with a selective SSM (S6) significantly improves performance, validating", "type": "text" } ], "index": 37 }, { "bbox": [ 113, 596, 226, 608 ], "spans": [ { "bbox": [ 113, 596, 226, 608 ], "score": 1.0, "content": "the motivation of Section 3.", "type": "text" } ], "index": 38 }, { "bbox": [ 104, 607, 505, 623 ], "spans": [ { "bbox": [ 104, 607, 505, 623 ], "score": 1.0, "content": "• The Mamba architecture performs similarly to the H3 architecture (and seems slightly better when", "type": "text" } ], "index": 39 }, { "bbox": [ 114, 620, 209, 632 ], "spans": [ { "bbox": [ 114, 620, 209, 632 ], "score": 1.0, "content": "using a selective layer).", "type": "text" } ], "index": 40 } ], "index": 36 }, { "type": "text", "bbox": [ 107, 640, 505, 673 ], "lines": [ { "bbox": [ 105, 640, 506, 653 ], "spans": [ { "bbox": [ 105, 640, 461, 653 ], "score": 1.0, "content": "Table 6 ablates the selective SSM layer by considering different combinations of selective", "type": "text" }, { "bbox": [ 462, 641, 485, 651 ], "score": 0.63, "content": "\\Delta , B", "type": "inline_equation" }, { "bbox": [ 485, 640, 506, 653 ], "score": 1.0, "content": ", and", "type": "text" } ], "index": 41 }, { "bbox": [ 107, 651, 505, 663 ], "spans": [ { "bbox": [ 107, 652, 117, 662 ], "score": 0.77, "content": "C", "type": "inline_equation" }, { "bbox": [ 117, 651, 279, 663 ], "score": 1.0, "content": "parameters (Algorithm 2), showing that", "type": "text" }, { "bbox": [ 279, 652, 289, 661 ], "score": 0.81, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 289, 651, 505, 663 ], "score": 1.0, "content": "is the most important parameter due to its connection", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 662, 221, 674 ], "spans": [ { "bbox": [ 106, 662, 221, 674 ], "score": 1.0, "content": "to RNN gating (Theorem 1).", "type": "text" } ], "index": 43 } ], "index": 42 }, { "type": "text", "bbox": [ 107, 677, 504, 732 ], "lines": [ { "bbox": [ 105, 677, 505, 690 ], "spans": [ { "bbox": [ 105, 677, 505, 690 ], "score": 1.0, "content": "Table 7 considers different initializations of the SSM, which have been shown to make a large", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 687, 506, 701 ], "spans": [ { "bbox": [ 105, 687, 506, 701 ], "score": 1.0, "content": "difference in some data modalities and settings (Gu et al., 2022a;b). On language modeling, we", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 699, 505, 711 ], "spans": [ { "bbox": [ 105, 699, 505, 711 ], "score": 1.0, "content": "find that simpler real-valued diagonal initializations (S4D-Real, row 3) instead of more standard", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 710, 506, 722 ], "spans": [ { "bbox": [ 105, 710, 506, 722 ], "score": 1.0, "content": "complex-valued parameterizations (S4D-Lin, row 1) perform better. Random initializations also work", "type": "text" } ], "index": 47 }, { "bbox": [ 106, 720, 369, 732 ], "spans": [ { "bbox": [ 106, 720, 369, 732 ], "score": 1.0, "content": "well, consistent with findings from prior work (Mehta et al., 2023).", "type": "text" } ], "index": 48 } ], "index": 46 } ], "page_idx": 19, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 304, 37 ], "lines": [ { "bbox": [ 106, 25, 305, 38 ], "spans": [ { "bbox": [ 106, 25, 305, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 298, 750, 313, 764 ], "spans": [ { "bbox": [ 298, 750, 313, 764 ], "score": 1.0, "content": "", "type": "text", "height": 14, "width": 15 } ] } ] }, { "type": "discarded", "bbox": [ 107, 82, 241, 93 ], "lines": [ { "bbox": [ 105, 81, 243, 95 ], "spans": [ { "bbox": [ 105, 81, 243, 95 ], "score": 1.0, "content": "E.2.1 SCALING: MODEL SIZE", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 105, 103, 504, 125 ], "lines": [ { "bbox": [ 105, 102, 505, 115 ], "spans": [ { "bbox": [ 105, 102, 505, 115 ], "score": 1.0, "content": "In this experiment, we investigate the scaling properties of genomics foundation models with various", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 114, 240, 126 ], "spans": [ { "bbox": [ 106, 114, 240, 126 ], "score": 1.0, "content": "model backbones (Figure 5 Left).", "type": "text" } ], "index": 1 } ], "index": 0.5, "bbox_fs": [ 105, 102, 505, 126 ] }, { "type": "text", "bbox": [ 107, 129, 505, 172 ], "lines": [ { "bbox": [ 105, 128, 505, 142 ], "spans": [ { "bbox": [ 105, 128, 505, 142 ], "score": 1.0, "content": "Training To advantage the baselines, we train on a short sequence length of 1024; as shown in", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 139, 506, 153 ], "spans": [ { "bbox": [ 105, 139, 506, 153 ], "score": 1.0, "content": "Appendix E.2.2, we expect results to favor Mamba even more at longer sequence lengths. We fix", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 149, 504, 164 ], "spans": [ { "bbox": [ 105, 149, 270, 164 ], "score": 1.0, "content": "a global batch size of 1024, for a total of", "type": "text" }, { "bbox": [ 271, 150, 312, 161 ], "score": 0.92, "content": "2 ^ { 2 0 } \\approx 1 M", "type": "inline_equation" }, { "bbox": [ 312, 149, 483, 164 ], "score": 1.0, "content": "tokens per batch. Models were trained for", "type": "text" }, { "bbox": [ 484, 151, 504, 161 ], "score": 0.78, "content": "1 0 K", "type": "inline_equation" } ], "index": 4 }, { "bbox": [ 105, 161, 264, 174 ], "spans": [ { "bbox": [ 105, 161, 213, 174 ], "score": 1.0, "content": "gradient steps for a total of", "type": "text" }, { "bbox": [ 213, 162, 232, 172 ], "score": 0.82, "content": "1 0 B", "type": "inline_equation" }, { "bbox": [ 233, 161, 264, 174 ], "score": 1.0, "content": "tokens.", "type": "text" } ], "index": 5 } ], "index": 3.5, "bbox_fs": [ 105, 128, 506, 174 ] }, { "type": "text", "bbox": [ 107, 177, 505, 221 ], "lines": [ { "bbox": [ 105, 177, 505, 190 ], "spans": [ { "bbox": [ 105, 177, 505, 190 ], "score": 1.0, "content": "Results Figure 5 (Left) shows that Mamba’s pretraining perplexity improves smoothly with model", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 188, 505, 200 ], "spans": [ { "bbox": [ 105, 188, 407, 200 ], "score": 1.0, "content": "size, and that Mamba scales better than both HyenaDNA and Transformer", "type": "text" }, { "bbox": [ 407, 190, 419, 198 ], "score": 0.56, "content": "^ { + + }", "type": "inline_equation" }, { "bbox": [ 420, 188, 505, 200 ], "score": 1.0, "content": ". For example, at the", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 199, 507, 211 ], "spans": [ { "bbox": [ 105, 199, 190, 211 ], "score": 1.0, "content": "largest model size of", "type": "text" }, { "bbox": [ 190, 199, 221, 209 ], "score": 0.89, "content": "{ \\approx } 4 0 M", "type": "inline_equation" }, { "bbox": [ 222, 199, 507, 211 ], "score": 1.0, "content": "parameters, the curve shows that Mamba can match the Transformer++", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 210, 367, 222 ], "spans": [ { "bbox": [ 106, 210, 255, 222 ], "score": 1.0, "content": "and HyenaDNA models with roughly", "type": "text" }, { "bbox": [ 255, 210, 270, 220 ], "score": 0.87, "content": "3 \\times", "type": "inline_equation" }, { "bbox": [ 270, 210, 280, 222 ], "score": 1.0, "content": "to", "type": "text" }, { "bbox": [ 280, 210, 294, 220 ], "score": 0.87, "content": "4 \\times", "type": "inline_equation" }, { "bbox": [ 294, 210, 367, 222 ], "score": 1.0, "content": "fewer parameters.", "type": "text" } ], "index": 9 } ], "index": 7.5, "bbox_fs": [ 105, 177, 507, 222 ] }, { "type": "title", "bbox": [ 108, 230, 266, 241 ], "lines": [ { "bbox": [ 105, 228, 268, 243 ], "spans": [ { "bbox": [ 105, 228, 268, 243 ], "score": 1.0, "content": "E.2.2 SCALING: CONTEXT LENGTH", "type": "text" } ], "index": 10 } ], "index": 10 }, { "type": "text", "bbox": [ 107, 250, 505, 326 ], "lines": [ { "bbox": [ 105, 249, 506, 264 ], "spans": [ { "bbox": [ 105, 249, 506, 264 ], "score": 1.0, "content": "In the next DNA experiment, we investigate the scaling properties of models with respect to sequence", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 262, 505, 274 ], "spans": [ { "bbox": [ 105, 262, 505, 274 ], "score": 1.0, "content": "length. We only compare the HyenaDNA and Mamba models, as quadratic attention becomes", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 272, 505, 284 ], "spans": [ { "bbox": [ 105, 272, 505, 284 ], "score": 1.0, "content": "prohibitively expensive at longer sequence lengths. We pretrain models on sequence lengths", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 280, 506, 295 ], "spans": [ { "bbox": [ 106, 282, 152, 293 ], "score": 0.88, "content": "\\dot { 2 } ^ { 1 0 } = 1 0 2 4", "type": "inline_equation" }, { "bbox": [ 153, 280, 156, 295 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 156, 282, 202, 294 ], "score": 0.86, "content": "2 ^ { 1 2 } = 4 0 9 6", "type": "inline_equation" }, { "bbox": [ 203, 280, 206, 295 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 206, 282, 257, 293 ], "score": 0.86, "content": "2 ^ { 1 4 } = \\bar { 1 } 6 3 8 4", "type": "inline_equation" }, { "bbox": [ 257, 280, 261, 295 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 261, 282, 312, 293 ], "score": 0.87, "content": "2 ^ { 1 6 } = 6 5 5 \\bar { 3 6 }", "type": "inline_equation" }, { "bbox": [ 312, 280, 315, 295 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 316, 282, 371, 293 ], "score": 0.84, "content": "2 ^ { 1 8 } = 2 \\dot { 6 } 2 1 4 4", "type": "inline_equation" }, { "bbox": [ 372, 280, 375, 295 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 376, 282, 436, 293 ], "score": 0.88, "content": "2 ^ { 2 0 } = 1 0 4 8 5 7 6", "type": "inline_equation" }, { "bbox": [ 437, 280, 506, 295 ], "score": 1.0, "content": ". We fix a model", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 293, 505, 306 ], "spans": [ { "bbox": [ 105, 293, 448, 306 ], "score": 1.0, "content": "size of 6 layers by width 128 (about 1.3M-1.4M parameters). Models were trained for", "type": "text" }, { "bbox": [ 448, 294, 469, 304 ], "score": 0.83, "content": "2 0 K", "type": "inline_equation" }, { "bbox": [ 469, 293, 505, 306 ], "score": 1.0, "content": "gradient", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 303, 506, 317 ], "spans": [ { "bbox": [ 105, 303, 178, 317 ], "score": 1.0, "content": "steps for a total of", "type": "text" }, { "bbox": [ 178, 304, 211, 315 ], "score": 0.91, "content": "{ \\approx } 3 3 0 B", "type": "inline_equation" }, { "bbox": [ 211, 303, 506, 317 ], "score": 1.0, "content": "tokens. The longer sequence lengths used sequence length warmup similar", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 315, 204, 327 ], "spans": [ { "bbox": [ 105, 315, 204, 327 ], "score": 1.0, "content": "to (Nguyen et al., 2023).", "type": "text" } ], "index": 17 } ], "index": 14, "bbox_fs": [ 105, 249, 506, 327 ] }, { "type": "text", "bbox": [ 107, 331, 505, 407 ], "lines": [ { "bbox": [ 106, 330, 505, 343 ], "spans": [ { "bbox": [ 106, 330, 505, 343 ], "score": 1.0, "content": "Results Figure 5 (Right) shows that Mamba is able to make use of longer context even up to extremely", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 341, 506, 355 ], "spans": [ { "bbox": [ 105, 341, 506, 355 ], "score": 1.0, "content": "long sequences of length 1M, and its pretraining perplexity improves as the context increases. On the", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 353, 505, 365 ], "spans": [ { "bbox": [ 105, 353, 505, 365 ], "score": 1.0, "content": "other hand, the HyenaDNA model gets worse with sequence length. This is intuitive from the discussion", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 362, 505, 375 ], "spans": [ { "bbox": [ 105, 362, 505, 375 ], "score": 1.0, "content": "in Section 3.5 that motivated the selection mechanism. In particular, LTI models cannot selectively", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 373, 505, 387 ], "spans": [ { "bbox": [ 105, 373, 505, 387 ], "score": 1.0, "content": "ignore information; from a convolutional perspective, a very long convolution kernel is aggregating", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 384, 506, 398 ], "spans": [ { "bbox": [ 105, 384, 506, 398 ], "score": 1.0, "content": "all information across a long sequence which may be very noisy. Note that while Nguyen et al. (2023)", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 396, 492, 408 ], "spans": [ { "bbox": [ 105, 396, 492, 408 ], "score": 1.0, "content": "claims that Hyena improves with longer context, their results do not control for computation time.", "type": "text" } ], "index": 24 } ], "index": 21, "bbox_fs": [ 105, 330, 506, 408 ] }, { "type": "title", "bbox": [ 108, 416, 302, 427 ], "lines": [ { "bbox": [ 105, 415, 304, 429 ], "spans": [ { "bbox": [ 105, 415, 304, 429 ], "score": 1.0, "content": "E.2.3 SYNTHETIC SPECIES CLASSIFICATION", "type": "text" } ], "index": 25 } ], "index": 25 }, { "type": "text", "bbox": [ 107, 437, 505, 480 ], "lines": [ { "bbox": [ 106, 437, 505, 448 ], "spans": [ { "bbox": [ 106, 437, 505, 448 ], "score": 1.0, "content": "We evaluate models on a downstream task of classifying between 5 different species by randomly", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 447, 505, 459 ], "spans": [ { "bbox": [ 105, 447, 505, 459 ], "score": 1.0, "content": "sampling a contiguous segment of their DNA. This task is adapted from HyenaDNA, which used the", "type": "text" } ], "index": 27 }, { "bbox": [ 104, 457, 506, 472 ], "spans": [ { "bbox": [ 104, 457, 137, 472 ], "score": 1.0, "content": "species", "type": "text" }, { "bbox": [ 137, 458, 143, 470 ], "score": 0.55, "content": "\\{", "type": "inline_equation" }, { "bbox": [ 144, 457, 275, 472 ], "score": 1.0, "content": "human, lemur, mouse, pig, hippo", "type": "text" }, { "bbox": [ 275, 458, 281, 470 ], "score": 0.51, "content": "\\}", "type": "inline_equation" }, { "bbox": [ 281, 457, 506, 472 ], "score": 1.0, "content": ". We modify the task to be significantly more challenging", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 469, 503, 482 ], "spans": [ { "bbox": [ 105, 469, 299, 482 ], "score": 1.0, "content": "by classifying between the five great apes species", "type": "text" }, { "bbox": [ 299, 469, 306, 481 ], "score": 0.26, "content": "\\{", "type": "inline_equation" }, { "bbox": [ 306, 469, 492, 482 ], "score": 1.0, "content": "human, chimpanzee, gorilla, orangutan, bonobo", "type": "text" }, { "bbox": [ 493, 469, 498, 481 ], "score": 0.41, "content": "\\}", "type": "inline_equation" }, { "bbox": [ 499, 469, 503, 482 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 29 } ], "index": 27.5, "bbox_fs": [ 104, 437, 506, 482 ] }, { "type": "title", "bbox": [ 107, 489, 217, 501 ], "lines": [ { "bbox": [ 105, 488, 219, 502 ], "spans": [ { "bbox": [ 105, 488, 219, 502 ], "score": 1.0, "content": "E.3 MODEL ABLATIONS", "type": "text" } ], "index": 30 } ], "index": 30 }, { "type": "text", "bbox": [ 105, 505, 504, 517 ], "lines": [ { "bbox": [ 106, 505, 506, 519 ], "spans": [ { "bbox": [ 106, 505, 506, 519 ], "score": 1.0, "content": "Table 5 investigates the effects of the architecture (block) and its inner SSM layer (Figure 2). We find that", "type": "text" } ], "index": 31 } ], "index": 31, "bbox_fs": [ 106, 505, 506, 519 ] }, { "type": "list", "bbox": [ 106, 525, 505, 631 ], "lines": [ { "bbox": [ 104, 525, 505, 540 ], "spans": [ { "bbox": [ 104, 525, 505, 540 ], "score": 1.0, "content": "• Among previous non-selective (LTI) SSMs, which are equivalent to global convolutions, performance", "type": "text" } ], "index": 32, "is_list_start_line": true }, { "bbox": [ 112, 536, 175, 550 ], "spans": [ { "bbox": [ 112, 536, 175, 550 ], "score": 1.0, "content": "is very similar.", "type": "text" } ], "index": 33, "is_list_end_line": true }, { "bbox": [ 104, 549, 505, 563 ], "spans": [ { "bbox": [ 104, 549, 505, 563 ], "score": 1.0, "content": "• Replacing the complex-valued S4 variant from previous work with a real-valued one does not affect", "type": "text" } ], "index": 34, "is_list_start_line": true }, { "bbox": [ 113, 561, 505, 573 ], "spans": [ { "bbox": [ 113, 561, 505, 573 ], "score": 1.0, "content": "performance much, suggesting that (at least for LM) real-valued SSMs may be a better choice when", "type": "text" } ], "index": 35 }, { "bbox": [ 113, 572, 256, 585 ], "spans": [ { "bbox": [ 113, 572, 256, 585 ], "score": 1.0, "content": "accounting for hardware efficiency.", "type": "text" } ], "index": 36, "is_list_end_line": true }, { "bbox": [ 105, 584, 505, 599 ], "spans": [ { "bbox": [ 105, 584, 505, 599 ], "score": 1.0, "content": "• Replacing any of these with a selective SSM (S6) significantly improves performance, validating", "type": "text" } ], "index": 37, "is_list_start_line": true }, { "bbox": [ 113, 596, 226, 608 ], "spans": [ { "bbox": [ 113, 596, 226, 608 ], "score": 1.0, "content": "the motivation of Section 3.", "type": "text" } ], "index": 38, "is_list_end_line": true }, { "bbox": [ 104, 607, 505, 623 ], "spans": [ { "bbox": [ 104, 607, 505, 623 ], "score": 1.0, "content": "• The Mamba architecture performs similarly to the H3 architecture (and seems slightly better when", "type": "text" } ], "index": 39, "is_list_start_line": true }, { "bbox": [ 114, 620, 209, 632 ], "spans": [ { "bbox": [ 114, 620, 209, 632 ], "score": 1.0, "content": "using a selective layer).", "type": "text" } ], "index": 40, "is_list_end_line": true } ], "index": 36, "bbox_fs": [ 104, 525, 505, 632 ] }, { "type": "text", "bbox": [ 107, 640, 505, 673 ], "lines": [ { "bbox": [ 105, 640, 506, 653 ], "spans": [ { "bbox": [ 105, 640, 461, 653 ], "score": 1.0, "content": "Table 6 ablates the selective SSM layer by considering different combinations of selective", "type": "text" }, { "bbox": [ 462, 641, 485, 651 ], "score": 0.63, "content": "\\Delta , B", "type": "inline_equation" }, { "bbox": [ 485, 640, 506, 653 ], "score": 1.0, "content": ", and", "type": "text" } ], "index": 41 }, { "bbox": [ 107, 651, 505, 663 ], "spans": [ { "bbox": [ 107, 652, 117, 662 ], "score": 0.77, "content": "C", "type": "inline_equation" }, { "bbox": [ 117, 651, 279, 663 ], "score": 1.0, "content": "parameters (Algorithm 2), showing that", "type": "text" }, { "bbox": [ 279, 652, 289, 661 ], "score": 0.81, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 289, 651, 505, 663 ], "score": 1.0, "content": "is the most important parameter due to its connection", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 662, 221, 674 ], "spans": [ { "bbox": [ 106, 662, 221, 674 ], "score": 1.0, "content": "to RNN gating (Theorem 1).", "type": "text" } ], "index": 43 } ], "index": 42, "bbox_fs": [ 105, 640, 506, 674 ] }, { "type": "text", "bbox": [ 107, 677, 504, 732 ], "lines": [ { "bbox": [ 105, 677, 505, 690 ], "spans": [ { "bbox": [ 105, 677, 505, 690 ], "score": 1.0, "content": "Table 7 considers different initializations of the SSM, which have been shown to make a large", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 687, 506, 701 ], "spans": [ { "bbox": [ 105, 687, 506, 701 ], "score": 1.0, "content": "difference in some data modalities and settings (Gu et al., 2022a;b). On language modeling, we", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 699, 505, 711 ], "spans": [ { "bbox": [ 105, 699, 505, 711 ], "score": 1.0, "content": "find that simpler real-valued diagonal initializations (S4D-Real, row 3) instead of more standard", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 710, 506, 722 ], "spans": [ { "bbox": [ 105, 710, 506, 722 ], "score": 1.0, "content": "complex-valued parameterizations (S4D-Lin, row 1) perform better. Random initializations also work", "type": "text" } ], "index": 47 }, { "bbox": [ 106, 720, 369, 732 ], "spans": [ { "bbox": [ 106, 720, 369, 732 ], "score": 1.0, "content": "well, consistent with findings from prior work (Mehta et al., 2023).", "type": "text" } ], "index": 48 } ], "index": 46, "bbox_fs": [ 105, 677, 506, 732 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 208, 129, 399, 244 ], "blocks": [ { "type": "table_caption", "bbox": [ 107, 79, 504, 120 ], "group_id": 0, "lines": [ { "bbox": [ 105, 78, 506, 93 ], "spans": [ { "bbox": [ 105, 78, 506, 93 ], "score": 1.0, "content": "Table 5: (Ablations: Architecture and SSM layer.) The Mamba block performs similarly to H3 while being", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 90, 505, 101 ], "spans": [ { "bbox": [ 105, 90, 505, 101 ], "score": 1.0, "content": "simpler. 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MODELBLOCKSSMLAYERPERPLEXITY
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-H3S4 (real)10.34
H3H3S4 (complex)10.30
H3S68.95
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MambaS4 (real)10.56
MambaS4 (complex) 10.54
MambaMambaS68.69
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Selective△ Selective B Selective C Perplexity
X10.93
Xxx<x> 10.15
x√9.98
x<xx> √9.81 8.71
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An InitializationFieldPerplexity
A=+miComplex 9.16
An=-Real8.85
An=-(n+1)Real8.71
An ~exp(N (0,1))Real8.71
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The selection mecha-", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 403, 229, 414 ], "spans": [ { "bbox": [ 106, 403, 135, 414 ], "score": 1.0, "content": "nism of", "type": "text" }, { "bbox": [ 135, 404, 144, 412 ], "score": 0.79, "content": "\\Delta", "type": "inline_equation" }, { "bbox": [ 144, 403, 229, 414 ], "score": 1.0, "content": "constructs it with a low-", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 413, 229, 424 ], "spans": [ { "bbox": [ 105, 413, 229, 424 ], "score": 1.0, "content": "rank projection of the input. Pro-", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 423, 228, 433 ], "spans": [ { "bbox": [ 105, 423, 228, 433 ], "score": 1.0, "content": "jecting it even to rank 1 provides", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 433, 228, 443 ], "spans": [ { "bbox": [ 105, 433, 228, 443 ], "score": 1.0, "content": "a large increase in performance;", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 443, 228, 453 ], "spans": [ { "bbox": [ 106, 443, 228, 453 ], "score": 1.0, "content": "increasing it further provides fur-", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 452, 228, 463 ], "spans": [ { "bbox": [ 106, 452, 228, 463 ], "score": 1.0, "content": "ther improvements at the cost of a", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 462, 218, 473 ], "spans": [ { "bbox": [ 106, 462, 218, 473 ], "score": 1.0, "content": "modest increase in parameters.", "type": "text" } ], "index": 42 } ], "index": 38 }, { "type": "table", "bbox": [ 304, 444, 443, 581 ], "blocks": [ { "type": "table_caption", "bbox": [ 244, 385, 504, 434 ], "group_id": 2, "lines": [ { "bbox": [ 244, 385, 505, 397 ], "spans": [ { "bbox": [ 244, 385, 399, 397 ], "score": 1.0, "content": "Table 9: (Ablations: SSM state dimension", "type": "text" }, { "bbox": [ 400, 386, 410, 395 ], "score": 0.64, "content": "N _ { \\cdot }", "type": "inline_equation" }, { "bbox": [ 410, 385, 479, 397 ], "score": 1.0, "content": ".) 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Increasing the state dimension, which", "type": "text" } ], "index": 44 }, { "bbox": [ 245, 406, 505, 415 ], "spans": [ { "bbox": [ 245, 406, 505, 415 ], "score": 1.0, "content": "can be viewed as an expansion factor on the dimension of the recurrent", "type": "text" } ], "index": 45 }, { "bbox": [ 244, 414, 506, 427 ], "spans": [ { "bbox": [ 244, 414, 506, 427 ], "score": 1.0, "content": "state, can significantly improve performance for a negligible cost in pa-", "type": "text" } ], "index": 46 }, { "bbox": [ 244, 424, 460, 436 ], "spans": [ { "bbox": [ 244, 424, 360, 436 ], "score": 1.0, "content": "rameters/FLOPs, but only when", "type": "text" }, { "bbox": [ 360, 425, 369, 433 ], "score": 0.76, "content": "_ B", "type": "inline_equation" }, { "bbox": [ 370, 424, 385, 436 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 385, 425, 394, 433 ], "score": 0.81, "content": "_ { C }", "type": "inline_equation" }, { "bbox": [ 395, 424, 460, 436 ], "score": 1.0, "content": "are also selective.", "type": "text" } ], "index": 47 } ], "index": 45 }, { "type": "table_body", "bbox": [ 304, 444, 443, 581 ], "group_id": 2, "lines": [ { "bbox": [ 304, 444, 443, 581 ], "spans": [ { "bbox": [ 304, 444, 443, 581 ], "score": 0.973, "html": "
State dim. N Params Perplexity
1367.1M 9.88
2367.4M 9.86
4368.0M 9.82
8369.1M 9.82
16371.5M 9.81
1367.1M 9.73
2367.4M 9.40
4368.0M 9.09
8369.1M 8.84
16371.5M 8.71
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Rank of△ proj. ParamsPerplexity
= 358.9M 9.12
1 359.1M 8.97
2 359.3M 8.97
359.7M 8.91
4 8 360.5M 8.83
16 362.1M 8.84
32 365.2M 8.80
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H3H3S4 (complex)10.30
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MambaS4 (real)10.56
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MambaMambaS68.69
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Selective△ Selective B Selective C Perplexity
X10.93
Xxx<x> 10.15
x√9.98
x<xx> √9.81 8.71
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An InitializationFieldPerplexity
A=+miComplex 9.16
An=-Real8.85
An=-(n+1)Real8.71
An ~exp(N (0,1))Real8.71
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State dim. N Params Perplexity
1367.1M 9.88
2367.4M 9.86
4368.0M 9.82
8369.1M 9.82
16371.5M 9.81
1367.1M 9.73
2367.4M 9.40
4368.0M 9.09
8369.1M 8.84
16371.5M 8.71
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Rank of△ proj. ParamsPerplexity
= 358.9M 9.12
1 359.1M 8.97
2 359.3M 8.97
359.7M 8.91
4 8 360.5M 8.83
16 362.1M 8.84
32 365.2M 8.80
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Models are trained on sequence length", "type": "text" }, { "bbox": [ 356, 80, 389, 91 ], "score": 0.91, "content": "2 ^ { 8 } = 2 5 6", "type": "inline_equation" }, { "bbox": [ 389, 79, 506, 93 ], "score": 1.0, "content": ", and tested on various sequence", "type": "text" } ], "index": 0 }, { "bbox": [ 104, 88, 507, 103 ], "spans": [ { "bbox": [ 104, 88, 145, 103 ], "score": 1.0, "content": "lengths of", "type": "text" }, { "bbox": [ 146, 90, 177, 100 ], "score": 0.69, "content": "2 ^ { 6 } = 6 4", "type": "inline_equation" }, { "bbox": [ 177, 88, 199, 103 ], "score": 1.0, "content": "up to", "type": "text" }, { "bbox": [ 199, 90, 257, 101 ], "score": 0.88, "content": "2 ^ { 2 0 } = 1 0 4 8 5 7 6", "type": "inline_equation" }, { "bbox": [ 257, 88, 261, 103 ], "score": 1.0, "content": ".", "type": "text" }, { "bbox": [ 261, 91, 270, 100 ], "score": 0.27, "content": "\\checkmark", "type": "inline_equation" }, { "bbox": [ 271, 88, 442, 103 ], "score": 1.0, "content": "denotes perfect generalization accuracy, while", "type": "text" }, { "bbox": [ 442, 92, 450, 100 ], "score": 0.63, "content": "\\pmb { \\chi }", "type": "inline_equation" }, { "bbox": [ 450, 88, 507, 103 ], "score": 1.0, "content": "denotes out of", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 100, 142, 113 ], "spans": [ { "bbox": [ 105, 100, 142, 113 ], "score": 1.0, "content": "memory.", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "table_body", "bbox": [ 110, 120, 498, 216 ], "group_id": 0, "lines": [ { "bbox": [ 110, 120, 498, 216 ], "spans": [ { "bbox": [ 110, 120, 498, 216 ], "score": 0.98, "html": "
ModelParamsTest Accuracy (%) at Sequence Length
26272829210211212213214 215 216 217 218 219220
MHA-Abs137K99.6100.0 58.6 26.6 18.8 9.810.97.8XX×XX×
MHA-RoPE 137K√100.0 83.6 31.318.4 8.69.05.5X×XXX
MHA-xPos137K√100.0 99.6 67.6 25.4 7.09.07.8 x×X×X
H3153K100.080.9 39.5 23.814.8 8.25.9 6.68.24.78.2 6.3 7.4
Hyena69M*97.7100.044.1 12.5 6.65.17.0 5.9 6.6 6.6 5.96.3 9.8
Mamba74K100.0√√√√√√
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Params n_layers dmodel n_heads /d_head Training steps Learning Rate Batch SizeTokens
125M1276812/6448006e-40.5M tokens2.5B
350M24102416/64135003e-40.5Mtokens7B
760M24153616/96290002.5e-40.5Mtokens15B
1.3B24204832/64500002e-40.5Mtokens26B
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We choose an “epoch” size of 8192 steps, and track the accuracy on fixed validation sets (also", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 362, 505, 375 ], "spans": [ { "bbox": [ 105, 362, 505, 375 ], "score": 1.0, "content": "randomly generated) of each target sequence length. For the MHA-Abs and Mamba models, results", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 373, 505, 385 ], "spans": [ { "bbox": [ 105, 373, 242, 385 ], "score": 1.0, "content": "are reported after the 25th epoch", "type": "text" }, { "bbox": [ 242, 374, 325, 384 ], "score": 0.9, "content": "\\mathrm { 8 1 9 2 \\times 2 5 = 2 0 4 8 0 0 }", "type": "inline_equation" }, { "bbox": [ 325, 373, 505, 385 ], "score": 1.0, "content": "steps). For the MHA-RoPE and MHA-xPos", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 384, 504, 396 ], "spans": [ { "bbox": [ 106, 385, 307, 396 ], "score": 1.0, "content": "models, results are reported after the 50th epoch", "type": "text" }, { "bbox": [ 307, 384, 393, 395 ], "score": 0.9, "content": "( 8 1 9 2 \\times 5 0 = 4 0 9 6 0 0", "type": "inline_equation" }, { "bbox": [ 393, 385, 504, 396 ], "score": 1.0, "content": "steps). For the LTI H3 and", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 395, 505, 407 ], "spans": [ { "bbox": [ 106, 395, 505, 407 ], "score": 1.0, "content": "Hyena models, results are reported after the 10th epoch (81920 steps) because they had converged", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 406, 255, 417 ], "spans": [ { "bbox": [ 106, 406, 255, 417 ], "score": 1.0, "content": "by then and failed to improve further.", "type": "text" } ], "index": 18 } ], "index": 15 }, { "type": "text", "bbox": [ 107, 421, 505, 476 ], "lines": [ { "bbox": [ 105, 420, 506, 434 ], "spans": [ { "bbox": [ 105, 420, 506, 434 ], "score": 1.0, "content": "We use the Adam optimizer with no weight decay. All models are trained at constant learning rates", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 431, 505, 445 ], "spans": [ { "bbox": [ 106, 432, 133, 443 ], "score": 0.8, "content": "2 e - 4", "type": "inline_equation" }, { "bbox": [ 133, 431, 151, 445 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 152, 433, 178, 443 ], "score": 0.81, "content": "1 e - 3", "type": "inline_equation" }, { "bbox": [ 179, 431, 388, 445 ], "score": 1.0, "content": ", and the better results are reported for each model", "type": "text" }, { "bbox": [ 388, 433, 416, 443 ], "score": 0.81, "content": "2 e - 4", "type": "inline_equation" }, { "bbox": [ 416, 431, 505, 445 ], "score": 1.0, "content": "for all models except", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 443, 505, 455 ], "spans": [ { "bbox": [ 105, 443, 360, 455 ], "score": 1.0, "content": "Mamba). 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Mamba learned at", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 464, 340, 477 ], "spans": [ { "bbox": [ 105, 464, 311, 477 ], "score": 1.0, "content": "both LRs, but extrapolated better at the larger LR of", "type": "text" }, { "bbox": [ 312, 465, 335, 475 ], "score": 0.8, "content": "1 e - 3", "type": "inline_equation" }, { "bbox": [ 336, 464, 340, 477 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 23 } ], "index": 21 }, { "type": "title", "bbox": [ 108, 484, 231, 495 ], "lines": [ { "bbox": [ 105, 484, 231, 497 ], "spans": [ { "bbox": [ 105, 484, 231, 497 ], "score": 1.0, "content": "F.2 LANGUAGE MODELING", "type": "text" } ], "index": 24 } ], "index": 24 }, { "type": "text", "bbox": [ 107, 500, 241, 511 ], "lines": [ { "bbox": [ 105, 498, 242, 513 ], "spans": [ { "bbox": [ 105, 498, 242, 513 ], "score": 1.0, "content": "F.2.1 SCALING LAW DETAILS", "type": "text" } ], "index": 25 } ], "index": 25 }, { "type": "text", "bbox": [ 108, 519, 248, 531 ], "lines": [ { "bbox": [ 106, 519, 249, 532 ], "spans": [ { "bbox": [ 106, 519, 249, 532 ], "score": 1.0, "content": "All models were trained on the Pile.", "type": "text" } ], "index": 26 } ], "index": 26 }, { "type": "text", "bbox": [ 106, 540, 505, 605 ], "lines": [ { "bbox": [ 105, 539, 505, 552 ], "spans": [ { "bbox": [ 105, 539, 505, 552 ], "score": 1.0, "content": "Model sizes. Table 11 specifies the model sizes we use for scaling laws. This is taken directly", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 549, 505, 563 ], "spans": [ { "bbox": [ 105, 549, 505, 563 ], "score": 1.0, "content": "from the GPT3 specifications (Brown et al., 2020), with very minor modifications. First, we changed", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 560, 505, 574 ], "spans": [ { "bbox": [ 105, 560, 326, 574 ], "score": 1.0, "content": "the batch size of the 1.3B model from 1M tokens to", "type": "text" }, { "bbox": [ 326, 561, 350, 572 ], "score": 0.29, "content": "0 . 5 \\mathbf { M }", "type": "inline_equation" }, { "bbox": [ 351, 560, 505, 574 ], "score": 1.0, "content": "tokens, since we did not use enough", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 572, 506, 585 ], "spans": [ { "bbox": [ 105, 572, 506, 585 ], "score": 1.0, "content": "parallelization to require the larger batch size. Second, we changed the number of training steps and", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 583, 506, 595 ], "spans": [ { "bbox": [ 105, 583, 506, 595 ], "score": 1.0, "content": "total tokens to roughly match Chinchilla scaling laws (Hoffmann et al., 2022), which specify that", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 594, 347, 606 ], "spans": [ { "bbox": [ 106, 594, 347, 606 ], "score": 1.0, "content": "training tokens should increase proportionally to model size.", "type": "text" } ], "index": 32 } ], "index": 29.5 }, { "type": "text", "bbox": [ 107, 613, 363, 625 ], "lines": [ { "bbox": [ 105, 612, 364, 627 ], "spans": [ { "bbox": [ 105, 612, 364, 627 ], "score": 1.0, "content": "Training recipes. All models used the AdamW optimizer with", "type": "text" } ], "index": 33 } ], "index": 33 }, { "type": "text", "bbox": [ 134, 632, 326, 687 ], "lines": [ { "bbox": [ 132, 632, 232, 644 ], "spans": [ { "bbox": [ 132, 632, 232, 644 ], "score": 1.0, "content": "• gradient clip value 1.0", "type": "text" } ], "index": 34 }, { "bbox": [ 133, 646, 211, 659 ], "spans": [ { "bbox": [ 133, 646, 211, 659 ], "score": 1.0, "content": "• weight decay 0.1", "type": "text" } ], "index": 35 }, { "bbox": [ 133, 660, 189, 674 ], "spans": [ { "bbox": [ 133, 660, 189, 674 ], "score": 1.0, "content": "• no dropout", "type": "text" } ], "index": 36 }, { "bbox": [ 133, 674, 326, 689 ], "spans": [ { "bbox": [ 133, 674, 326, 689 ], "score": 1.0, "content": "• linear learning rate warmup with cosine decay", "type": "text" } ], "index": 37 } ], "index": 35.5 }, { "type": "text", "bbox": [ 107, 694, 342, 705 ], "lines": [ { "bbox": [ 106, 693, 343, 708 ], "spans": [ { "bbox": [ 106, 693, 343, 708 ], "score": 1.0, "content": "By default, the peak learning rate is the GPT3 specification.", "type": "text" } ], "index": 38 } ], "index": 38 }, { "type": "text", "bbox": [ 105, 710, 506, 732 ], "lines": [ { "bbox": [ 105, 708, 505, 723 ], "spans": [ { "bbox": [ 105, 708, 505, 723 ], "score": 1.0, "content": "We give several models an “improved recipe”, inspired by changes adopted by popular large language", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 720, 494, 732 ], "spans": [ { "bbox": [ 105, 720, 494, 732 ], "score": 1.0, "content": "models such as PaLM (Chowdhery et al., 2022) and LLaMa (Touvron et al., 2023). These include:", "type": "text" } ], "index": 40 } ], "index": 39.5 } ], "page_idx": 21, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 304, 37 ], "lines": [ { "bbox": [ 106, 26, 305, 38 ], "spans": [ { "bbox": [ 106, 26, 305, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "22", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 110, 120, 498, 216 ], "blocks": [ { "type": "table_caption", "bbox": [ 107, 81, 505, 111 ], "group_id": 0, "lines": [ { "bbox": [ 104, 79, 506, 93 ], "spans": [ { "bbox": [ 104, 79, 355, 93 ], "score": 1.0, "content": "Table 10: (Induction heads.) Models are trained on sequence length", "type": "text" }, { "bbox": [ 356, 80, 389, 91 ], "score": 0.91, "content": "2 ^ { 8 } = 2 5 6", "type": "inline_equation" }, { "bbox": [ 389, 79, 506, 93 ], "score": 1.0, "content": ", and tested on various sequence", "type": "text" } ], "index": 0 }, { "bbox": [ 104, 88, 507, 103 ], "spans": [ { "bbox": [ 104, 88, 145, 103 ], "score": 1.0, "content": "lengths of", "type": "text" }, { "bbox": [ 146, 90, 177, 100 ], "score": 0.69, "content": "2 ^ { 6 } = 6 4", "type": "inline_equation" }, { "bbox": [ 177, 88, 199, 103 ], "score": 1.0, "content": "up to", "type": "text" }, { "bbox": [ 199, 90, 257, 101 ], "score": 0.88, "content": "2 ^ { 2 0 } = 1 0 4 8 5 7 6", "type": "inline_equation" }, { "bbox": [ 257, 88, 261, 103 ], "score": 1.0, "content": ".", "type": "text" }, { "bbox": [ 261, 91, 270, 100 ], "score": 0.27, "content": "\\checkmark", "type": "inline_equation" }, { "bbox": [ 271, 88, 442, 103 ], "score": 1.0, "content": "denotes perfect generalization accuracy, while", "type": "text" }, { "bbox": [ 442, 92, 450, 100 ], "score": 0.63, "content": "\\pmb { \\chi }", "type": "inline_equation" }, { "bbox": [ 450, 88, 507, 103 ], "score": 1.0, "content": "denotes out of", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 100, 142, 113 ], "spans": [ { "bbox": [ 105, 100, 142, 113 ], "score": 1.0, "content": "memory.", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "table_body", "bbox": [ 110, 120, 498, 216 ], "group_id": 0, "lines": [ { "bbox": [ 110, 120, 498, 216 ], "spans": [ { "bbox": [ 110, 120, 498, 216 ], "score": 0.98, "html": "
ModelParamsTest Accuracy (%) at Sequence Length
26272829210211212213214 215 216 217 218 219220
MHA-Abs137K99.6100.0 58.6 26.6 18.8 9.810.97.8XX×XX×
MHA-RoPE 137K√100.0 83.6 31.318.4 8.69.05.5X×XXX
MHA-xPos137K√100.0 99.6 67.6 25.4 7.09.07.8 x×X×X
H3153K100.080.9 39.5 23.814.8 8.25.9 6.68.24.78.2 6.3 7.4
Hyena69M*97.7100.044.1 12.5 6.65.17.0 5.9 6.6 6.6 5.96.3 9.8
Mamba74K100.0√√√√√√
", "type": "table", "image_path": "3e5ab4e41e9cbdd9618b51c8e245e3618367993a0d541f8e2027c9f9a4a51469.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 110, 120, 498, 152.0 ], "spans": [], "index": 3 }, { "bbox": [ 110, 152.0, 498, 184.0 ], "spans": [], "index": 4 }, { "bbox": [ 110, 184.0, 498, 216.0 ], "spans": [], "index": 5 } ] }, { "type": "table_footnote", "bbox": [ 113, 216, 337, 226 ], "group_id": 0, "lines": [ { "bbox": [ 112, 213, 338, 228 ], "spans": [ { "bbox": [ 112, 213, 338, 228 ], "score": 1.0, "content": "∗ Most of the parameters are in learnable positional encodings.", "type": "text" } ], "index": 6 } ], "index": 6 } ], "index": 4 }, { "type": "table", "bbox": [ 117, 266, 492, 328 ], "blocks": [ { "type": "table_caption", "bbox": [ 107, 236, 504, 257 ], "group_id": 1, "lines": [ { "bbox": [ 105, 236, 505, 248 ], "spans": [ { "bbox": [ 105, 236, 505, 248 ], "score": 1.0, "content": "Table 11: (Scaling Law Model Sizes.) Our model sizes and hyperparameters for scaling experiments. (Model", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 246, 354, 257 ], "spans": [ { "bbox": [ 106, 246, 354, 257 ], "score": 1.0, "content": "dimension and number of heads applies only to Transformer models.)", "type": "text" } ], "index": 8 } ], "index": 7.5 }, { "type": "table_body", "bbox": [ 117, 266, 492, 328 ], "group_id": 1, "lines": [ { "bbox": [ 117, 266, 492, 328 ], "spans": [ { "bbox": [ 117, 266, 492, 328 ], "score": 0.98, "html": "
Params n_layers dmodel n_heads /d_head Training steps Learning Rate Batch SizeTokens
125M1276812/6448006e-40.5M tokens2.5B
350M24102416/64135003e-40.5Mtokens7B
760M24153616/96290002.5e-40.5Mtokens15B
1.3B24204832/64500002e-40.5Mtokens26B
", "type": "table", "image_path": "759d3e6e46218a9cf9203fbfcef0527c9a99037c9441814bfe77e2c513c54f42.jpg" } ] } ], "index": 10, "virtual_lines": [ { "bbox": [ 117, 266, 492, 286.6666666666667 ], "spans": [], "index": 9 }, { "bbox": [ 117, 286.6666666666667, 492, 307.33333333333337 ], "spans": [], "index": 10 }, { "bbox": [ 117, 307.33333333333337, 492, 328.00000000000006 ], "spans": [], "index": 11 } ] } ], "index": 8.75 }, { "type": "text", "bbox": [ 106, 341, 505, 417 ], "lines": [ { "bbox": [ 105, 341, 506, 354 ], "spans": [ { "bbox": [ 105, 341, 506, 354 ], "score": 1.0, "content": "Induction Heads Training consists of randomly generating data every step, with a batch size of", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 352, 505, 365 ], "spans": [ { "bbox": [ 106, 352, 505, 365 ], "score": 1.0, "content": "8. We choose an “epoch” size of 8192 steps, and track the accuracy on fixed validation sets (also", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 362, 505, 375 ], "spans": [ { "bbox": [ 105, 362, 505, 375 ], "score": 1.0, "content": "randomly generated) of each target sequence length. For the MHA-Abs and Mamba models, results", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 373, 505, 385 ], "spans": [ { "bbox": [ 105, 373, 242, 385 ], "score": 1.0, "content": "are reported after the 25th epoch", "type": "text" }, { "bbox": [ 242, 374, 325, 384 ], "score": 0.9, "content": "\\mathrm { 8 1 9 2 \\times 2 5 = 2 0 4 8 0 0 }", "type": "inline_equation" }, { "bbox": [ 325, 373, 505, 385 ], "score": 1.0, "content": "steps). For the MHA-RoPE and MHA-xPos", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 384, 504, 396 ], "spans": [ { "bbox": [ 106, 385, 307, 396 ], "score": 1.0, "content": "models, results are reported after the 50th epoch", "type": "text" }, { "bbox": [ 307, 384, 393, 395 ], "score": 0.9, "content": "( 8 1 9 2 \\times 5 0 = 4 0 9 6 0 0", "type": "inline_equation" }, { "bbox": [ 393, 385, 504, 396 ], "score": 1.0, "content": "steps). For the LTI H3 and", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 395, 505, 407 ], "spans": [ { "bbox": [ 106, 395, 505, 407 ], "score": 1.0, "content": "Hyena models, results are reported after the 10th epoch (81920 steps) because they had converged", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 406, 255, 417 ], "spans": [ { "bbox": [ 106, 406, 255, 417 ], "score": 1.0, "content": "by then and failed to improve further.", "type": "text" } ], "index": 18 } ], "index": 15, "bbox_fs": [ 105, 341, 506, 417 ] }, { "type": "text", "bbox": [ 107, 421, 505, 476 ], "lines": [ { "bbox": [ 105, 420, 506, 434 ], "spans": [ { "bbox": [ 105, 420, 506, 434 ], "score": 1.0, "content": "We use the Adam optimizer with no weight decay. All models are trained at constant learning rates", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 431, 505, 445 ], "spans": [ { "bbox": [ 106, 432, 133, 443 ], "score": 0.8, "content": "2 e - 4", "type": "inline_equation" }, { "bbox": [ 133, 431, 151, 445 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 152, 433, 178, 443 ], "score": 0.81, "content": "1 e - 3", "type": "inline_equation" }, { "bbox": [ 179, 431, 388, 445 ], "score": 1.0, "content": ", and the better results are reported for each model", "type": "text" }, { "bbox": [ 388, 433, 416, 443 ], "score": 0.81, "content": "2 e - 4", "type": "inline_equation" }, { "bbox": [ 416, 431, 505, 445 ], "score": 1.0, "content": "for all models except", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 443, 505, 455 ], "spans": [ { "bbox": [ 105, 443, 360, 455 ], "score": 1.0, "content": "Mamba). The attention and Hyena models did not learn at LR", "type": "text" }, { "bbox": [ 360, 443, 385, 453 ], "score": 0.8, "content": "1 e - 3", "type": "inline_equation" }, { "bbox": [ 386, 443, 505, 455 ], "score": 1.0, "content": ". H3 learned at both LRs, but", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 453, 506, 466 ], "spans": [ { "bbox": [ 105, 453, 400, 466 ], "score": 1.0, "content": "interestingly generalized better to shorter sequences at the smaller LR of", "type": "text" }, { "bbox": [ 400, 454, 425, 464 ], "score": 0.85, "content": "2 e - 4", "type": "inline_equation" }, { "bbox": [ 426, 453, 506, 466 ], "score": 1.0, "content": ". Mamba learned at", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 464, 340, 477 ], "spans": [ { "bbox": [ 105, 464, 311, 477 ], "score": 1.0, "content": "both LRs, but extrapolated better at the larger LR of", "type": "text" }, { "bbox": [ 312, 465, 335, 475 ], "score": 0.8, "content": "1 e - 3", "type": "inline_equation" }, { "bbox": [ 336, 464, 340, 477 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 23 } ], "index": 21, "bbox_fs": [ 105, 420, 506, 477 ] }, { "type": "title", "bbox": [ 108, 484, 231, 495 ], "lines": [ { "bbox": [ 105, 484, 231, 497 ], "spans": [ { "bbox": [ 105, 484, 231, 497 ], "score": 1.0, "content": "F.2 LANGUAGE MODELING", "type": "text" } ], "index": 24 } ], "index": 24 }, { "type": "text", "bbox": [ 107, 500, 241, 511 ], "lines": [ { "bbox": [ 105, 498, 242, 513 ], "spans": [ { "bbox": [ 105, 498, 242, 513 ], "score": 1.0, "content": "F.2.1 SCALING LAW DETAILS", "type": "text" } ], "index": 25 } ], "index": 25, "bbox_fs": [ 105, 498, 242, 513 ] }, { "type": "text", "bbox": [ 108, 519, 248, 531 ], "lines": [ { "bbox": [ 106, 519, 249, 532 ], "spans": [ { "bbox": [ 106, 519, 249, 532 ], "score": 1.0, "content": "All models were trained on the Pile.", "type": "text" } ], "index": 26 } ], "index": 26, "bbox_fs": [ 106, 519, 249, 532 ] }, { "type": "text", "bbox": [ 106, 540, 505, 605 ], "lines": [ { "bbox": [ 105, 539, 505, 552 ], "spans": [ { "bbox": [ 105, 539, 505, 552 ], "score": 1.0, "content": "Model sizes. Table 11 specifies the model sizes we use for scaling laws. This is taken directly", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 549, 505, 563 ], "spans": [ { "bbox": [ 105, 549, 505, 563 ], "score": 1.0, "content": "from the GPT3 specifications (Brown et al., 2020), with very minor modifications. First, we changed", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 560, 505, 574 ], "spans": [ { "bbox": [ 105, 560, 326, 574 ], "score": 1.0, "content": "the batch size of the 1.3B model from 1M tokens to", "type": "text" }, { "bbox": [ 326, 561, 350, 572 ], "score": 0.29, "content": "0 . 5 \\mathbf { M }", "type": "inline_equation" }, { "bbox": [ 351, 560, 505, 574 ], "score": 1.0, "content": "tokens, since we did not use enough", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 572, 506, 585 ], "spans": [ { "bbox": [ 105, 572, 506, 585 ], "score": 1.0, "content": "parallelization to require the larger batch size. Second, we changed the number of training steps and", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 583, 506, 595 ], "spans": [ { "bbox": [ 105, 583, 506, 595 ], "score": 1.0, "content": "total tokens to roughly match Chinchilla scaling laws (Hoffmann et al., 2022), which specify that", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 594, 347, 606 ], "spans": [ { "bbox": [ 106, 594, 347, 606 ], "score": 1.0, "content": "training tokens should increase proportionally to model size.", "type": "text" } ], "index": 32 } ], "index": 29.5, "bbox_fs": [ 105, 539, 506, 606 ] }, { "type": "text", "bbox": [ 107, 613, 363, 625 ], "lines": [ { "bbox": [ 105, 612, 364, 627 ], "spans": [ { "bbox": [ 105, 612, 364, 627 ], "score": 1.0, "content": "Training recipes. All models used the AdamW optimizer with", "type": "text" } ], "index": 33 } ], "index": 33, "bbox_fs": [ 105, 612, 364, 627 ] }, { "type": "list", "bbox": [ 134, 632, 326, 687 ], "lines": [ { "bbox": [ 132, 632, 232, 644 ], "spans": [ { "bbox": [ 132, 632, 232, 644 ], "score": 1.0, "content": "• gradient clip value 1.0", "type": "text" } ], "index": 34, "is_list_start_line": true }, { "bbox": [ 133, 646, 211, 659 ], "spans": [ { "bbox": [ 133, 646, 211, 659 ], "score": 1.0, "content": "• weight decay 0.1", "type": "text" } ], "index": 35, "is_list_start_line": true }, { "bbox": [ 133, 660, 189, 674 ], "spans": [ { "bbox": [ 133, 660, 189, 674 ], "score": 1.0, "content": "• no dropout", "type": "text" } ], "index": 36, "is_list_start_line": true }, { "bbox": [ 133, 674, 326, 689 ], "spans": [ { "bbox": [ 133, 674, 326, 689 ], "score": 1.0, "content": "• linear learning rate warmup with cosine decay", "type": "text" } ], "index": 37, "is_list_start_line": true } ], "index": 35.5, "bbox_fs": [ 132, 632, 326, 689 ] }, { "type": "text", "bbox": [ 107, 694, 342, 705 ], "lines": [ { "bbox": [ 106, 693, 343, 708 ], "spans": [ { "bbox": [ 106, 693, 343, 708 ], "score": 1.0, "content": "By default, the peak learning rate is the GPT3 specification.", "type": "text" } ], "index": 38 } ], "index": 38, "bbox_fs": [ 106, 693, 343, 708 ] }, { "type": "text", "bbox": [ 105, 710, 506, 732 ], "lines": [ { "bbox": [ 105, 708, 505, 723 ], "spans": [ { "bbox": [ 105, 708, 505, 723 ], "score": 1.0, "content": "We give several models an “improved recipe”, inspired by changes adopted by popular large language", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 720, 494, 732 ], "spans": [ { "bbox": [ 105, 720, 494, 732 ], "score": 1.0, "content": "models such as PaLM (Chowdhery et al., 2022) and LLaMa (Touvron et al., 2023). These include:", "type": "text" } ], "index": 40 } ], "index": 39.5, "bbox_fs": [ 105, 708, 505, 732 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 133, 82, 505, 162 ], "lines": [ { "bbox": [ 132, 82, 505, 95 ], "spans": [ { "bbox": [ 132, 82, 339, 95 ], "score": 1.0, "content": "• linear learning rate warmup with cosine decay to", "type": "text" }, { "bbox": [ 339, 83, 363, 93 ], "score": 0.72, "content": "1 e - 5", "type": "inline_equation" }, { "bbox": [ 364, 82, 449, 95 ], "score": 1.0, "content": ", with a peak value of", "type": "text" }, { "bbox": [ 449, 83, 463, 93 ], "score": 0.85, "content": "5 \\times", "type": "inline_equation" }, { "bbox": [ 464, 82, 505, 95 ], "score": 1.0, "content": "the GPT3", "type": "text" } ], "index": 0 }, { "bbox": [ 141, 93, 167, 105 ], "spans": [ { "bbox": [ 141, 93, 167, 105 ], "score": 1.0, "content": "value", "type": "text" } ], "index": 1 }, { "bbox": [ 133, 109, 222, 120 ], "spans": [ { "bbox": [ 133, 109, 222, 120 ], "score": 1.0, "content": "• no linear bias terms", "type": "text" } ], "index": 2 }, { "bbox": [ 132, 122, 279, 136 ], "spans": [ { "bbox": [ 132, 122, 279, 136 ], "score": 1.0, "content": "• RMSNorm instead of LayerNorm", "type": "text" } ], "index": 3 }, { "bbox": [ 132, 137, 506, 152 ], "spans": [ { "bbox": [ 132, 137, 245, 152 ], "score": 1.0, "content": "• AdamW hyperparameter", "type": "text" }, { "bbox": [ 245, 139, 295, 151 ], "score": 0.89, "content": "\\beta = ( . 9 , . 9 5 )", "type": "inline_equation" }, { "bbox": [ 296, 137, 506, 152 ], "score": 1.0, "content": "(the GPT3 value) instead of the PyTorch default of", "type": "text" } ], "index": 4 }, { "bbox": [ 142, 150, 195, 162 ], "spans": [ { "bbox": [ 142, 150, 195, 162 ], "score": 0.85, "content": "\\beta = ( . 9 , . 9 9 9 )", "type": "inline_equation" } ], "index": 5 } ], "index": 2.5 }, { "type": "text", "bbox": [ 106, 170, 322, 181 ], "lines": [ { "bbox": [ 106, 168, 322, 183 ], "spans": [ { "bbox": [ 106, 168, 322, 183 ], "score": 1.0, "content": "Architecture and training details. Our models are:", "type": "text" } ], "index": 6 } ], "index": 6 }, { "type": "text", "bbox": [ 105, 189, 506, 355 ], "lines": [ { "bbox": [ 105, 189, 387, 201 ], "spans": [ { "bbox": [ 105, 189, 387, 201 ], "score": 1.0, "content": "• Transformer: The standard Transformer based on GPT3 (Table 11).", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 201, 505, 216 ], "spans": [ { "bbox": [ 105, 201, 169, 216 ], "score": 1.0, "content": "• Transformer", "type": "text" }, { "bbox": [ 169, 204, 181, 213 ], "score": 0.65, "content": "^ { + + }", "type": "inline_equation" }, { "bbox": [ 181, 201, 505, 216 ], "score": 1.0, "content": ": A Transformer with an improved architecture, namely rotary positional encodings", "type": "text" } ], "index": 8 }, { "bbox": [ 113, 213, 480, 226 ], "spans": [ { "bbox": [ 113, 213, 480, 226 ], "score": 1.0, "content": "(Su et al., 2021) and SwiGLU MLP (Shazeer, 2020), and the improved training recipe above.", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 226, 505, 239 ], "spans": [ { "bbox": [ 106, 226, 505, 239 ], "score": 1.0, "content": "• Hyena: Interleaving a Hyena block (the H3 block with S4 replaced by a global convolution param-", "type": "text" } ], "index": 10 }, { "bbox": [ 114, 237, 505, 250 ], "spans": [ { "bbox": [ 114, 237, 505, 250 ], "score": 1.0, "content": "eterized by an MLP) with standard MLP blocks. The MLP blocks have expansion factor 2 instead", "type": "text" } ], "index": 11 }, { "bbox": [ 113, 248, 495, 261 ], "spans": [ { "bbox": [ 113, 248, 358, 261 ], "score": 1.0, "content": "of 4 and the number of layers is correspondingly increased by", "type": "text" }, { "bbox": [ 358, 248, 380, 259 ], "score": 0.86, "content": "1 . 5 \\times", "type": "inline_equation" }, { "bbox": [ 380, 248, 495, 261 ], "score": 1.0, "content": "to preserve parameter count.", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 260, 506, 273 ], "spans": [ { "bbox": [ 106, 260, 114, 273 ], "score": 1.0, "content": "•", "type": "text" }, { "bbox": [ 114, 261, 141, 272 ], "score": 0.85, "content": "\\mathbf { H } 3 { + } +", "type": "inline_equation" }, { "bbox": [ 141, 260, 506, 273 ], "score": 1.0, "content": ": The H3 architecture with a few modifications, including (i) using the same “thin” Hyena", "type": "text" } ], "index": 13 }, { "bbox": [ 113, 272, 507, 284 ], "spans": [ { "bbox": [ 113, 272, 507, 284 ], "score": 1.0, "content": "dimensions above (ii) the improved training recipe above (iii) a linear attention head dimension of 8.", "type": "text" } ], "index": 14 }, { "bbox": [ 107, 284, 505, 297 ], "spans": [ { "bbox": [ 107, 284, 505, 297 ], "score": 1.0, "content": "• RWKV: The default RWKV model from Peng et al. (2023), including its modified MLP block. We", "type": "text" } ], "index": 15 }, { "bbox": [ 113, 296, 505, 308 ], "spans": [ { "bbox": [ 113, 296, 505, 308 ], "score": 1.0, "content": "also used as much of its specified training recipe as possible, such as increasing the learning rates", "type": "text" } ], "index": 16 }, { "bbox": [ 114, 307, 257, 318 ], "spans": [ { "bbox": [ 114, 307, 127, 318 ], "score": 1.0, "content": "by", "type": "text" }, { "bbox": [ 127, 307, 141, 317 ], "score": 0.86, "content": "2 \\times", "type": "inline_equation" }, { "bbox": [ 141, 307, 151, 318 ], "score": 1.0, "content": "or", "type": "text" }, { "bbox": [ 152, 307, 166, 317 ], "score": 0.87, "content": "3 \\times", "type": "inline_equation" }, { "bbox": [ 167, 307, 257, 318 ], "score": 1.0, "content": "on certain parameters.", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 318, 505, 332 ], "spans": [ { "bbox": [ 106, 318, 505, 332 ], "score": 1.0, "content": "• RetNet: The default RetNet model from Sun et al. (2023). We also gave it the improved training", "type": "text" } ], "index": 18 }, { "bbox": [ 113, 330, 170, 342 ], "spans": [ { "bbox": [ 113, 330, 170, 342 ], "score": 1.0, "content": "recipe above.", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 341, 426, 356 ], "spans": [ { "bbox": [ 105, 341, 426, 356 ], "score": 1.0, "content": "• Mamba: The standard Mamba architecture, with the improved training recipe.", "type": "text" } ], "index": 20 } ], "index": 13.5 }, { "type": "title", "bbox": [ 106, 363, 301, 374 ], "lines": [ { "bbox": [ 105, 362, 301, 376 ], "spans": [ { "bbox": [ 105, 362, 301, 376 ], "score": 1.0, "content": "F.2.2 DOWNSTREAM EVALUATION DETAILS", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "text", "bbox": [ 107, 383, 505, 427 ], "lines": [ { "bbox": [ 105, 383, 506, 396 ], "spans": [ { "bbox": [ 105, 383, 506, 396 ], "score": 1.0, "content": "This pretraining procedure is the same as the scaling law protocol, but extended to 300B tokens. For", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 394, 505, 406 ], "spans": [ { "bbox": [ 106, 394, 505, 406 ], "score": 1.0, "content": "the 1.3B model, we use a batch size of 1M tokens to be consistent with the GPT3 specifications. We", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 405, 506, 417 ], "spans": [ { "bbox": [ 105, 405, 506, 417 ], "score": 1.0, "content": "report the perplexity on the Pile validation set, and for this metric only compare to models trained", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 416, 426, 427 ], "spans": [ { "bbox": [ 105, 416, 426, 427 ], "score": 1.0, "content": "on the same dataset and with the same tokenizer, in particular Pythia and RWKV.", "type": "text" } ], "index": 25 } ], "index": 23.5 }, { "type": "text", "bbox": [ 107, 431, 505, 464 ], "lines": [ { "bbox": [ 105, 430, 507, 444 ], "spans": [ { "bbox": [ 105, 430, 507, 444 ], "score": 1.0, "content": "For downstream evaluation, we use the LM evaluation harness from EleutherAI (Gao et al., 2021),", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 442, 505, 453 ], "spans": [ { "bbox": [ 106, 442, 505, 453 ], "score": 1.0, "content": "as done by most work in this area. We evaluate on the following tasks/datasets that measure common", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 453, 174, 466 ], "spans": [ { "bbox": [ 105, 453, 174, 466 ], "score": 1.0, "content": "sense reasoning:", "type": "text" } ], "index": 28 } ], "index": 27 }, { "type": "text", "bbox": [ 133, 472, 323, 560 ], "lines": [ { "bbox": [ 134, 472, 282, 485 ], "spans": [ { "bbox": [ 134, 472, 282, 485 ], "score": 1.0, "content": "• LAMBADA (Paperno et al., 2016).", "type": "text" } ], "index": 29 }, { "bbox": [ 133, 487, 277, 500 ], "spans": [ { "bbox": [ 133, 487, 277, 500 ], "score": 1.0, "content": "• HellaSwag (Paperno et al., 2016).", "type": "text" } ], "index": 30 }, { "bbox": [ 133, 502, 242, 515 ], "spans": [ { "bbox": [ 133, 502, 242, 515 ], "score": 1.0, "content": "• PIQA (Bisk et al., 2020).", "type": "text" } ], "index": 31 }, { "bbox": [ 133, 516, 324, 531 ], "spans": [ { "bbox": [ 133, 516, 324, 531 ], "score": 1.0, "content": "• ARC-easy: an easy subset of ARC-challenge.", "type": "text" } ], "index": 32 }, { "bbox": [ 134, 533, 284, 545 ], "spans": [ { "bbox": [ 134, 533, 284, 545 ], "score": 1.0, "content": "• ARC-challenge (Clark et al., 2018).", "type": "text" } ], "index": 33 }, { "bbox": [ 133, 547, 293, 561 ], "spans": [ { "bbox": [ 133, 547, 293, 561 ], "score": 1.0, "content": "• WinoGrande (Sakaguchi et al., 2021).", "type": "text" } ], "index": 34 } ], "index": 31.5 }, { "type": "text", "bbox": [ 108, 568, 505, 601 ], "lines": [ { "bbox": [ 106, 568, 505, 581 ], "spans": [ { "bbox": [ 106, 568, 505, 581 ], "score": 1.0, "content": "We report accuracy for LAMBADA, WinoGrande, PIQA, and ARC-easy, and accuracy normalized", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 579, 505, 592 ], "spans": [ { "bbox": [ 106, 579, 505, 592 ], "score": 1.0, "content": "by sequence length for HellaSwag and ARC-challenge (since normalized accuracy is higher for almost", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 590, 210, 602 ], "spans": [ { "bbox": [ 106, 590, 210, 602 ], "score": 1.0, "content": "all models for these task).", "type": "text" } ], "index": 37 } ], "index": 36 }, { "type": "title", "bbox": [ 107, 610, 205, 621 ], "lines": [ { "bbox": [ 106, 610, 205, 623 ], "spans": [ { "bbox": [ 106, 610, 205, 623 ], "score": 1.0, "content": "F.3 DNA MODELING", "type": "text" } ], "index": 38 } ], "index": 38 }, { "type": "title", "bbox": [ 107, 626, 240, 637 ], "lines": [ { "bbox": [ 105, 625, 240, 639 ], "spans": [ { "bbox": [ 105, 625, 240, 639 ], "score": 1.0, "content": "F.3.1 PRETRAINING DETAILS", "type": "text" } ], "index": 39 } ], "index": 39 }, { "type": "text", "bbox": [ 105, 646, 464, 658 ], "lines": [ { "bbox": [ 105, 645, 465, 659 ], "spans": [ { "bbox": [ 105, 645, 465, 659 ], "score": 1.0, "content": "We describe the dataset and training procedure of the HG38 pretraining task in more detail.", "type": "text" } ], "index": 40 } ], "index": 40 }, { "type": "text", "bbox": [ 107, 662, 505, 705 ], "lines": [ { "bbox": [ 105, 661, 505, 674 ], "spans": [ { "bbox": [ 105, 661, 505, 674 ], "score": 1.0, "content": "The dataset follows the splits from the prior Enformer work on genomics (Avsec et al., 2021); the", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 671, 506, 685 ], "spans": [ { "bbox": [ 105, 671, 228, 685 ], "score": 1.0, "content": "training split contains a total of", "type": "text" }, { "bbox": [ 228, 673, 271, 683 ], "score": 0.9, "content": "S { = } 3 4 0 2 1", "type": "inline_equation" }, { "bbox": [ 271, 671, 347, 685 ], "score": 1.0, "content": "segments of length", "type": "text" }, { "bbox": [ 347, 672, 401, 683 ], "score": 0.9, "content": "2 ^ { 1 7 } = 1 3 1 0 7 2", "type": "inline_equation" }, { "bbox": [ 402, 671, 506, 685 ], "score": 1.0, "content": "that cover the genome, for", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 684, 505, 696 ], "spans": [ { "bbox": [ 105, 684, 505, 696 ], "score": 1.0, "content": "a total of approximately 4.5 billion tokens (DNA base pairs). 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On the other hand, we use the entire training data:", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "text", "bbox": [ 106, 108, 506, 192 ], "lines": [ { "bbox": [ 104, 106, 507, 123 ], "spans": [ { "bbox": [ 104, 106, 218, 123 ], "score": 1.0, "content": "• When the context length", "type": "text" }, { "bbox": [ 219, 110, 227, 119 ], "score": 0.77, "content": "L", "type": "inline_equation" }, { "bbox": [ 228, 106, 332, 123 ], "score": 1.0, "content": "is less than (or equal to)", "type": "text" }, { "bbox": [ 332, 108, 347, 119 ], "score": 0.82, "content": "2 ^ { 1 7 }", "type": "inline_equation" }, { "bbox": [ 347, 106, 507, 123 ], "score": 1.0, "content": ", we divide up each segment into non-", "type": "text" } ], "index": 2 }, { "bbox": [ 110, 115, 504, 137 ], "spans": [ { "bbox": [ 110, 115, 261, 137 ], "score": 1.0, "content": "overlapping sub-segments of length", "type": "text" }, { "bbox": [ 261, 123, 269, 132 ], "score": 0.78, "content": "L", "type": "inline_equation" }, { "bbox": [ 269, 115, 338, 137 ], "score": 1.0, "content": ", so that there are", "type": "text" }, { "bbox": [ 338, 119, 369, 135 ], "score": 0.93, "content": "\\begin{array} { r } { S \\times \\frac { 2 ^ { 1 7 } } { L } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 369, 115, 442, 137 ], "score": 1.0, "content": "total samples and", "type": "text" }, { "bbox": [ 442, 121, 504, 133 ], "score": 0.91, "content": "S \\times 2 ^ { 1 7 } \\approx 4 . 5 B", "type": "inline_equation" } ], "index": 3 }, { "bbox": [ 114, 133, 186, 145 ], "spans": [ { "bbox": [ 114, 133, 186, 145 ], "score": 1.0, "content": "tokens per epoch.", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 146, 506, 163 ], "spans": [ { "bbox": [ 106, 146, 216, 163 ], "score": 1.0, "content": "• When the context length", "type": "text" }, { "bbox": [ 217, 150, 225, 159 ], "score": 0.78, "content": "L", "type": "inline_equation" }, { "bbox": [ 225, 146, 286, 163 ], "score": 1.0, "content": "is greater than", "type": "text" }, { "bbox": [ 286, 149, 300, 159 ], "score": 0.82, "content": "2 ^ { 1 7 }", "type": "inline_equation" }, { "bbox": [ 301, 146, 506, 163 ], "score": 1.0, "content": ", we turn each segment into two samples, one that", "type": "text" } ], "index": 5 }, { "bbox": [ 114, 159, 506, 172 ], "spans": [ { "bbox": [ 114, 159, 506, 172 ], "score": 1.0, "content": "begins with the prescribed segment and one that ends with the prescribed segment. Thus each epoch", "type": "text" } ], "index": 6 }, { "bbox": [ 113, 168, 506, 184 ], "spans": [ { "bbox": [ 113, 168, 131, 184 ], "score": 1.0, "content": "has", "type": "text" }, { "bbox": [ 131, 171, 144, 181 ], "score": 0.77, "content": "2 S", "type": "inline_equation" }, { "bbox": [ 144, 168, 185, 184 ], "score": 1.0, "content": "items and", "type": "text" }, { "bbox": [ 186, 171, 205, 181 ], "score": 0.79, "content": "2 S L", "type": "inline_equation" }, { "bbox": [ 205, 168, 411, 184 ], "score": 1.0, "content": "tokens per epoch. For example, at sequence length", "type": "text" }, { "bbox": [ 412, 170, 467, 181 ], "score": 0.9, "content": "2 ^ { 1 8 } = 2 6 2 1 4 4", "type": "inline_equation" }, { "bbox": [ 468, 168, 506, 184 ], "score": 1.0, "content": "there are", "type": "text" } ], "index": 7 }, { "bbox": [ 114, 179, 476, 195 ], "spans": [ { "bbox": [ 114, 182, 128, 192 ], "score": 0.85, "content": "4 \\times", "type": "inline_equation" }, { "bbox": [ 129, 179, 339, 195 ], "score": 1.0, "content": "as many tokens as the default, and at sequence length", "type": "text" }, { "bbox": [ 339, 181, 353, 191 ], "score": 0.86, "content": "2 ^ { 2 0 }", "type": "inline_equation" }, { "bbox": [ 354, 179, 390, 195 ], "score": 1.0, "content": "there are", "type": "text" }, { "bbox": [ 390, 182, 410, 192 ], "score": 0.87, "content": "1 6 \\times", "type": "inline_equation" }, { "bbox": [ 410, 179, 476, 195 ], "score": 1.0, "content": "as many tokens.", "type": "text" } ], "index": 8 } ], "index": 5 }, { "type": "text", "bbox": [ 107, 197, 505, 230 ], "lines": [ { "bbox": [ 105, 196, 505, 210 ], "spans": [ { "bbox": [ 105, 196, 505, 210 ], "score": 1.0, "content": "Other training details generally follow the same protocol as our language modeling experiments", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 207, 505, 222 ], "spans": [ { "bbox": [ 105, 207, 319, 222 ], "score": 1.0, "content": "(Appendix F.2). For example, we use the AdamW with", "type": "text" }, { "bbox": [ 320, 208, 402, 220 ], "score": 0.92, "content": "( \\beta _ { 1 } , \\beta _ { 2 } ) = ( 0 . 9 , 0 . 9 5 ) ^ { \\mathrm { ~ ~ } }", "type": "inline_equation" }, { "bbox": [ 402, 207, 505, 222 ], "score": 1.0, "content": ", no dropout, weight decay", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 217, 449, 232 ], "spans": [ { "bbox": [ 105, 217, 373, 232 ], "score": 1.0, "content": "0.1. We use a cosine learning rate scheduler with linear warmup for", "type": "text" }, { "bbox": [ 373, 219, 392, 228 ], "score": 0.83, "content": "10 \\%", "type": "inline_equation" }, { "bbox": [ 393, 217, 449, 232 ], "score": 1.0, "content": "of total steps.", "type": "text" } ], "index": 11 } ], "index": 10 }, { "type": "title", "bbox": [ 107, 239, 279, 250 ], "lines": [ { "bbox": [ 105, 237, 280, 252 ], "spans": [ { "bbox": [ 105, 237, 280, 252 ], "score": 1.0, "content": "F.3.2 SCALING: MODEL SIZE DETAILS", "type": "text" } ], "index": 12 } ], "index": 12 }, { "type": "text", "bbox": [ 107, 259, 219, 271 ], "lines": [ { "bbox": [ 106, 258, 221, 272 ], "spans": [ { "bbox": [ 106, 258, 221, 272 ], "score": 1.0, "content": "The models we consider are:", "type": "text" } ], "index": 13 } ], "index": 13 }, { "type": "text", "bbox": [ 106, 280, 506, 361 ], "lines": [ { "bbox": [ 105, 278, 505, 292 ], "spans": [ { "bbox": [ 105, 278, 505, 292 ], "score": 1.0, "content": "• Transformer++: a Transformer with improved architecture, notably the usage of RoPE positional", "type": "text" } ], "index": 14 }, { "bbox": [ 114, 290, 505, 302 ], "spans": [ { "bbox": [ 114, 290, 505, 302 ], "score": 1.0, "content": "encodings (Su et al., 2021). Informally, we found these to be noticeably better than vanilla positional", "type": "text" } ], "index": 15 }, { "bbox": [ 114, 302, 268, 313 ], "spans": [ { "bbox": [ 114, 302, 268, 313 ], "score": 1.0, "content": "encodings from (Vaswani et al., 2017).", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 313, 506, 327 ], "spans": [ { "bbox": [ 105, 313, 506, 327 ], "score": 1.0, "content": "• HyenaDNA: the Hyena model from Poli et al. (2023); Nguyen et al. (2023), which is roughly a", "type": "text" } ], "index": 17 }, { "bbox": [ 112, 323, 506, 338 ], "spans": [ { "bbox": [ 112, 323, 506, 338 ], "score": 1.0, "content": "Transformer with the MHA block replaced by an H3 block using a global convolution parameterized", "type": "text" } ], "index": 18 }, { "bbox": [ 114, 336, 163, 348 ], "spans": [ { "bbox": [ 114, 336, 163, 348 ], "score": 1.0, "content": "by an MLP.", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 349, 285, 361 ], "spans": [ { "bbox": [ 105, 349, 285, 361 ], "score": 1.0, "content": "• Mamba: the standard Mamba architecture.", "type": "text" } ], "index": 20 } ], "index": 17 }, { "type": "text", "bbox": [ 106, 370, 241, 381 ], "lines": [ { "bbox": [ 106, 368, 242, 383 ], "spans": [ { "bbox": [ 106, 368, 242, 383 ], "score": 1.0, "content": "We use the following model sizes.", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "table", "bbox": [ 167, 390, 443, 430 ], "blocks": [ { "type": "table_body", "bbox": [ 167, 390, 443, 430 ], "group_id": 0, "lines": [ { "bbox": [ 167, 390, 443, 430 ], "spans": [ { "bbox": [ 167, 390, 443, 430 ], "score": 0.966, "html": "
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The optimal Transformer and HyenaDNA learning rates were 2e-3 across all sizes.", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 486, 506, 499 ], "spans": [ { "bbox": [ 106, 486, 506, 499 ], "score": 1.0, "content": "The optimal Mamba learning rate was 8e-3; note that Mamba performed better than baselines with", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 496, 506, 510 ], "spans": [ { "bbox": [ 106, 496, 506, 510 ], "score": 1.0, "content": "matched learning rates (2e-3), but was more stable and improved even more at higher learning rates. (Fur-", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 507, 506, 521 ], "spans": [ { "bbox": [ 106, 507, 506, 521 ], "score": 1.0, "content": "thermore, as this LR is on the upper range of the sweep, it is possible that our results are still suboptimal.)", "type": "text" } ], "index": 31 } ], "index": 29 }, { "type": "text", "bbox": [ 107, 523, 505, 556 ], "lines": [ { "bbox": [ 106, 523, 505, 535 ], "spans": [ { "bbox": [ 106, 523, 505, 535 ], "score": 1.0, "content": "Note that, in contrast to standard LM scaling laws (Table 11), our LR held constant across model sizes", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 534, 506, 546 ], "spans": [ { "bbox": [ 105, 534, 506, 546 ], "score": 1.0, "content": "for simplicity. The optimal LR should go down for larger models, but we didn’t find a noticeable effect", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 545, 400, 557 ], "spans": [ { "bbox": [ 105, 545, 400, 557 ], "score": 1.0, "content": "at the small model sizes (at most a few million parameters) we considered.", "type": "text" } ], "index": 34 } ], "index": 33 }, { "type": "title", "bbox": [ 108, 565, 304, 577 ], "lines": [ { "bbox": [ 105, 564, 306, 578 ], "spans": [ { "bbox": [ 105, 564, 306, 578 ], "score": 1.0, "content": "F.3.3 SCALING: CONTEXT LENGTH DETAILS", "type": "text" } ], "index": 35 } ], "index": 35 }, { "type": "text", "bbox": [ 107, 586, 505, 641 ], "lines": [ { "bbox": [ 105, 585, 507, 601 ], "spans": [ { "bbox": [ 105, 585, 219, 601 ], "score": 1.0, "content": "We use a total batch size of", "type": "text" }, { "bbox": [ 219, 586, 267, 597 ], "score": 0.92, "content": "2 ^ { 2 4 } \\approx 1 6 M", "type": "inline_equation" }, { "bbox": [ 267, 585, 507, 601 ], "score": 1.0, "content": "tokens per training step, for every sequence length (e.g. at", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 595, 506, 612 ], "spans": [ { "bbox": [ 105, 595, 133, 612 ], "score": 1.0, "content": "length", "type": "text" }, { "bbox": [ 133, 597, 147, 608 ], "score": 0.85, "content": "2 ^ { 2 0 }", "type": "inline_equation" }, { "bbox": [ 148, 595, 324, 612 ], "score": 1.0, "content": "there are 16 segments per batch and at length", "type": "text" }, { "bbox": [ 324, 597, 339, 608 ], "score": 0.82, "content": "2 ^ { 1 0 }", "type": "inline_equation" }, { "bbox": [ 339, 595, 506, 612 ], "score": 1.0, "content": "there are 16384 segments per batch). This", "type": "text" } ], "index": 37 }, { "bbox": [ 104, 607, 504, 621 ], "spans": [ { "bbox": [ 104, 607, 489, 621 ], "score": 1.0, "content": "is a large batch size relative to the model size by usual LM standards, but note that a batch size of", "type": "text" }, { "bbox": [ 490, 608, 504, 619 ], "score": 0.84, "content": "2 ^ { 2 3 }", "type": "inline_equation" } ], "index": 38 }, { "bbox": [ 105, 619, 506, 631 ], "spans": [ { "bbox": [ 105, 619, 403, 631 ], "score": 1.0, "content": "is the minimum possible on a machine with 8 GPUs and sequence length of", "type": "text" }, { "bbox": [ 404, 619, 419, 630 ], "score": 0.89, "content": "2 ^ { 2 } 0", "type": "inline_equation" }, { "bbox": [ 419, 619, 506, 631 ], "score": 1.0, "content": ", and that HyenaDNA", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 629, 236, 644 ], "spans": [ { "bbox": [ 105, 629, 236, 644 ], "score": 1.0, "content": "used much larger batches of 228.", "type": "text" } ], "index": 40 } ], "index": 38 }, { "type": "text", "bbox": [ 108, 645, 504, 679 ], "lines": [ { "bbox": [ 105, 645, 505, 659 ], "spans": [ { "bbox": [ 105, 645, 217, 659 ], "score": 1.0, "content": "The learning rate used was", "type": "text" }, { "bbox": [ 217, 646, 243, 657 ], "score": 0.75, "content": "8 e - 3", "type": "inline_equation" }, { "bbox": [ 243, 645, 309, 659 ], "score": 1.0, "content": "for Mamba and", "type": "text" }, { "bbox": [ 309, 646, 335, 657 ], "score": 0.79, "content": "1 e - 3", "type": "inline_equation" }, { "bbox": [ 335, 645, 505, 659 ], "score": 1.0, "content": "for HyenaDNA; we initially attempted to", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 657, 505, 669 ], "spans": [ { "bbox": [ 105, 657, 254, 669 ], "score": 1.0, "content": "use the same optimal learning rate of", "type": "text" }, { "bbox": [ 254, 657, 279, 667 ], "score": 0.75, "content": "2 e - 3", "type": "inline_equation" }, { "bbox": [ 279, 657, 505, 669 ], "score": 1.0, "content": "from the previous section for HyenaDNA, but found that", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 668, 281, 680 ], "spans": [ { "bbox": [ 105, 668, 281, 680 ], "score": 1.0, "content": "it was unstable at the longest context length.", "type": "text" } ], "index": 43 } ], "index": 42 }, { "type": "text", "bbox": [ 107, 688, 505, 732 ], "lines": [ { "bbox": [ 105, 687, 505, 702 ], "spans": [ { "bbox": [ 105, 687, 505, 702 ], "score": 1.0, "content": "Sequence Length Warmup. Following (Nguyen et al., 2023), we use sequence length warmup", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 699, 505, 712 ], "spans": [ { "bbox": [ 105, 699, 505, 712 ], "score": 1.0, "content": "(SLW) during pretraining. We choose a simple schedule of 2 epochs at each power-of-two sequence", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 709, 505, 723 ], "spans": [ { "bbox": [ 105, 709, 189, 723 ], "score": 1.0, "content": "length starting from", "type": "text" }, { "bbox": [ 189, 709, 234, 720 ], "score": 0.9, "content": "2 ^ { 1 0 } = 1 0 2 4", "type": "inline_equation" }, { "bbox": [ 235, 709, 505, 723 ], "score": 1.0, "content": ". (Note that because of how data is curated, at the longest sequence", "type": "text" } ], "index": 46 }, { "bbox": [ 104, 720, 504, 734 ], "spans": [ { "bbox": [ 104, 720, 489, 734 ], "score": 1.0, "content": "lengths more steps and tokens are spent proportionally. In particular, each stage up to length", "type": "text" }, { "bbox": [ 489, 720, 504, 730 ], "score": 0.79, "content": "2 ^ { 1 7 }", "type": "inline_equation" } ], "index": 47 } ], "index": 45.5 } ], "page_idx": 23, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 304, 37 ], "lines": [ { "bbox": [ 106, 25, 305, 38 ], "spans": [ { "bbox": [ 106, 25, 305, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 298, 750, 313, 764 ], "spans": [ { "bbox": [ 298, 750, 313, 764 ], "score": 1.0, "content": "", "type": "text", "height": 14, "width": 15 } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 105, 82, 504, 105 ], "lines": [], "index": 0.5, "bbox_fs": [ 105, 82, 505, 106 ], "lines_deleted": true }, { "type": "text", "bbox": [ 106, 108, 506, 192 ], "lines": [ { "bbox": [ 104, 106, 507, 123 ], "spans": [ { "bbox": [ 104, 106, 218, 123 ], "score": 1.0, "content": "• When the context length", "type": "text" }, { "bbox": [ 219, 110, 227, 119 ], "score": 0.77, "content": "L", "type": "inline_equation" }, { "bbox": [ 228, 106, 332, 123 ], "score": 1.0, "content": "is less than (or equal to)", "type": "text" }, { "bbox": [ 332, 108, 347, 119 ], "score": 0.82, "content": "2 ^ { 1 7 }", "type": "inline_equation" }, { "bbox": [ 347, 106, 507, 123 ], "score": 1.0, "content": ", we divide up each segment into non-", "type": "text" } ], "index": 2 }, { "bbox": [ 110, 115, 504, 137 ], "spans": [ { "bbox": [ 110, 115, 261, 137 ], "score": 1.0, "content": "overlapping sub-segments of length", "type": "text" }, { "bbox": [ 261, 123, 269, 132 ], "score": 0.78, "content": "L", "type": "inline_equation" }, { "bbox": [ 269, 115, 338, 137 ], "score": 1.0, "content": ", so that there are", "type": "text" }, { "bbox": [ 338, 119, 369, 135 ], "score": 0.93, "content": "\\begin{array} { r } { S \\times \\frac { 2 ^ { 1 7 } } { L } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 369, 115, 442, 137 ], "score": 1.0, "content": "total samples and", "type": "text" }, { "bbox": [ 442, 121, 504, 133 ], "score": 0.91, "content": "S \\times 2 ^ { 1 7 } \\approx 4 . 5 B", "type": "inline_equation" } ], "index": 3 }, { "bbox": [ 114, 133, 186, 145 ], "spans": [ { "bbox": [ 114, 133, 186, 145 ], "score": 1.0, "content": "tokens per epoch.", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 146, 506, 163 ], "spans": [ { "bbox": [ 106, 146, 216, 163 ], "score": 1.0, "content": "• When the context length", "type": "text" }, { "bbox": [ 217, 150, 225, 159 ], "score": 0.78, "content": "L", "type": "inline_equation" }, { "bbox": [ 225, 146, 286, 163 ], "score": 1.0, "content": "is greater than", "type": "text" }, { "bbox": [ 286, 149, 300, 159 ], "score": 0.82, "content": "2 ^ { 1 7 }", "type": "inline_equation" }, { "bbox": [ 301, 146, 506, 163 ], "score": 1.0, "content": ", we turn each segment into two samples, one that", "type": "text" } ], "index": 5 }, { "bbox": [ 114, 159, 506, 172 ], "spans": [ { "bbox": [ 114, 159, 506, 172 ], "score": 1.0, "content": "begins with the prescribed segment and one that ends with the prescribed segment. 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For example, we use the AdamW with", "type": "text" }, { "bbox": [ 320, 208, 402, 220 ], "score": 0.92, "content": "( \\beta _ { 1 } , \\beta _ { 2 } ) = ( 0 . 9 , 0 . 9 5 ) ^ { \\mathrm { ~ ~ } }", "type": "inline_equation" }, { "bbox": [ 402, 207, 505, 222 ], "score": 1.0, "content": ", no dropout, weight decay", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 217, 449, 232 ], "spans": [ { "bbox": [ 105, 217, 373, 232 ], "score": 1.0, "content": "0.1. We use a cosine learning rate scheduler with linear warmup for", "type": "text" }, { "bbox": [ 373, 219, 392, 228 ], "score": 0.83, "content": "10 \\%", "type": "inline_equation" }, { "bbox": [ 393, 217, 449, 232 ], "score": 1.0, "content": "of total steps.", "type": "text" } ], "index": 11 } ], "index": 10, "bbox_fs": [ 105, 196, 505, 232 ] }, { "type": "title", "bbox": [ 107, 239, 279, 250 ], "lines": [ { "bbox": [ 105, 237, 280, 252 ], "spans": [ { "bbox": [ 105, 237, 280, 252 ], "score": 1.0, "content": "F.3.2 SCALING: MODEL SIZE DETAILS", "type": "text" } ], "index": 12 } ], "index": 12 }, { "type": "text", "bbox": [ 107, 259, 219, 271 ], "lines": [ { "bbox": [ 106, 258, 221, 272 ], "spans": [ { "bbox": [ 106, 258, 221, 272 ], "score": 1.0, "content": "The models we consider are:", "type": "text" } ], "index": 13 } ], "index": 13, "bbox_fs": [ 106, 258, 221, 272 ] }, { "type": "list", "bbox": [ 106, 280, 506, 361 ], "lines": [ { "bbox": [ 105, 278, 505, 292 ], "spans": [ { "bbox": [ 105, 278, 505, 292 ], "score": 1.0, "content": "• Transformer++: a Transformer with improved architecture, notably the usage of RoPE positional", "type": "text" } ], "index": 14, "is_list_start_line": true }, { "bbox": [ 114, 290, 505, 302 ], "spans": [ { "bbox": [ 114, 290, 505, 302 ], "score": 1.0, "content": "encodings (Su et al., 2021). 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Blocks456781012
Model Dim.6496128192256384512
Params (Appr0X.） 250K 700K 1.4M 3.5M 7.0M 19.3M 40.7M
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This", "type": "text" } ], "index": 37 }, { "bbox": [ 104, 607, 504, 621 ], "spans": [ { "bbox": [ 104, 607, 489, 621 ], "score": 1.0, "content": "is a large batch size relative to the model size by usual LM standards, but note that a batch size of", "type": "text" }, { "bbox": [ 490, 608, 504, 619 ], "score": 0.84, "content": "2 ^ { 2 3 }", "type": "inline_equation" } ], "index": 38 }, { "bbox": [ 105, 619, 506, 631 ], "spans": [ { "bbox": [ 105, 619, 403, 631 ], "score": 1.0, "content": "is the minimum possible on a machine with 8 GPUs and sequence length of", "type": "text" }, { "bbox": [ 404, 619, 419, 630 ], "score": 0.89, "content": "2 ^ { 2 } 0", "type": "inline_equation" }, { "bbox": [ 419, 619, 506, 631 ], "score": 1.0, "content": ", and that HyenaDNA", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 629, 236, 644 ], "spans": [ { "bbox": [ 105, 629, 236, 644 ], "score": 1.0, "content": "used much larger batches of 228.", "type": "text" } ], "index": 40 } ], "index": 38, "bbox_fs": [ 104, 585, 507, 644 ] }, { "type": "text", "bbox": [ 108, 645, 504, 679 ], "lines": [ { "bbox": [ 105, 645, 505, 659 ], "spans": [ { "bbox": [ 105, 645, 217, 659 ], "score": 1.0, "content": "The learning rate used was", "type": "text" }, { "bbox": [ 217, 646, 243, 657 ], "score": 0.75, "content": "8 e - 3", "type": "inline_equation" }, { "bbox": [ 243, 645, 309, 659 ], "score": 1.0, "content": "for Mamba and", "type": "text" }, { "bbox": [ 309, 646, 335, 657 ], "score": 0.79, "content": "1 e - 3", "type": "inline_equation" }, { "bbox": [ 335, 645, 505, 659 ], "score": 1.0, "content": "for HyenaDNA; we initially attempted to", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 657, 505, 669 ], "spans": [ { "bbox": [ 105, 657, 254, 669 ], "score": 1.0, "content": "use the same optimal learning rate of", "type": "text" }, { "bbox": [ 254, 657, 279, 667 ], "score": 0.75, "content": "2 e - 3", "type": "inline_equation" }, { "bbox": [ 279, 657, 505, 669 ], "score": 1.0, "content": "from the previous section for HyenaDNA, but found that", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 668, 281, 680 ], "spans": [ { "bbox": [ 105, 668, 281, 680 ], "score": 1.0, "content": "it was unstable at the longest context length.", "type": "text" } ], "index": 43 } ], "index": 42, "bbox_fs": [ 105, 645, 505, 680 ] }, { "type": "text", "bbox": [ 107, 688, 505, 732 ], "lines": [ { "bbox": [ 105, 687, 505, 702 ], "spans": [ { "bbox": [ 105, 687, 505, 702 ], "score": 1.0, "content": "Sequence Length Warmup. 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In particular, each stage up to length", "type": "text" }, { "bbox": [ 489, 720, 504, 730 ], "score": 0.79, "content": "2 ^ { 1 7 }", "type": "inline_equation" } ], "index": 47 }, { "bbox": [ 105, 202, 505, 219 ], "spans": [ { "bbox": [ 105, 202, 270, 219 ], "score": 1.0, "content": "processes the same number of tokens, but", "type": "text", "cross_page": true }, { "bbox": [ 270, 205, 284, 215 ], "score": 0.87, "content": "4 \\times", "type": "inline_equation", "cross_page": true }, { "bbox": [ 285, 202, 438, 219 ], "score": 1.0, "content": "as many tokens are processed at length", "type": "text", "cross_page": true }, { "bbox": [ 438, 204, 452, 215 ], "score": 0.75, "content": "2 ^ { 1 8 }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 452, 202, 455, 219 ], "score": 1.0, "content": ",", "type": "text", "cross_page": true }, { "bbox": [ 456, 205, 470, 216 ], "score": 0.84, "content": "8 \\times", "type": "inline_equation", "cross_page": true }, { "bbox": [ 470, 202, 505, 219 ], "score": 1.0, "content": "as many", "type": "text", "cross_page": true } ], "index": 5 }, { "bbox": [ 105, 213, 289, 228 ], "spans": [ { "bbox": [ 105, 213, 142, 228 ], "score": 1.0, "content": "at length", "type": "text", "cross_page": true }, { "bbox": [ 142, 215, 157, 226 ], "score": 0.74, "content": "2 ^ { 1 9 }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 157, 213, 177, 228 ], "score": 1.0, "content": ", and", "type": "text", "cross_page": true }, { "bbox": [ 177, 216, 196, 226 ], "score": 0.88, "content": "1 6 \\times", "type": "inline_equation", "cross_page": true }, { "bbox": [ 196, 213, 267, 228 ], "score": 1.0, "content": "as many at length", "type": "text", "cross_page": true }, { "bbox": [ 267, 216, 281, 226 ], "score": 0.85, "content": "2 ^ { 2 0 }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 282, 213, 289, 228 ], "score": 1.0, "content": ".)", "type": "text", "cross_page": true } ], "index": 6 } ], "index": 45.5, "bbox_fs": [ 104, 687, 505, 734 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 164, 110, 444, 188 ], "blocks": [ { "type": "table_caption", "bbox": [ 106, 81, 504, 102 ], "group_id": 0, "lines": [ { "bbox": [ 104, 79, 506, 92 ], "spans": [ { "bbox": [ 104, 79, 443, 92 ], "score": 1.0, "content": "Table 12: (Great Apes DNA Classification.) Accuracy after fine-tuning on sequences of length", "type": "text" }, { "bbox": [ 443, 80, 485, 91 ], "score": 0.86, "content": "2 ^ { 1 0 } = 1 0 2 4", "type": "inline_equation" }, { "bbox": [ 485, 79, 506, 92 ], "score": 1.0, "content": "up to", "type": "text" } ], "index": 0 }, { "bbox": [ 107, 88, 441, 103 ], "spans": [ { "bbox": [ 107, 90, 161, 101 ], "score": 0.89, "content": "2 ^ { 2 0 } = 1 0 4 \\dot { 8 } 5 7 6", "type": "inline_equation" }, { "bbox": [ 162, 88, 419, 103 ], "score": 1.0, "content": "with pre-trained models of the same context length. Random guessing is", "type": "text" }, { "bbox": [ 419, 91, 437, 100 ], "score": 0.86, "content": "20 \\%", "type": "inline_equation" }, { "bbox": [ 437, 88, 441, 103 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "table_body", "bbox": [ 164, 110, 444, 188 ], "group_id": 0, "lines": [ { "bbox": [ 164, 110, 444, 188 ], "spans": [ { "bbox": [ 164, 110, 444, 188 ], "score": 0.981, "html": "
ModelParams Accuracy (%) at Sequence Length
210212214216218220
HyenaDNA 1.4M28.04 28.43 41.17 42.22 31.10 54.87
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We use a model with 3 blocks per stage", "type": "text" }, { "bbox": [ 312, 678, 356, 689 ], "score": 0.88, "content": "3 \\times 5 = 1 5", "type": "inline_equation" }, { "bbox": [ 356, 677, 505, 691 ], "score": 1.0, "content": "total Mamba blocks), pooling factor", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 688, 361, 702 ], "spans": [ { "bbox": [ 106, 689, 133, 700 ], "score": 0.92, "content": "p { = } 1 6", "type": "inline_equation" }, { "bbox": [ 133, 688, 218, 702 ], "score": 1.0, "content": ", and outer dimension", "type": "text" }, { "bbox": [ 218, 689, 248, 699 ], "score": 0.89, "content": "D = 6 4", "type": "inline_equation" }, { "bbox": [ 248, 688, 289, 702 ], "score": 1.0, "content": ", for about", "type": "text" }, { "bbox": [ 289, 689, 312, 699 ], "score": 0.5, "content": "3 . 5 \\mathrm { M }", "type": "inline_equation" }, { "bbox": [ 312, 688, 361, 702 ], "score": 1.0, "content": "parameters.", "type": "text" } ], "index": 43 } ], "index": 42.5 }, { "type": "text", "bbox": [ 106, 709, 504, 731 ], "lines": [ { "bbox": [ 105, 709, 505, 722 ], "spans": [ { "bbox": [ 105, 709, 505, 722 ], "score": 1.0, "content": "Dataset. The data is mu-law encoded at 8 bits, so the model is modeling discrete tokens with a vocab", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 720, 155, 732 ], "spans": [ { "bbox": [ 105, 720, 155, 732 ], "score": 1.0, "content": "size of 256.", "type": "text" } ], "index": 45 } ], "index": 44.5 } ], "page_idx": 24, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 304, 37 ], "lines": [ { "bbox": [ 106, 25, 305, 39 ], "spans": [ { "bbox": [ 106, 25, 305, 39 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 298, 750, 313, 764 ], "spans": [ { "bbox": [ 298, 750, 313, 764 ], "score": 1.0, "content": "", "type": "text", "height": 14, "width": 15 } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 164, 110, 444, 188 ], "blocks": [ { "type": "table_caption", "bbox": [ 106, 81, 504, 102 ], "group_id": 0, "lines": [ { "bbox": [ 104, 79, 506, 92 ], "spans": [ { "bbox": [ 104, 79, 443, 92 ], "score": 1.0, "content": "Table 12: (Great Apes DNA Classification.) Accuracy after fine-tuning on sequences of length", "type": "text" }, { "bbox": [ 443, 80, 485, 91 ], "score": 0.86, "content": "2 ^ { 1 0 } = 1 0 2 4", "type": "inline_equation" }, { "bbox": [ 485, 79, 506, 92 ], "score": 1.0, "content": "up to", "type": "text" } ], "index": 0 }, { "bbox": [ 107, 88, 441, 103 ], "spans": [ { "bbox": [ 107, 90, 161, 101 ], "score": 0.89, "content": "2 ^ { 2 0 } = 1 0 4 \\dot { 8 } 5 7 6", "type": "inline_equation" }, { "bbox": [ 162, 88, 419, 103 ], "score": 1.0, "content": "with pre-trained models of the same context length. Random guessing is", "type": "text" }, { "bbox": [ 419, 91, 437, 100 ], "score": 0.86, "content": "20 \\%", "type": "inline_equation" }, { "bbox": [ 437, 88, 441, 103 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "table_body", "bbox": [ 164, 110, 444, 188 ], "group_id": 0, "lines": [ { "bbox": [ 164, 110, 444, 188 ], "spans": [ { "bbox": [ 164, 110, 444, 188 ], "score": 0.981, "html": "
ModelParams Accuracy (%) at Sequence Length
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Note that we control for the total number of elements in", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 352, 506, 367 ], "spans": [ { "bbox": [ 105, 352, 506, 367 ], "score": 1.0, "content": "the loss function per gradient step. The pretraining objective includes all positions across the sequence", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 363, 506, 377 ], "spans": [ { "bbox": [ 105, 363, 220, 377 ], "score": 1.0, "content": "length, so that batch size", "type": "text" }, { "bbox": [ 221, 365, 230, 374 ], "score": 0.74, "content": "\\times", "type": "inline_equation" }, { "bbox": [ 231, 363, 506, 377 ], "score": 1.0, "content": "sequence length is held constant; in other words, the batch size", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 375, 505, 387 ], "spans": [ { "bbox": [ 106, 375, 505, 387 ], "score": 1.0, "content": "decreases as the sequence length increases. However, for a classification task, since only the last", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 385, 506, 399 ], "spans": [ { "bbox": [ 105, 385, 506, 399 ], "score": 1.0, "content": "position enters the loss, the batch size itself is held constant. Note that this also means that fine-tuning", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 397, 396, 409 ], "spans": [ { "bbox": [ 105, 397, 396, 409 ], "score": 1.0, "content": "models with longer sequence lengths is more computationally expensive.", "type": "text" } ], "index": 20 } ], "index": 17, "bbox_fs": [ 105, 331, 506, 409 ] }, { "type": "text", "bbox": [ 107, 412, 505, 445 ], "lines": [ { "bbox": [ 105, 411, 505, 425 ], "spans": [ { "bbox": [ 105, 411, 505, 425 ], "score": 1.0, "content": "Training consists of 10 epochs, each of which has 1024 gradient steps. Each gradient step uses batch", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 423, 505, 436 ], "spans": [ { "bbox": [ 105, 423, 505, 436 ], "score": 1.0, "content": "size 64, which are all independently randomly drawn by uniformly picking a species, uniformly", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 433, 434, 446 ], "spans": [ { "bbox": [ 105, 433, 434, 446 ], "score": 1.0, "content": "picking a chromosome, and then uniformly picking a contiguous segment of DNA.", "type": "text" } ], "index": 23 } ], "index": 22, "bbox_fs": [ 105, 411, 505, 446 ] }, { "type": "text", "bbox": [ 107, 449, 505, 504 ], "lines": [ { "bbox": [ 105, 448, 505, 461 ], "spans": [ { "bbox": [ 105, 448, 453, 461 ], "score": 1.0, "content": "Following (Nguyen et al., 2023), models with a maximum context length greater than", "type": "text" }, { "bbox": [ 454, 448, 505, 459 ], "score": 0.89, "content": "2 ^ { 1 4 } = 1 6 3 8 4", "type": "inline_equation" } ], "index": 24 }, { "bbox": [ 105, 459, 506, 472 ], "spans": [ { "bbox": [ 105, 459, 317, 472 ], "score": 1.0, "content": "use sequence length warmup with 1 epoch at length", "type": "text" }, { "bbox": [ 317, 460, 368, 471 ], "score": 0.86, "content": "2 ^ { 1 4 } = 1 6 3 8 4", "type": "inline_equation" }, { "bbox": [ 369, 459, 443, 472 ], "score": 1.0, "content": ", 1 epoch at length", "type": "text" }, { "bbox": [ 443, 460, 495, 470 ], "score": 0.83, "content": "2 ^ { 1 5 } = 3 2 7 6 8", "type": "inline_equation" }, { "bbox": [ 495, 459, 506, 472 ], "score": 1.0, "content": ", 1", "type": "text" } ], "index": 25 }, { "bbox": [ 104, 468, 506, 484 ], "spans": [ { "bbox": [ 104, 468, 169, 484 ], "score": 1.0, "content": "epoch at length", "type": "text" }, { "bbox": [ 169, 470, 218, 481 ], "score": 0.9, "content": "2 ^ { \\mathrm { 1 6 } } = 6 5 5 3 6", "type": "inline_equation" }, { "bbox": [ 218, 468, 506, 484 ], "score": 1.0, "content": ", and so on up to the maximum sequence length. For example, the model", "type": "text" } ], "index": 26 }, { "bbox": [ 104, 478, 506, 496 ], "spans": [ { "bbox": [ 104, 478, 126, 496 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 127, 481, 187, 492 ], "score": 0.9, "content": "2 ^ { 2 0 } = \\bar { 1 0 } 4 8 5 7 6", "type": "inline_equation" }, { "bbox": [ 188, 478, 506, 496 ], "score": 1.0, "content": "context undergoes 6 epochs of sequence length warmup before 4 more epochs", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 492, 237, 505 ], "spans": [ { "bbox": [ 105, 492, 237, 505 ], "score": 1.0, "content": "at its maximum sequence length.", "type": "text" } ], "index": 28 } ], "index": 26, "bbox_fs": [ 104, 448, 506, 505 ] }, { "type": "text", "bbox": [ 106, 507, 506, 616 ], "lines": [ { "bbox": [ 105, 507, 507, 520 ], "spans": [ { "bbox": [ 105, 507, 284, 520 ], "score": 1.0, "content": "The learning rate for all Hyena models is", "type": "text" }, { "bbox": [ 285, 509, 315, 519 ], "score": 0.84, "content": "4 e \\mathrm { ~ - ~ } 5", "type": "inline_equation" }, { "bbox": [ 315, 507, 507, 520 ], "score": 1.0, "content": ", while the learning rate for all Mamba mod-", "type": "text" } ], "index": 29 }, { "bbox": [ 104, 516, 506, 533 ], "spans": [ { "bbox": [ 104, 516, 132, 533 ], "score": 1.0, "content": "els is", "type": "text" }, { "bbox": [ 132, 519, 163, 529 ], "score": 0.85, "content": "1 e \\mathrm { ~ - ~ } 4", "type": "inline_equation" }, { "bbox": [ 164, 516, 506, 533 ], "score": 1.0, "content": ". These were found by performing learning rate sweeps for each model among", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 523, 508, 545 ], "spans": [ { "bbox": [ 106, 529, 254, 541 ], "score": 0.9, "content": "\\{ 1 e - 5 , 2 e - 5 , 4 e - 5 , 1 e - 4 , 2 e - 4 \\}", "type": "inline_equation" }, { "bbox": [ 254, 523, 389, 545 ], "score": 1.0, "content": "for the smaller sequence lengths", "type": "text" }, { "bbox": [ 389, 529, 460, 541 ], "score": 0.92, "content": "( \\dot { 2 } ^ { 1 0 } , 2 ^ { 1 2 } , 2 ^ { 1 4 } , 2 ^ { 1 6 } )", "type": "inline_equation" }, { "bbox": [ 461, 523, 508, 545 ], "score": 1.0, "content": ", and these", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 538, 506, 554 ], "spans": [ { "bbox": [ 105, 538, 506, 554 ], "score": 1.0, "content": "values were consistently found to be the best for each model. An abridged learning rate sweep was", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 550, 506, 564 ], "spans": [ { "bbox": [ 105, 550, 165, 564 ], "score": 1.0, "content": "done at length", "type": "text" }, { "bbox": [ 165, 550, 179, 561 ], "score": 0.78, "content": "2 ^ { 1 8 }", "type": "inline_equation" }, { "bbox": [ 180, 550, 413, 564 ], "score": 1.0, "content": ", which agreed with these values, and a single run at length", "type": "text" }, { "bbox": [ 414, 550, 428, 561 ], "score": 0.87, "content": "2 ^ { 2 0 }", "type": "inline_equation" }, { "bbox": [ 428, 550, 506, 564 ], "score": 1.0, "content": "was performed (as", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 561, 506, 574 ], "spans": [ { "bbox": [ 105, 561, 506, 574 ], "score": 1.0, "content": "described above, the computational cost of these experiments is proportional to the sequence length).", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 572, 506, 585 ], "spans": [ { "bbox": [ 105, 572, 506, 585 ], "score": 1.0, "content": "The learning rate followed a cosine decay schedule with warmup with 5 epochs of linear warmup to", "type": "text" } ], "index": 35 }, { "bbox": [ 104, 582, 506, 597 ], "spans": [ { "bbox": [ 104, 582, 364, 597 ], "score": 1.0, "content": "the maximum learning rate, and 5 epochs of cosine decay down to", "type": "text" }, { "bbox": [ 365, 584, 389, 594 ], "score": 0.62, "content": "1 e { - } 6", "type": "inline_equation" }, { "bbox": [ 389, 582, 506, 597 ], "score": 1.0, "content": ". The unusually long learning", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 594, 505, 607 ], "spans": [ { "bbox": [ 105, 594, 505, 607 ], "score": 1.0, "content": "rate warmup schedule was chosen because the sequence length warmup was also long (e.g. comprising", "type": "text" } ], "index": 37 }, { "bbox": [ 104, 603, 492, 618 ], "spans": [ { "bbox": [ 104, 603, 312, 618 ], "score": 1.0, "content": "6 out of 10 epochs for the model with context length", "type": "text" }, { "bbox": [ 313, 604, 327, 615 ], "score": 0.85, "content": "2 ^ { 2 0 }", "type": "inline_equation" }, { "bbox": [ 327, 603, 492, 618 ], "score": 1.0, "content": "); we did not experiment with this choice.", "type": "text" } ], "index": 38 } ], "index": 33.5, "bbox_fs": [ 104, 507, 508, 618 ] }, { "type": "text", "bbox": [ 107, 620, 331, 631 ], "lines": [ { "bbox": [ 106, 620, 332, 633 ], "spans": [ { "bbox": [ 106, 620, 332, 633 ], "score": 1.0, "content": "Results for the Species classification task are in Table 12.", "type": "text" } ], "index": 39 } ], "index": 39, "bbox_fs": [ 106, 620, 332, 633 ] }, { "type": "title", "bbox": [ 108, 641, 200, 653 ], "lines": [ { "bbox": [ 105, 640, 201, 654 ], "spans": [ { "bbox": [ 105, 640, 201, 654 ], "score": 1.0, "content": "F.4 AUDIO DETAILS", "type": "text" } ], "index": 40 } ], "index": 40 }, { "type": "title", "bbox": [ 106, 657, 295, 669 ], "lines": [ { "bbox": [ 106, 657, 297, 669 ], "spans": [ { "bbox": [ 106, 657, 297, 669 ], "score": 1.0, "content": "F.4.1 YOUTUBEMIX AUDIO PRETRAINING", "type": "text" } ], "index": 41 } ], "index": 41 }, { "type": "text", "bbox": [ 105, 678, 505, 700 ], "lines": [ { "bbox": [ 105, 677, 505, 691 ], "spans": [ { "bbox": [ 105, 677, 312, 691 ], "score": 1.0, "content": "Model. We use a model with 3 blocks per stage", "type": "text" }, { "bbox": [ 312, 678, 356, 689 ], "score": 0.88, "content": "3 \\times 5 = 1 5", "type": "inline_equation" }, { "bbox": [ 356, 677, 505, 691 ], "score": 1.0, "content": "total Mamba blocks), pooling factor", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 688, 361, 702 ], "spans": [ { "bbox": [ 106, 689, 133, 700 ], "score": 0.92, "content": "p { = } 1 6", "type": "inline_equation" }, { "bbox": [ 133, 688, 218, 702 ], "score": 1.0, "content": ", and outer dimension", "type": "text" }, { "bbox": [ 218, 689, 248, 699 ], "score": 0.89, "content": "D = 6 4", "type": "inline_equation" }, { "bbox": [ 248, 688, 289, 702 ], "score": 1.0, "content": ", for about", "type": "text" }, { "bbox": [ 289, 689, 312, 699 ], "score": 0.5, "content": "3 . 5 \\mathrm { M }", "type": "inline_equation" }, { "bbox": [ 312, 688, 361, 702 ], "score": 1.0, "content": "parameters.", "type": "text" } ], "index": 43 } ], "index": 42.5, "bbox_fs": [ 105, 677, 505, 702 ] }, { "type": "text", "bbox": [ 106, 709, 504, 731 ], "lines": [ { "bbox": [ 105, 709, 505, 722 ], "spans": [ { "bbox": [ 105, 709, 505, 722 ], "score": 1.0, "content": "Dataset. The data is mu-law encoded at 8 bits, so the model is modeling discrete tokens with a vocab", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 720, 155, 732 ], "spans": [ { "bbox": [ 105, 720, 155, 732 ], "score": 1.0, "content": "size of 256.", "type": "text" } ], "index": 45 } ], "index": 44.5, "bbox_fs": [ 105, 709, 505, 732 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 201, 99, 407, 209 ], "blocks": [ { "type": "table_caption", "bbox": [ 177, 80, 433, 91 ], "group_id": 0, "lines": [ { "bbox": [ 177, 79, 434, 92 ], "spans": [ { "bbox": [ 177, 79, 434, 92 ], "score": 1.0, "content": "Table 13: YouTubeMix length scaling sequence lengths and batch sizes.", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table_body", "bbox": [ 201, 99, 407, 209 ], "group_id": 0, "lines": [ { "bbox": [ 201, 99, 407, 209 ], "spans": [ { "bbox": [ 201, 99, 407, 209 ], "score": 0.98, "html": "
Sequence length Batch size Tokens per batch
468×2048=958464 1958464
234×2048=479232 2958464
117×2048=239616 4958464
59×2048=120832 8966656
30×2048=61440 16983040
15×2048=30720 32983040
8×2048=16384 641048576
4×2048=8192 1281048576
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Since the architecture involves two stages of pooling by", "type": "text" } ], "index": 10 }, { "bbox": [ 104, 246, 507, 263 ], "spans": [ { "bbox": [ 104, 246, 507, 263 ], "score": 1.0, "content": "a factor of 16, and we want the resulting sequence length to be a a multiple of 8 for hardware efficiency,", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 259, 505, 272 ], "spans": [ { "bbox": [ 106, 259, 236, 272 ], "score": 1.0, "content": "the longest possible sequence is", "type": "text" }, { "bbox": [ 236, 260, 322, 270 ], "score": 0.91, "content": "4 6 8 \\times 2 0 4 8 = 9 5 8 4 6 4", "type": "inline_equation" }, { "bbox": [ 322, 259, 505, 272 ], "score": 1.0, "content": ". The rest of our sequence lengths are defined", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 270, 409, 282 ], "spans": [ { "bbox": [ 105, 270, 409, 282 ], "score": 1.0, "content": "by successively halving this and rounding up to the nearest multiple of 2048.", "type": "text" } ], "index": 13 } ], "index": 11 }, { "type": "text", "bbox": [ 107, 285, 505, 330 ], "lines": [ { "bbox": [ 105, 285, 505, 298 ], "spans": [ { "bbox": [ 105, 285, 505, 298 ], "score": 1.0, "content": "Table 13 lists the specifications used in Figure 7. Beyond the varying batch sizes, the number of valid", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 296, 506, 311 ], "spans": [ { "bbox": [ 105, 296, 506, 311 ], "score": 1.0, "content": "segments in the training set varied between different sequence lengths (e.g. the number of training", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 308, 505, 320 ], "spans": [ { "bbox": [ 105, 308, 505, 320 ], "score": 1.0, "content": "steps per epoch was not constant for different points in the graph), which may have contributed to", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 318, 215, 331 ], "spans": [ { "bbox": [ 105, 318, 215, 331 ], "score": 1.0, "content": "kinks in the scaling curves.", "type": "text" } ], "index": 17 } ], "index": 15.5 }, { "type": "text", "bbox": [ 106, 339, 505, 362 ], "lines": [ { "bbox": [ 106, 339, 504, 352 ], "spans": [ { "bbox": [ 106, 339, 249, 352 ], "score": 1.0, "content": "Training. Models were trained for", "type": "text" }, { "bbox": [ 249, 340, 275, 350 ], "score": 0.87, "content": "2 0 0 K", "type": "inline_equation" }, { "bbox": [ 275, 339, 483, 352 ], "score": 1.0, "content": "training steps with a maximum learning rate of 0.002,", "type": "text" }, { "bbox": [ 483, 339, 504, 350 ], "score": 0.54, "content": "2 0 K", "type": "inline_equation" } ], "index": 18 }, { "bbox": [ 107, 350, 506, 363 ], "spans": [ { "bbox": [ 107, 351, 132, 362 ], "score": 0.79, "content": "( 1 0 \\% )", "type": "inline_equation" }, { "bbox": [ 132, 350, 506, 363 ], "score": 1.0, "content": "warmup steps, and weight decay 0.1 (similar to our general pretraining recipe across domains).", "type": "text" } ], "index": 19 } ], "index": 18.5 }, { "type": "title", "bbox": [ 108, 371, 259, 383 ], "lines": [ { "bbox": [ 105, 370, 260, 384 ], "spans": [ { "bbox": [ 105, 370, 260, 384 ], "score": 1.0, "content": "F.4.2 SC09 SPEECH GENERATION", "type": "text" } ], "index": 20 } ], "index": 20 }, { "type": "text", "bbox": [ 106, 392, 486, 404 ], "lines": [ { "bbox": [ 105, 390, 488, 407 ], "spans": [ { "bbox": [ 105, 390, 488, 407 ], "score": 1.0, "content": "Autoregressive training largely followed the autoregressive language modeling protocol, such as", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "text", "bbox": [ 133, 413, 317, 476 ], "lines": [ { "bbox": [ 133, 412, 213, 427 ], "spans": [ { "bbox": [ 133, 412, 213, 427 ], "score": 1.0, "content": "• Weight decay 0.1", "type": "text" } ], "index": 22 }, { "bbox": [ 133, 430, 318, 443 ], "spans": [ { "bbox": [ 133, 430, 245, 443 ], "score": 1.0, "content": "• Learning rate warmup for", "type": "text" }, { "bbox": [ 246, 430, 265, 441 ], "score": 0.86, "content": "10 \\%", "type": "inline_equation" }, { "bbox": [ 265, 430, 318, 443 ], "score": 1.0, "content": "of total steps", "type": "text" } ], "index": 23 }, { "bbox": [ 133, 447, 295, 460 ], "spans": [ { "bbox": [ 133, 447, 237, 460 ], "score": 1.0, "content": "• AdamW optimizer with", "type": "text" }, { "bbox": [ 238, 447, 295, 460 ], "score": 0.8, "content": "\\beta = ( 0 . 9 , 0 . 9 5 )", "type": "inline_equation" } ], "index": 24 }, { "bbox": [ 133, 464, 234, 477 ], "spans": [ { "bbox": [ 133, 464, 234, 477 ], "score": 1.0, "content": "• Gradient clip value 0.1", "type": "text" } ], "index": 25 } ], "index": 23.5 }, { "type": "text", "bbox": [ 107, 485, 418, 497 ], "lines": [ { "bbox": [ 106, 484, 419, 498 ], "spans": [ { "bbox": [ 106, 484, 419, 498 ], "score": 1.0, "content": "We used a learning rate of 0.002 and 200000 training steps at a batch size of 16.", "type": "text" } ], "index": 26 } ], "index": 26 }, { "type": "text", "bbox": [ 107, 501, 506, 545 ], "lines": [ { "bbox": [ 105, 500, 506, 514 ], "spans": [ { "bbox": [ 105, 500, 456, 514 ], "score": 1.0, "content": "The large Mamba model in Table 3 has 15 layers per stage with an outer dimension of", "type": "text" }, { "bbox": [ 456, 501, 487, 511 ], "score": 0.9, "content": "D = 9 6", "type": "inline_equation" }, { "bbox": [ 487, 500, 506, 514 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 511, 505, 525 ], "spans": [ { "bbox": [ 105, 511, 505, 525 ], "score": 1.0, "content": "pooling factor 4. We note that this dataset is small (training went through 100 epochs) and for this large", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 522, 505, 535 ], "spans": [ { "bbox": [ 106, 522, 505, 535 ], "score": 1.0, "content": "model, there was significant overfitting of the BPB or NLL. However, automated metrics of generated", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 532, 312, 547 ], "spans": [ { "bbox": [ 105, 532, 312, 547 ], "score": 1.0, "content": "samples continually improving throughout training.", "type": "text" } ], "index": 30 } ], "index": 28.5 }, { "type": "text", "bbox": [ 106, 549, 505, 593 ], "lines": [ { "bbox": [ 105, 549, 506, 561 ], "spans": [ { "bbox": [ 105, 549, 506, 561 ], "score": 1.0, "content": "The models in the architecture ablations in Table 4 all have 8 layers per stage with an outer dimension of", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 559, 506, 572 ], "spans": [ { "bbox": [ 106, 560, 133, 570 ], "score": 0.88, "content": "\\mathtt { D } = 6 4", "type": "inline_equation" }, { "bbox": [ 134, 559, 234, 572 ], "score": 1.0, "content": "and pooling factor 4. The", "type": "text" }, { "bbox": [ 234, 560, 271, 570 ], "score": 0.73, "content": "_ { \\mathbf { S 4 + M L P } }", "type": "inline_equation" }, { "bbox": [ 271, 559, 343, 572 ], "score": 1.0, "content": "block has roughly", "type": "text" }, { "bbox": [ 343, 559, 389, 570 ], "score": 0.92, "content": "\\dot { 2 } D ^ { 2 } \\dot { + } 4 D ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 389, 559, 506, 572 ], "score": 1.0, "content": "parameters (expansion factor", "type": "text" } ], "index": 32 }, { "bbox": [ 104, 570, 507, 583 ], "spans": [ { "bbox": [ 104, 570, 278, 583 ], "score": 1.0, "content": "2 in the MLP). The Transformer block has", "type": "text" }, { "bbox": [ 279, 570, 325, 581 ], "score": 0.92, "content": "4 D ^ { 2 } + 2 D ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 326, 570, 507, 583 ], "score": 1.0, "content": "parameters (expansion factor 1 in the MLP).", "type": "text" } ], "index": 33 }, { "bbox": [ 104, 579, 493, 594 ], "spans": [ { "bbox": [ 104, 579, 232, 594 ], "score": 1.0, "content": "The Mamba block has the usual", "type": "text" }, { "bbox": [ 232, 581, 261, 591 ], "score": 0.9, "content": "{ \\approx } 6 D ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 261, 579, 493, 594 ], "score": 1.0, "content": "parameters. All models have roughly 6M total parameters.", "type": "text" } ], "index": 34 } ], "index": 32.5 }, { "type": "title", "bbox": [ 107, 602, 218, 614 ], "lines": [ { "bbox": [ 105, 602, 219, 615 ], "spans": [ { "bbox": [ 105, 602, 219, 615 ], "score": 1.0, "content": "F.5 SPEED BENCHMARK", "type": "text" } ], "index": 35 } ], "index": 35 }, { "type": "text", "bbox": [ 107, 618, 504, 640 ], "lines": [ { "bbox": [ 106, 618, 505, 631 ], "spans": [ { "bbox": [ 106, 618, 505, 631 ], "score": 1.0, "content": "Scan operation. We compare the core operation of selective SSMs, which is the parallel scan", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 630, 458, 641 ], "spans": [ { "bbox": [ 105, 630, 458, 641 ], "score": 1.0, "content": "(Section 3.3), against convolution and attention, measured on an A100 80GB PCIe GPU.", "type": "text" } ], "index": 37 } ], "index": 36.5 }, { "type": "text", "bbox": [ 107, 644, 504, 668 ], "lines": [ { "bbox": [ 106, 644, 505, 657 ], "spans": [ { "bbox": [ 106, 644, 505, 657 ], "score": 1.0, "content": "As a baseline, we implement a standard parallel scan in PyTorch with no kernel fusion. This requires", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 655, 292, 669 ], "spans": [ { "bbox": [ 105, 657, 223, 669 ], "score": 1.0, "content": "materializing the parameters", "type": "text" }, { "bbox": [ 223, 655, 254, 668 ], "score": 0.34, "content": "{ \\overline { { A } } } , { \\overline { { B } } } , C", "type": "inline_equation" }, { "bbox": [ 255, 657, 292, 669 ], "score": 1.0, "content": "in HBM.", "type": "text" } ], "index": 39 } ], "index": 38.5 }, { "type": "text", "bbox": [ 108, 672, 504, 695 ], "lines": [ { "bbox": [ 106, 672, 505, 685 ], "spans": [ { "bbox": [ 106, 672, 505, 685 ], "score": 1.0, "content": "Our scan implementation fuses the discretization step and the parallel scan, avoiding the cost of", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 683, 292, 695 ], "spans": [ { "bbox": [ 105, 683, 292, 695 ], "score": 1.0, "content": "materializing all the large parameters in HBM.", "type": "text" } ], "index": 41 } ], "index": 40.5 }, { "type": "text", "bbox": [ 108, 699, 504, 732 ], "lines": [ { "bbox": [ 105, 699, 506, 712 ], "spans": [ { "bbox": [ 105, 699, 506, 712 ], "score": 1.0, "content": "For convolution, we use the standard implementation in PyTorch, which separately performs FFTs on", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 709, 506, 722 ], "spans": [ { "bbox": [ 105, 709, 506, 722 ], "score": 1.0, "content": "the inputs and the filters, multiply them in frequency domain, then performs an inverse FFT to obtain", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 720, 408, 734 ], "spans": [ { "bbox": [ 105, 720, 265, 734 ], "score": 1.0, "content": "the result. The theoretical complexity is", "type": "text" }, { "bbox": [ 265, 721, 315, 732 ], "score": 0.92, "content": "O ( L \\log ( L ) )", "type": "inline_equation" }, { "bbox": [ 316, 720, 396, 734 ], "score": 1.0, "content": "for sequence length", "type": "text" }, { "bbox": [ 396, 721, 403, 730 ], "score": 0.74, "content": "L", "type": "inline_equation" }, { "bbox": [ 404, 720, 408, 734 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 44 } ], "index": 43 } ], "page_idx": 25, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 298, 749, 313, 764 ], "spans": [ { "bbox": [ 298, 749, 313, 764 ], "score": 1.0, "content": "", "type": "text", "height": 15, "width": 15 } ] } ] }, { "type": "discarded", "bbox": [ 107, 27, 304, 37 ], "lines": [ { "bbox": [ 107, 25, 304, 38 ], "spans": [ { "bbox": [ 107, 25, 304, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 201, 99, 407, 209 ], "blocks": [ { "type": "table_caption", "bbox": [ 177, 80, 433, 91 ], "group_id": 0, "lines": [ { "bbox": [ 177, 79, 434, 92 ], "spans": [ { "bbox": [ 177, 79, 434, 92 ], "score": 1.0, "content": "Table 13: YouTubeMix length scaling sequence lengths and batch sizes.", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table_body", "bbox": [ 201, 99, 407, 209 ], "group_id": 0, "lines": [ { "bbox": [ 201, 99, 407, 209 ], "spans": [ { "bbox": [ 201, 99, 407, 209 ], "score": 0.98, "html": "
Sequence length Batch size Tokens per batch
468×2048=958464 1958464
234×2048=479232 2958464
117×2048=239616 4958464
59×2048=120832 8966656
30×2048=61440 16983040
15×2048=30720 32983040
8×2048=16384 641048576
4×2048=8192 1281048576
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Since the architecture involves two stages of pooling by", "type": "text" } ], "index": 10 }, { "bbox": [ 104, 246, 507, 263 ], "spans": [ { "bbox": [ 104, 246, 507, 263 ], "score": 1.0, "content": "a factor of 16, and we want the resulting sequence length to be a a multiple of 8 for hardware efficiency,", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 259, 505, 272 ], "spans": [ { "bbox": [ 106, 259, 236, 272 ], "score": 1.0, "content": "the longest possible sequence is", "type": "text" }, { "bbox": [ 236, 260, 322, 270 ], "score": 0.91, "content": "4 6 8 \\times 2 0 4 8 = 9 5 8 4 6 4", "type": "inline_equation" }, { "bbox": [ 322, 259, 505, 272 ], "score": 1.0, "content": ". The rest of our sequence lengths are defined", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 270, 409, 282 ], "spans": [ { "bbox": [ 105, 270, 409, 282 ], "score": 1.0, "content": "by successively halving this and rounding up to the nearest multiple of 2048.", "type": "text" } ], "index": 13 } ], "index": 11, "bbox_fs": [ 104, 227, 507, 282 ] }, { "type": "text", "bbox": [ 107, 285, 505, 330 ], "lines": [ { "bbox": [ 105, 285, 505, 298 ], "spans": [ { "bbox": [ 105, 285, 505, 298 ], "score": 1.0, "content": "Table 13 lists the specifications used in Figure 7. Beyond the varying batch sizes, the number of valid", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 296, 506, 311 ], "spans": [ { "bbox": [ 105, 296, 506, 311 ], "score": 1.0, "content": "segments in the training set varied between different sequence lengths (e.g. the number of training", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 308, 505, 320 ], "spans": [ { "bbox": [ 105, 308, 505, 320 ], "score": 1.0, "content": "steps per epoch was not constant for different points in the graph), which may have contributed to", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 318, 215, 331 ], "spans": [ { "bbox": [ 105, 318, 215, 331 ], "score": 1.0, "content": "kinks in the scaling curves.", "type": "text" } ], "index": 17 } ], "index": 15.5, "bbox_fs": [ 105, 285, 506, 331 ] }, { "type": "text", "bbox": [ 106, 339, 505, 362 ], "lines": [ { "bbox": [ 106, 339, 504, 352 ], "spans": [ { "bbox": [ 106, 339, 249, 352 ], "score": 1.0, "content": "Training. Models were trained for", "type": "text" }, { "bbox": [ 249, 340, 275, 350 ], "score": 0.87, "content": "2 0 0 K", "type": "inline_equation" }, { "bbox": [ 275, 339, 483, 352 ], "score": 1.0, "content": "training steps with a maximum learning rate of 0.002,", "type": "text" }, { "bbox": [ 483, 339, 504, 350 ], "score": 0.54, "content": "2 0 K", "type": "inline_equation" } ], "index": 18 }, { "bbox": [ 107, 350, 506, 363 ], "spans": [ { "bbox": [ 107, 351, 132, 362 ], "score": 0.79, "content": "( 1 0 \\% )", "type": "inline_equation" }, { "bbox": [ 132, 350, 506, 363 ], "score": 1.0, "content": "warmup steps, and weight decay 0.1 (similar to our general pretraining recipe across domains).", "type": "text" } ], "index": 19 } ], "index": 18.5, "bbox_fs": [ 106, 339, 506, 363 ] }, { "type": "title", "bbox": [ 108, 371, 259, 383 ], "lines": [ { "bbox": [ 105, 370, 260, 384 ], "spans": [ { "bbox": [ 105, 370, 260, 384 ], "score": 1.0, "content": "F.4.2 SC09 SPEECH GENERATION", "type": "text" } ], "index": 20 } ], "index": 20 }, { "type": "text", "bbox": [ 106, 392, 486, 404 ], "lines": [ { "bbox": [ 105, 390, 488, 407 ], "spans": [ { "bbox": [ 105, 390, 488, 407 ], "score": 1.0, "content": "Autoregressive training largely followed the autoregressive language modeling protocol, such as", "type": "text" } ], "index": 21 } ], "index": 21, "bbox_fs": [ 105, 390, 488, 407 ] }, { "type": "list", "bbox": [ 133, 413, 317, 476 ], "lines": [ { "bbox": [ 133, 412, 213, 427 ], "spans": [ { "bbox": [ 133, 412, 213, 427 ], "score": 1.0, "content": "• Weight decay 0.1", "type": "text" } ], "index": 22, "is_list_start_line": true }, { "bbox": [ 133, 430, 318, 443 ], "spans": [ { "bbox": [ 133, 430, 245, 443 ], "score": 1.0, "content": "• Learning rate warmup for", "type": "text" }, { "bbox": [ 246, 430, 265, 441 ], "score": 0.86, "content": "10 \\%", "type": "inline_equation" }, { "bbox": [ 265, 430, 318, 443 ], "score": 1.0, "content": "of total steps", "type": "text" } ], "index": 23, "is_list_start_line": true }, { "bbox": [ 133, 447, 295, 460 ], "spans": [ { "bbox": [ 133, 447, 237, 460 ], "score": 1.0, "content": "• AdamW optimizer with", "type": "text" }, { "bbox": [ 238, 447, 295, 460 ], "score": 0.8, "content": "\\beta = ( 0 . 9 , 0 . 9 5 )", "type": "inline_equation" } ], "index": 24, "is_list_start_line": true }, { "bbox": [ 133, 464, 234, 477 ], "spans": [ { "bbox": [ 133, 464, 234, 477 ], "score": 1.0, "content": "• Gradient clip value 0.1", "type": "text" } ], "index": 25, "is_list_start_line": true } ], "index": 23.5, "bbox_fs": [ 133, 412, 318, 477 ] }, { "type": "text", "bbox": [ 107, 485, 418, 497 ], "lines": [ { "bbox": [ 106, 484, 419, 498 ], "spans": [ { "bbox": [ 106, 484, 419, 498 ], "score": 1.0, "content": "We used a learning rate of 0.002 and 200000 training steps at a batch size of 16.", "type": "text" } ], "index": 26 } ], "index": 26, "bbox_fs": [ 106, 484, 419, 498 ] }, { "type": "text", "bbox": [ 107, 501, 506, 545 ], "lines": [ { "bbox": [ 105, 500, 506, 514 ], "spans": [ { "bbox": [ 105, 500, 456, 514 ], "score": 1.0, "content": "The large Mamba model in Table 3 has 15 layers per stage with an outer dimension of", "type": "text" }, { "bbox": [ 456, 501, 487, 511 ], "score": 0.9, "content": "D = 9 6", "type": "inline_equation" }, { "bbox": [ 487, 500, 506, 514 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 511, 505, 525 ], "spans": [ { "bbox": [ 105, 511, 505, 525 ], "score": 1.0, "content": "pooling factor 4. We note that this dataset is small (training went through 100 epochs) and for this large", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 522, 505, 535 ], "spans": [ { "bbox": [ 106, 522, 505, 535 ], "score": 1.0, "content": "model, there was significant overfitting of the BPB or NLL. However, automated metrics of generated", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 532, 312, 547 ], "spans": [ { "bbox": [ 105, 532, 312, 547 ], "score": 1.0, "content": "samples continually improving throughout training.", "type": "text" } ], "index": 30 } ], "index": 28.5, "bbox_fs": [ 105, 500, 506, 547 ] }, { "type": "text", "bbox": [ 106, 549, 505, 593 ], "lines": [ { "bbox": [ 105, 549, 506, 561 ], "spans": [ { "bbox": [ 105, 549, 506, 561 ], "score": 1.0, "content": "The models in the architecture ablations in Table 4 all have 8 layers per stage with an outer dimension of", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 559, 506, 572 ], "spans": [ { "bbox": [ 106, 560, 133, 570 ], "score": 0.88, "content": "\\mathtt { D } = 6 4", "type": "inline_equation" }, { "bbox": [ 134, 559, 234, 572 ], "score": 1.0, "content": "and pooling factor 4. The", "type": "text" }, { "bbox": [ 234, 560, 271, 570 ], "score": 0.73, "content": "_ { \\mathbf { S 4 + M L P } }", "type": "inline_equation" }, { "bbox": [ 271, 559, 343, 572 ], "score": 1.0, "content": "block has roughly", "type": "text" }, { "bbox": [ 343, 559, 389, 570 ], "score": 0.92, "content": "\\dot { 2 } D ^ { 2 } \\dot { + } 4 D ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 389, 559, 506, 572 ], "score": 1.0, "content": "parameters (expansion factor", "type": "text" } ], "index": 32 }, { "bbox": [ 104, 570, 507, 583 ], "spans": [ { "bbox": [ 104, 570, 278, 583 ], "score": 1.0, "content": "2 in the MLP). The Transformer block has", "type": "text" }, { "bbox": [ 279, 570, 325, 581 ], "score": 0.92, "content": "4 D ^ { 2 } + 2 D ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 326, 570, 507, 583 ], "score": 1.0, "content": "parameters (expansion factor 1 in the MLP).", "type": "text" } ], "index": 33 }, { "bbox": [ 104, 579, 493, 594 ], "spans": [ { "bbox": [ 104, 579, 232, 594 ], "score": 1.0, "content": "The Mamba block has the usual", "type": "text" }, { "bbox": [ 232, 581, 261, 591 ], "score": 0.9, "content": "{ \\approx } 6 D ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 261, 579, 493, 594 ], "score": 1.0, "content": "parameters. All models have roughly 6M total parameters.", "type": "text" } ], "index": 34 } ], "index": 32.5, "bbox_fs": [ 104, 549, 507, 594 ] }, { "type": "title", "bbox": [ 107, 602, 218, 614 ], "lines": [ { "bbox": [ 105, 602, 219, 615 ], "spans": [ { "bbox": [ 105, 602, 219, 615 ], "score": 1.0, "content": "F.5 SPEED BENCHMARK", "type": "text" } ], "index": 35 } ], "index": 35 }, { "type": "text", "bbox": [ 107, 618, 504, 640 ], "lines": [ { "bbox": [ 106, 618, 505, 631 ], "spans": [ { "bbox": [ 106, 618, 505, 631 ], "score": 1.0, "content": "Scan operation. We compare the core operation of selective SSMs, which is the parallel scan", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 630, 458, 641 ], "spans": [ { "bbox": [ 105, 630, 458, 641 ], "score": 1.0, "content": "(Section 3.3), against convolution and attention, measured on an A100 80GB PCIe GPU.", "type": "text" } ], "index": 37 } ], "index": 36.5, "bbox_fs": [ 105, 618, 505, 641 ] }, { "type": "text", "bbox": [ 107, 644, 504, 668 ], "lines": [ { "bbox": [ 106, 644, 505, 657 ], "spans": [ { "bbox": [ 106, 644, 505, 657 ], "score": 1.0, "content": "As a baseline, we implement a standard parallel scan in PyTorch with no kernel fusion. This requires", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 655, 292, 669 ], "spans": [ { "bbox": [ 105, 657, 223, 669 ], "score": 1.0, "content": "materializing the parameters", "type": "text" }, { "bbox": [ 223, 655, 254, 668 ], "score": 0.34, "content": "{ \\overline { { A } } } , { \\overline { { B } } } , C", "type": "inline_equation" }, { "bbox": [ 255, 657, 292, 669 ], "score": 1.0, "content": "in HBM.", "type": "text" } ], "index": 39 } ], "index": 38.5, "bbox_fs": [ 105, 644, 505, 669 ] }, { "type": "text", "bbox": [ 108, 672, 504, 695 ], "lines": [ { "bbox": [ 106, 672, 505, 685 ], "spans": [ { "bbox": [ 106, 672, 505, 685 ], "score": 1.0, "content": "Our scan implementation fuses the discretization step and the parallel scan, avoiding the cost of", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 683, 292, 695 ], "spans": [ { "bbox": [ 105, 683, 292, 695 ], "score": 1.0, "content": "materializing all the large parameters in HBM.", "type": "text" } ], "index": 41 } ], "index": 40.5, "bbox_fs": [ 105, 672, 505, 695 ] }, { "type": "text", "bbox": [ 108, 699, 504, 732 ], "lines": [ { "bbox": [ 105, 699, 506, 712 ], "spans": [ { "bbox": [ 105, 699, 506, 712 ], "score": 1.0, "content": "For convolution, we use the standard implementation in PyTorch, which separately performs FFTs on", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 709, 506, 722 ], "spans": [ { "bbox": [ 105, 709, 506, 722 ], "score": 1.0, "content": "the inputs and the filters, multiply them in frequency domain, then performs an inverse FFT to obtain", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 720, 408, 734 ], "spans": [ { "bbox": [ 105, 720, 265, 734 ], "score": 1.0, "content": "the result. The theoretical complexity is", "type": "text" }, { "bbox": [ 265, 721, 315, 732 ], "score": 0.92, "content": "O ( L \\log ( L ) )", "type": "inline_equation" }, { "bbox": [ 316, 720, 396, 734 ], "score": 1.0, "content": "for sequence length", "type": "text" }, { "bbox": [ 396, 721, 403, 730 ], "score": 0.74, "content": "L", "type": "inline_equation" }, { "bbox": [ 404, 720, 408, 734 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 44 } ], "index": 43, "bbox_fs": [ 105, 699, 506, 734 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 82, 504, 115 ], "lines": [ { "bbox": [ 106, 82, 506, 94 ], "spans": [ { "bbox": [ 106, 82, 506, 94 ], "score": 1.0, "content": "For attention, we compare against the fastest implementation that we are aware of (FlashAttention-", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 93, 506, 105 ], "spans": [ { "bbox": [ 106, 93, 458, 105 ], "score": 1.0, "content": "2 (Dao, 2023)), with causal mask. Note that FlashAttention-2 with causal mask is about", "type": "text" }, { "bbox": [ 458, 93, 480, 104 ], "score": 0.9, "content": "1 . 7 \\times", "type": "inline_equation" }, { "bbox": [ 480, 93, 506, 105 ], "score": 1.0, "content": "faster", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 104, 478, 116 ], "spans": [ { "bbox": [ 105, 104, 478, 116 ], "score": 1.0, "content": "than without causal mask, since approximately only half of the attention entries are computed.", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "text", "bbox": [ 107, 119, 505, 163 ], "lines": [ { "bbox": [ 105, 119, 505, 132 ], "spans": [ { "bbox": [ 105, 119, 505, 132 ], "score": 1.0, "content": "We use batch size of 1 and increase the sequence length from 512, 1K, 2K, ... to 512K (some of the", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 130, 506, 142 ], "spans": [ { "bbox": [ 105, 130, 445, 142 ], "score": 1.0, "content": "baselines run out of memory before reaching 512K). We use a model dimension of", "type": "text" }, { "bbox": [ 445, 131, 487, 141 ], "score": 0.91, "content": "D = 1 0 2 4", "type": "inline_equation" }, { "bbox": [ 487, 130, 506, 142 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 140, 506, 154 ], "spans": [ { "bbox": [ 105, 140, 171, 154 ], "score": 1.0, "content": "state dimension", "type": "text" }, { "bbox": [ 171, 141, 201, 151 ], "score": 0.89, "content": "N { = } 1 6", "type": "inline_equation" }, { "bbox": [ 202, 140, 506, 154 ], "score": 1.0, "content": ". We measure with BF16 inputs, which is the data type most commonly used", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 151, 200, 165 ], "spans": [ { "bbox": [ 105, 151, 200, 165 ], "score": 1.0, "content": "for large scale training.", "type": "text" } ], "index": 6 } ], "index": 4.5 }, { "type": "text", "bbox": [ 107, 171, 504, 205 ], "lines": [ { "bbox": [ 106, 172, 505, 183 ], "spans": [ { "bbox": [ 106, 172, 505, 183 ], "score": 1.0, "content": "End-to-end inference. We measure the inference throughput of a Mamba 1.4B model and an", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 182, 505, 194 ], "spans": [ { "bbox": [ 105, 182, 505, 194 ], "score": 1.0, "content": "untrained Mamba 6.9B model, against a standard Transformer (GPT3 architecture) at 1.3B and 6.7B", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 193, 505, 206 ], "spans": [ { "bbox": [ 105, 193, 505, 206 ], "score": 1.0, "content": "size. We use the standard Transformer implementation in the Huggingface transformers library.", "type": "text" } ], "index": 9 } ], "index": 8 }, { "type": "text", "bbox": [ 107, 209, 505, 252 ], "lines": [ { "bbox": [ 105, 209, 505, 222 ], "spans": [ { "bbox": [ 105, 209, 505, 222 ], "score": 1.0, "content": "We set the prompt length to be 2048 and the generation length to be 128. We vary the batch size from", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 220, 506, 232 ], "spans": [ { "bbox": [ 105, 220, 506, 232 ], "score": 1.0, "content": "1, 2, 4, 8, 16, 32, 64, to 128, and measure time time taken to generate 128 tokens. We then calculate", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 230, 506, 243 ], "spans": [ { "bbox": [ 105, 230, 245, 243 ], "score": 1.0, "content": "the throughput (tokens/s) as batch", "type": "text" }, { "bbox": [ 245, 231, 288, 241 ], "score": 0.58, "content": "{ \\mathrm { s i z e } } \\times 1 2 8 ", "type": "inline_equation" }, { "bbox": [ 288, 230, 506, 243 ], "score": 1.0, "content": "/time taken. We repeat the measurements 3 times and", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 242, 390, 253 ], "spans": [ { "bbox": [ 105, 242, 390, 253 ], "score": 1.0, "content": "take the average. Measurements are done on an A100 80GB PCIe GPU.", "type": "text" } ], "index": 13 } ], "index": 11.5 } ], "page_idx": 26, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 304, 37 ], "lines": [ { "bbox": [ 106, 26, 305, 38 ], "spans": [ { "bbox": [ 106, 26, 305, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 298, 750, 313, 765 ], "spans": [ { "bbox": [ 298, 750, 313, 765 ], "score": 1.0, "content": "", "type": "text", "height": 15, "width": 15 } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 82, 504, 115 ], "lines": [ { "bbox": [ 106, 82, 506, 94 ], "spans": [ { "bbox": [ 106, 82, 506, 94 ], "score": 1.0, "content": "For attention, we compare against the fastest implementation that we are aware of (FlashAttention-", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 93, 506, 105 ], "spans": [ { "bbox": [ 106, 93, 458, 105 ], "score": 1.0, "content": "2 (Dao, 2023)), with causal mask. Note that FlashAttention-2 with causal mask is about", "type": "text" }, { "bbox": [ 458, 93, 480, 104 ], "score": 0.9, "content": "1 . 7 \\times", "type": "inline_equation" }, { "bbox": [ 480, 93, 506, 105 ], "score": 1.0, "content": "faster", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 104, 478, 116 ], "spans": [ { "bbox": [ 105, 104, 478, 116 ], "score": 1.0, "content": "than without causal mask, since approximately only half of the attention entries are computed.", "type": "text" } ], "index": 2 } ], "index": 1, "bbox_fs": [ 105, 82, 506, 116 ] }, { "type": "text", "bbox": [ 107, 119, 505, 163 ], "lines": [ { "bbox": [ 105, 119, 505, 132 ], "spans": [ { "bbox": [ 105, 119, 505, 132 ], "score": 1.0, "content": "We use batch size of 1 and increase the sequence length from 512, 1K, 2K, ... to 512K (some of the", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 130, 506, 142 ], "spans": [ { "bbox": [ 105, 130, 445, 142 ], "score": 1.0, "content": "baselines run out of memory before reaching 512K). We use a model dimension of", "type": "text" }, { "bbox": [ 445, 131, 487, 141 ], "score": 0.91, "content": "D = 1 0 2 4", "type": "inline_equation" }, { "bbox": [ 487, 130, 506, 142 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 140, 506, 154 ], "spans": [ { "bbox": [ 105, 140, 171, 154 ], "score": 1.0, "content": "state dimension", "type": "text" }, { "bbox": [ 171, 141, 201, 151 ], "score": 0.89, "content": "N { = } 1 6", "type": "inline_equation" }, { "bbox": [ 202, 140, 506, 154 ], "score": 1.0, "content": ". We measure with BF16 inputs, which is the data type most commonly used", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 151, 200, 165 ], "spans": [ { "bbox": [ 105, 151, 200, 165 ], "score": 1.0, "content": "for large scale training.", "type": "text" } ], "index": 6 } ], "index": 4.5, "bbox_fs": [ 105, 119, 506, 165 ] }, { "type": "text", "bbox": [ 107, 171, 504, 205 ], "lines": [ { "bbox": [ 106, 172, 505, 183 ], "spans": [ { "bbox": [ 106, 172, 505, 183 ], "score": 1.0, "content": "End-to-end inference. We measure the inference throughput of a Mamba 1.4B model and an", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 182, 505, 194 ], "spans": [ { "bbox": [ 105, 182, 505, 194 ], "score": 1.0, "content": "untrained Mamba 6.9B model, against a standard Transformer (GPT3 architecture) at 1.3B and 6.7B", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 193, 505, 206 ], "spans": [ { "bbox": [ 105, 193, 505, 206 ], "score": 1.0, "content": "size. We use the standard Transformer implementation in the Huggingface transformers library.", "type": "text" } ], "index": 9 } ], "index": 8, "bbox_fs": [ 105, 172, 505, 206 ] }, { "type": "text", "bbox": [ 107, 209, 505, 252 ], "lines": [ { "bbox": [ 105, 209, 505, 222 ], "spans": [ { "bbox": [ 105, 209, 505, 222 ], "score": 1.0, "content": "We set the prompt length to be 2048 and the generation length to be 128. We vary the batch size from", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 220, 506, 232 ], "spans": [ { "bbox": [ 105, 220, 506, 232 ], "score": 1.0, "content": "1, 2, 4, 8, 16, 32, 64, to 128, and measure time time taken to generate 128 tokens. We then calculate", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 230, 506, 243 ], "spans": [ { "bbox": [ 105, 230, 245, 243 ], "score": 1.0, "content": "the throughput (tokens/s) as batch", "type": "text" }, { "bbox": [ 245, 231, 288, 241 ], "score": 0.58, "content": "{ \\mathrm { s i z e } } \\times 1 2 8 ", "type": "inline_equation" }, { "bbox": [ 288, 230, 506, 243 ], "score": 1.0, "content": "/time taken. We repeat the measurements 3 times and", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 242, 390, 253 ], "spans": [ { "bbox": [ 105, 242, 390, 253 ], "score": 1.0, "content": "take the average. Measurements are done on an A100 80GB PCIe GPU.", "type": "text" } ], "index": 13 } ], "index": 11.5, "bbox_fs": [ 105, 209, 506, 253 ] } ] } ], "_backend": "pipeline", "_version_name": "2.1.11" }