Authors:
(1) Albert Gu, Machine Learning Department, Carnegie Mellon University and with equal contribution;
(2) Tri Dao, Department of Computer Science, Princeton University and with equal contribution. Table of Links Abstract and 1 Introduction 2 State Space Models 3 Selective State Space Models and 3.1 Motivation: Selection as a Means of Compression 3.2 Improving SSMs with Selection 3.3 Efficient Implementation of Selective SSMs 3.4 A Simplified SSM Architecture 3.5 Properties of Selection Mechanisms 3.6 Additional Model Details 4 Empirical Evaluation and 4.1 Synthetic Tasks 4.2 Language Modeling 4.3 DNA Modeling 4.4 Audio Modeling and Generation 4.5 Speed and Memory Benchmarks 4.6 Model Ablations 5 Discussion 6 Conclusion and References A Discussion: Selection Mechanism B Related Work C Mechanics of Selective SSMs D Hardware-aware Algorithm For Selective SSMs E Experimental Details and Additional Results 2 State Space Models Thus far, all structured SSMs have been LTI (e.g. computed as convolutions) because of fundamental efficiency constraints, discussed in Section 3.3. However, a core insight of this work is that LTI models have fundamental limitations in modeling certain types of data, and our technical contributions involve removing the LTI constraint while overcoming the efficiency bottlenecks. Structure and Dimensions. Finally, we note that structured SSMs are so named because computing them efficiently also requires imposing structure on the A matrix. The most popular form of structure is diagonal (Gu, Gupta, et al. 2022; Gupta, Gu, and Berant 2022; Smith, Warrington, and Linderman 2023), which we also use. General State Space Models. We note that the term state space model has a very broad meaning which simply represents the notion of any recurrent process with a latent state. It has been used to refer to many disparate concepts in different disciplines, including Markov decision processes (MDP) (reinforcement learning (Hafner et al. 2020)), dynamic causal modeling (DCM) (computational neuroscience (Friston, Harrison, and Penny 2003)), Kalman filters (controls (Kalman 1960)), hidden Markov models (HMM) and linear dynamical systems (LDS) (machine learning), and recurrent (and sometimes convolutional) models at large (deep learning). Throughout this entire paper we use the term “SSM” to refer exclusively to the class of structured SSMs or S4 models (Gu, Goel, and Ré 2022; Gu, Gupta, et al. 2022; Gupta, Gu, and Berant 2022; Hasani et al. 2023; Ma et al. 2023; Smith, Warrington, and Linderman 2023) and use these terms interchangeably. For convenience we may also include derivatives of such models, such as those focusing on either the linear-recurrence or global-convolution viewpoints (Y. Li et al. 2023; Orvieto et al. 2023; Poli et al. 2023), and clarify nuances when necessary. SSM Architectures. SSMs are standalone sequence transformations that can be incorporated into end-to-end neural network architectures. (We also sometimes call SSM architectures SSNNs, which are to SSM layers as CNNs are to linear convolution layers.) We discuss some of the most well-known SSM architectures, many of which will also serve as our primary baselines. • Linear attention (Katharopoulos et al. 2020) is an approximation of self-attention involving a recurrence which can be viewed as a degenerate linear SSM. • H3 (Dao, Fu, Saab, et al. 2023) generalized this recurrence to use S4; it can be viewed as an architecture with an SSM sandwiched by two gated connections (Figure 3). H3 also inserts a standard local convolution, which they frame as a shift-SSM, before the main SSM layer. • Hyena (Poli et al. 2023) uses the same architecture as H3 but replaces the S4 layer with an MLP-parameterized global convolution (Romero et al. 2021). • RetNet (Y. Sun et al. 2023) adds an additional gate to the architecture and uses a simpler SSM, allowing an alternative parallelizable computation path, using a variant of multi-head attention (MHA) instead of convolutions. • RWKV (B. Peng et al. 2023) is a recent RNN designed for language modeling based on another linear attention approximation (attention-free Transformer (S. Zhai et al. 2021)). Its main “WKV” mechanism involves LTI recurrences and can be viewed as the ratio of two SSMs. Other closely related SSMs and architectures are discussed further in an extended related work (Appendix B). We highlight in particular S5 (Smith, Warrington, and Linderman 2023), QRNN (Bradbury et al. 2016), and SRU (Lei et al. 2017), which we view as the most closely related methods to our core selective SSM. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Albert Gu, Machine Learning Department, Carnegie Mellon University and with equal contribution; (2) Tri Dao, Department of Computer Science, Princeton University and with equal contribution. Authors: Authors: (1) Albert Gu, Machine Learning Department, Carnegie Mellon University and with equal contribution; (2) Tri Dao, Department of Computer Science, Princeton University and with equal contribution. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 State Space Models 2 State Space Models 3 Selective State Space Models and 3.1 Motivation: Selection as a Means of Compression 3 Selective State Space Models and 3.1 Motivation: Selection as a Means of Compression 3.2 Improving SSMs with Selection 3.2 Improving SSMs with Selection 3.3 Efficient Implementation of Selective SSMs 3.3 Efficient Implementation of Selective SSMs 3.4 A Simplified SSM Architecture 3.4 A Simplified SSM Architecture 3.5 Properties of Selection Mechanisms 3.5 Properties of Selection Mechanisms 3.6 Additional Model Details 3.6 Additional Model Details 4 Empirical Evaluation and 4.1 Synthetic Tasks 4 Empirical Evaluation and 4.1 Synthetic Tasks 4.2 Language Modeling 4.2 Language Modeling 4.3 DNA Modeling 4.3 DNA Modeling 4.4 Audio Modeling and Generation 4.4 Audio Modeling and Generation 4.5 Speed and Memory Benchmarks 4.5 Speed and Memory Benchmarks 4.6 Model Ablations 4.6 Model Ablations 5 Discussion 5 Discussion 6 Conclusion and References 6 Conclusion and References A Discussion: Selection Mechanism A Discussion: Selection Mechanism B Related Work B Related Work C Mechanics of Selective SSMs C Mechanics of Selective SSMs D Hardware-aware Algorithm For Selective SSMs D Hardware-aware Algorithm For Selective SSMs E Experimental Details and Additional Results E Experimental Details and Additional Results 2 State Space Models Thus far, all structured SSMs have been LTI (e.g. computed as convolutions) because of fundamental efficiency constraints, discussed in Section 3.3. However, a core insight of this work is that LTI models have fundamental limitations in modeling certain types of data, and our technical contributions involve removing the LTI constraint while overcoming the efficiency bottlenecks. Structure and Dimensions. Finally, we note that structured SSMs are so named because computing them efficiently also requires imposing structure on the A matrix. The most popular form of structure is diagonal (Gu, Gupta, et al. 2022; Gupta, Gu, and Berant 2022; Smith, Warrington, and Linderman 2023), which we also use. General State Space Models. We note that the term state space model has a very broad meaning which simply represents the notion of any recurrent process with a latent state. It has been used to refer to many disparate concepts in different disciplines, including Markov decision processes (MDP) (reinforcement learning (Hafner et al. 2020)), dynamic causal modeling (DCM) (computational neuroscience (Friston, Harrison, and Penny 2003)), Kalman filters (controls (Kalman 1960)), hidden Markov models (HMM) and linear dynamical systems (LDS) (machine learning), and recurrent (and sometimes convolutional) models at large (deep learning). General State Space Models. Throughout this entire paper we use the term “SSM” to refer exclusively to the class of structured SSMs or S4 models (Gu, Goel, and Ré 2022; Gu, Gupta, et al. 2022; Gupta, Gu, and Berant 2022; Hasani et al. 2023; Ma et al. 2023; Smith, Warrington, and Linderman 2023) and use these terms interchangeably. For convenience we may also include derivatives of such models, such as those focusing on either the linear-recurrence or global-convolution viewpoints (Y. Li et al. 2023; Orvieto et al. 2023; Poli et al. 2023), and clarify nuances when necessary. SSM Architectures. SSMs are standalone sequence transformations that can be incorporated into end-to-end neural network architectures. (We also sometimes call SSM architectures SSNNs, which are to SSM layers as CNNs are to linear convolution layers.) We discuss some of the most well-known SSM architectures, many of which will also serve as our primary baselines. SSM Architectures. • Linear attention (Katharopoulos et al. 2020) is an approximation of self-attention involving a recurrence which can be viewed as a degenerate linear SSM. • H3 (Dao, Fu, Saab, et al. 2023) generalized this recurrence to use S4; it can be viewed as an architecture with an SSM sandwiched by two gated connections (Figure 3). H3 also inserts a standard local convolution, which they frame as a shift-SSM, before the main SSM layer. • Hyena (Poli et al. 2023) uses the same architecture as H3 but replaces the S4 layer with an MLP-parameterized global convolution (Romero et al. 2021). • RetNet (Y. Sun et al. 2023) adds an additional gate to the architecture and uses a simpler SSM, allowing an alternative parallelizable computation path, using a variant of multi-head attention (MHA) instead of convolutions. • RWKV (B. Peng et al. 2023) is a recent RNN designed for language modeling based on another linear attention approximation (attention-free Transformer (S. Zhai et al. 2021)). Its main “WKV” mechanism involves LTI recurrences and can be viewed as the ratio of two SSMs. Other closely related SSMs and architectures are discussed further in an extended related work (Appendix B). We highlight in particular S5 (Smith, Warrington, and Linderman 2023), QRNN (Bradbury et al. 2016), and SRU (Lei et al. 2017), which we view as the most closely related methods to our core selective SSM. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

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