The Transformer architecture revolutionised natural language processing with its self-attention mechanism, enabling parallel computation and effective context retrieval. However, Transformers face significant limitations when processing longer sequences due to their quadratic computational complexity. Linear Recurrent Neural Networks (RNNs) have emerged as a promising alternative, offering parallel training capabilities while maintaining linear inference-time complexity. The expressivity of these models depends fundamentally on their state-transition matrices. The evolution of linear RNNs has progressed from early models with token-independent state-transition matrices to more powerful token-dependent designs. The field has further advanced with non-diagonal structures that allow simultaneous mixing of information across both tokens and channels, creating more expressive architectures. These developments address the critical challenge of efficiently processing long sequences while maintaining computational feasibility.
Linear RNNs face a fundamental trade-off between training efficiency and expressivity, determined by their state-transition matrix structure. Models with diagonal state-transition matrices like Mamba and GLA train efficiently but suffer from significant expressivity limitations, being unable to perform even basic operations like addition modulo 3 on arbitrary-length sequences in finite precision. Transformers encounter similar constraints, as they effectively function as special linear RNNs with identity state-transition matrices and infinite-dimensional states. DeltaNet partially addresses these limitations through generalized Householder matrices, achieving greater expressivity with modest training cost increases, though still requiring multiple layers for certain tasks. At the opposite end of the spectrum, linear RNNs with full state-transition matrices offer maximal expressivity and can recognize any regular language with a single layer, but their training costs become prohibitively expensive. This efficiency-expressivity trade-off represents a central challenge in the design of sequence models that must balance computational feasibility with model capability.
Researchers from the University of Freiburg, ELLIS Institute Tubingen, Microsoft Research, CSML, Istituto Italiano di Tecnologia, AI Centre, University College London present DeltaProduct that addresses the efficiency-expressivity trade-off in linear RNNs through a unique approach that balances computational feasibility with model capability. While DeltaNet performs a single gradient step per token on a linear key-to-value mapping, DeltaProduct takes multiple (nh) gradient steps using additional keys and values, creating state-transition matrices that are products of multiple generalized Householder matrices. This elegant connection between optimization steps and matrix structure provides a tunable mechanism to interpolate between diagonal and dense matrices—increasing gradient steps automatically increases the number of Householder matrices in the product, enhancing expressivity while maintaining computational efficiency. The method ensures stability during training on long sequences by precisely controlling the norm of state transition matrices to remain ≤ 1. DeltaProduct generalizes DeltaNet while offering theoretical advances in expressivity, capable of solving word problems for dihedral groups with just two layers. Empirical validation demonstrates DeltaProduct’s superior performance in complex state-tracking tasks, Chomsky hierarchy benchmarks, and language modeling with enhanced length extrapolation capabilities.
DeltaProduct generalizes DeltaNet by enhancing its expressivity through state transition matrices formed as products of generalized Householder matrices. While DeltaNet performs one step of online gradient descent per token, DeltaProduct refines the hidden state multiple times per token, naturally leading to more expressive state-transition matrices where each additional step expands the range of achievable linear transformations.
Beyond increasing the number of gradient steps per token, DeltaNet’s expressivity (equivalent to DeltaProduct with nh = 1) can also be enhanced by increasing the number of layers, though its theoretical limits remain partially unexplored. Recent research extends previous findings to demonstrate that a two-layer DeltaNet with extended eigenvalue range can solve not only cyclic group problems but also the more complex dihedral group word problems for any m ∈ N. Dihedral groups represent both rotations and reflections of regular polygons, with D3 being isomorphic to the symmetric group S3. This capability can be implemented using a two-layer DeltaNet with two heads in the first layer. The first layer computes parity for rotations and reflections separately, while the second layer’s recurrent state maintains multiple possible values decoded differently based on reflection parity. This construction demonstrates that even with minimal architecture complexity, DeltaNet possesses significant theoretical expressivity beyond what was previously established, offering insights into the model’s capabilities when multiple layers are employed.
Based on extensive evaluations, DeltaProduct consistently outperforms existing models across multiple benchmark tasks. In Chomsky hierarchy experiments, DeltaProductnh with nh ≥ 2 demonstrates superior expressivity compared to DeltaNet and other baselines, with the most pronounced improvement in complex tasks like modular arithmetic with brackets. This performance gain is particularly evident when using the extended eigenvalue range [−1, 1]. Analysis of the model’s behavior reveals that DeltaProduct2[−1, 1] successfully approximates rotations by combining two reflections, with beta values clustering near 2, confirming theoretical predictions about its operational mechanism. Also, PCA analysis of key vectors shows the model primarily operates in a three-dimensional subspace, aligning with the expected structure. For language modeling tasks, both DeltaProduct and Gated DeltaProduct outperform their baseline counterparts across benchmarks when increasing nh. Notably, DeltaProduct3[−1, 1] achieves comparable performance to Gated DeltaNet[−1, 1] despite lacking a forget gate mechanism. DeltaProduct also exhibits significantly better length extrapolation with higher nh values, showing minimal performance degradation across sequence lengths up to 32k tokens.
DeltaProduct extends DeltaNet by using products of Householder transformations as state-transition matrices, effectively bridging the gap between structured and dense matrices. Each recurrence step performs multiple gradient descent steps on an associative recall loss, compared to DeltaNet’s single-step approach. The number of Householder matrices (nh) serves as a tunable parameter that elegantly balances expressivity and computational efficiency. Experimental results demonstrate DeltaProduct’s superior performance across state tracking tasks, formal language recognition, and language modeling, with particularly impressive length extrapolation capabilities. The architecture represents a significant advancement toward developing sequence models that are both more capable and scalable. Despite its advantages, DeltaProduct has limitations, including increased computational resources and memory requirements that scale linearly with nh.
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