On the Expressivity of Selective State-Space Layers: A Multivariate Polynomial Approach

Recent advances in efficient sequence modeling have introduced selective state-space layers, a key component of the Mamba architecture, which have demonstrated remarkable success in a wide range of NLP and vision tasks. While Mamba's empirical performance has matched or surpassed SoTA transformers on such diverse benchmarks, the theoretical foundations underlying its powerful representational capabilities remain less explored. In this work, we investigate the expressivity of selective state-space layers using multivariate polynomials, and prove that they surpass linear transformers in expressiveness. Consequently, our findings reveal that Mamba offers superior representational power over linear attention-based models for long sequences, while not sacrificing their generalization. Our theoretical insights are validated by a comprehensive set of empirical experiments on various datasets.

Power-Softmax: Towards Secure LLM Inference over Encrypted Data

Modern cryptographic methods for implementing privacy-preserving LLMs such as Homomorphic Encryption (HE) require the LLMs to have a polynomial form. Forming such a representation is challenging because Transformers include non-polynomial components, such as Softmax and layer normalization. Previous approaches have either directly approximated pre-trained models with large-degree polynomials, which are less efficient over HE, or replaced non-polynomial components with easier-to-approximate primitives before training, e.g., Softmax with pointwise attention. The latter approach might introduce scalability challenges. We present a new HE-friendly variant of self-attention that offers a stable form for training and is easy to approximate with polynomials for secure inference. Our work introduces the first polynomial LLMs with 32 layers and over a billion parameters, exceeding the size of previous models by more than tenfold. The resulting models demonstrate reasoning and in-context learning (ICL) capabilities comparable to standard transformers of the same size, representing a breakthrough in the field. Finally, we provide a detailed latency breakdown for each computation over encrypted data, paving the way for further optimization, and explore the differences in inductive bias between transformers relying on our HE-friendly variant and standard transformers. Our code is attached as a supplement.

DeciMamba: Exploring the Length Extrapolation Potential of Mamba

Long-range sequence processing poses a significant challenge for Transformers due to their quadratic complexity in input length. A promising alternative is Mamba, which demonstrates high performance and achieves Transformer-level capabilities while requiring substantially fewer computational resources. In this paper we explore the length-generalization capabilities of Mamba, which we find to be relatively limited. Through a series of visualizations and analyses we identify that the limitations arise from a restricted effective receptive field, dictated by the sequence length used during training. To address this constraint, we introduce DeciMamba, a context-extension method specifically designed for Mamba. This mechanism, built on top of a hidden filtering mechanism embedded within the S6 layer, enables the trained model to extrapolate well even without additional training. Empirical experiments over real-world long-range NLP tasks show that DeciMamba can extrapolate to context lengths that are 25x times longer than the ones seen during training, and does so without utilizing additional computational resources. We will release our code and models.

A Unified Implicit Attention Formulation for Gated-Linear Recurrent Sequence Models

Recent advances in efficient sequence modeling have led to attention-free layers, such as Mamba, RWKV, and various gated RNNs, all featuring sub-quadratic complexity in sequence length and excellent scaling properties, enabling the construction of a new type of foundation models. In this paper, we present a unified view of these models, formulating such layers as implicit causal self-attention layers. The formulation includes most of their sub-components and is not limited to a specific part of the architecture. The framework compares the underlying mechanisms on similar grounds for different layers and provides a direct means for applying explainability methods. Our experiments show that our attention matrices and attribution method outperform an alternative and a more limited formulation that was recently proposed for Mamba. For the other architectures for which our method is the first to provide such a view, our method is effective and competitive in the relevant metrics compared to the results obtained by state-of-the-art transformer explainability methods. Our code is publicly available.

The Hidden Attention of Mamba Models

The Mamba layer offers an efficient selective state space model (SSM) that is highly effective in modeling multiple domains including NLP, long-range sequences processing, and computer vision. Selective SSMs are viewed as dual models, in which one trains in parallel on the entire sequence via IO-aware parallel scan, and deploys in an autoregressive manner. We add a third view and show that such models can be viewed as attention-driven models. This new perspective enables us to compare the underlying mechanisms to that of the self-attention layers in transformers and allows us to peer inside the inner workings of the Mamba model with explainability methods. Our code is publicly available.

On the Long Range Abilities of Transformers

Despite their dominance in modern DL and, especially, NLP domains, transformer architectures exhibit sub-optimal performance on long-range tasks compared to recent layers that are specifically designed for this purpose. In this work, drawing inspiration from key attributes of long-range layers, such as state-space layers, linear RNN layers, and global convolution layers, we demonstrate that minimal modifications to the transformer architecture can significantly enhance performance on the Long Range Arena (LRA) benchmark, thus narrowing the gap with these specialized layers. We identify that two key principles for long-range tasks are (i) incorporating an inductive bias towards smoothness, and (ii) locality. As we show, integrating these ideas into the attention mechanism improves results with a negligible amount of additional computation and without any additional trainable parameters. Our theory and experiments also shed light on the reasons for the inferior performance of transformers on long-range tasks and identify critical properties that are essential for successfully capturing long-range dependencies.

Converting Transformers to Polynomial Form for Secure Inference Over Homomorphic Encryption

Designing privacy-preserving deep learning models is a major challenge within the deep learning community. Homomorphic Encryption (HE) has emerged as one of the most promising approaches in this realm, enabling the decoupling of knowledge between the model owner and the data owner. Despite extensive research and application of this technology, primarily in convolutional neural networks, incorporating HE into transformer models has been challenging because of the difficulties in converting these models into a polynomial form. We break new ground by introducing the first polynomial transformer, providing the first demonstration of secure inference over HE with transformers. This includes a transformer architecture tailored for HE, alongside a novel method for converting operators to their polynomial equivalent. This innovation enables us to perform secure inference on LMs with WikiText-103. It also allows us to perform image classification with CIFAR-100 and Tiny-ImageNet. Our models yield results comparable to traditional methods, bridging the performance gap with transformers of similar scale and underscoring the viability of HE for state-of-the-art applications. Finally, we assess the stability of our models and conduct a series of ablations to quantify the contribution of each model component.

Multi-Dimensional Hyena for Spatial Inductive Bias

In recent years, Vision Transformers have attracted increasing interest from computer vision researchers. However, the advantage of these transformers over CNNs is only fully manifested when trained over a large dataset, mainly due to the reduced inductive bias towards spatial locality within the transformer's self-attention mechanism. In this work, we present a data-efficient vision transformer that does not rely on self-attention. Instead, it employs a novel generalization to multiple axes of the very recent Hyena layer. We propose several alternative approaches for obtaining this generalization and delve into their unique distinctions and considerations from both empirical and theoretical perspectives. Our empirical findings indicate that the proposed Hyena N-D layer boosts the performance of various Vision Transformer architectures, such as ViT, Swin, and DeiT across multiple datasets. Furthermore, in the small dataset regime, our Hyena-based ViT is favorable to ViT variants from the recent literature that are specifically designed for solving the same challenge, i.e., working with small datasets or incorporating image-specific inductive bias into the self-attention mechanism. Finally, we show that a hybrid approach that is based on Hyena N-D for the first layers in ViT, followed by layers that incorporate conventional attention, consistently boosts the performance of various vision transformer architectures.

Efficient Skip Connections Realization for Secure Inference on Encrypted Data

Homomorphic Encryption (HE) is a cryptographic tool that allows performing computation under encryption, which is used by many privacy-preserving machine learning solutions, for example, to perform secure classification. Modern deep learning applications yield good performance for example in image processing tasks benchmarks by including many skip connections. The latter appears to be very costly when attempting to execute model inference under HE. In this paper, we show that by replacing (mid-term) skip connections with (short-term) Dirac parameterization and (long-term) shared-source skip connection we were able to reduce the skip connections burden for HE-based solutions, achieving x1.3 computing power improvement for the same accuracy.

2-D SSM: A General Spatial Layer for Visual Transformers

A central objective in computer vision is to design models with appropriate 2-D inductive bias. Desiderata for 2D inductive bias include two-dimensional position awareness, dynamic spatial locality, and translation and permutation invariance. To address these goals, we leverage an expressive variation of the multidimensional State Space Model (SSM). Our approach introduces efficient parameterization, accelerated computation, and a suitable normalization scheme. Empirically, we observe that incorporating our layer at the beginning of each transformer block of Vision Transformers (ViT) significantly enhances performance for multiple ViT backbones and across datasets. The new layer is effective even with a negligible amount of additional parameters and inference time. Ablation studies and visualizations demonstrate that the layer has a strong 2-D inductive bias. For example, vision transformers equipped with our layer exhibit effective performance even without positional encoding