Parallel Training of Recurrent Networks

Looped Transformers increase computation per token by repeatedly applying a shared backbone, offering a way to improve language modeling and downstream task performance at a fixed backbone parameter count. However, each loop depends on the preceding loop, so increasing recurrent depth also increases decoding latency. Existing approaches include Parallel Loop Transformers, which reduce latency by overlapping loop computation across token positions, but yield rapidly diminishing marginal accuracy gains at larger depths. To address this challenge, we introduce LoopFold, which folds computation at multiple recurrent depths into parallel batches for the same token prefix. A lightweight predictor estimates inputs to later loops so they can start before earlier loops finish. A corrector then refines the outputs of these later loops using newly available results from earlier loops. All components are trained jointly using only the final next-token prediction loss. In experiments with 1.4B models, deeper LoopFold configurations achieve progressively higher average zero-shot accuracy across eight tasks, rising from 53.41% to 56.38%. Vanilla looped models and PLT show limited gains over the same depth range. Compared with a nine-loop vanilla model, LoopFold uses five sequential stages and additional parallel computation to reduce decoding latency by 46%.

A central aim of neuromorphic computing is to provide a viable alternative to highly energy-intensive Transformer-based AI. However, efficient alternatives struggle to capture the set of qualities that have secured the Transformer's status as the de facto standard in sequence modelling. Any realistic contender must be data-adaptive, able to capture long-range dependencies, and GPU-parallelisable, but also non-linearly recurrent to enable complex reasoning. Based on evidence suggesting the auditory cortex operates on fixed timescales, this work proposes the ADaptive with Prescriptive Timescales Network (ADPTNet) as a potential solution to achieving all four properties simultaneously. ADPTNet is built around local topological conjugates, obtained by a novel combination of linear attention and Riemannian optimisation, applied to static global dynamics. This enables non-linear yet predictable long-term behaviour. Dynamical systems theory proofs provide theoretical guarantees for the parametric control of ADPTNet's timescales (its Lyapunov spectrum). ADPTNet improves performance on Selective Copying over Hawk, the existing method balancing long-range memory and adaptability, while also improving state tracking over linear SSMs like Mamba. On sequential CIFAR-10, ADPTNet matches linear SSM accuracy and outperforms existing selective models (incl. the Transformer), using fewer parameters. We also introduce a neuromorphic SpikingADPTNet, which achieves a new state-of-the-art accuracy on the Spiking Speech Commands dataset ($83.56\%\pm0.15$). Finally, ADPTNet's constant timescales enable two efficient, Jacobian-free extensions to the DEER parallel simulation algorithm (Conv and Forward DEER) that retain the same average convergence. Conv DEER adds no computational overhead beyond the network's forward pass and enables non-linear RNN parallelisation via iterated convolutions for the first time.

State-space models (SSMs) provide a powerful theoretical framework to enable parallel training of recurrent networks. We expand on previous work adapting SSMs to spiking models to provide a novel interpretation of resonate-and-fire (R&F) neural networks which is compatible both with real and spiking inputs, parallel and recurrent execution, has clear connections to hyperdimensional (HD) computing, and maintains biologically-realistic features. We demonstrate an implementation of this approach which integrates an STFT, recurrent memory, and attentional features within a single spike-compatible network.

Training recurrent neural networks (RNNs) requires assigning credit across long sequences of computations. Standard backpropagation through time (BPTT) addresses this problem poorly: it is sequential in time, limiting parallelism, and suffers from vanishing or exploding gradients, making long-range associations difficult to learn. We propose Supervised Memory Training (SMT), a method for training nonlinear RNNs that sidesteps recurrent credit propagation entirely by reducing RNN training to supervised learning on one-step memory transition labels $(m_t, x_{t+1}) \rightarrow m_{t+1}$. SMT acquires these memory labels by training a Transformer-based encoder on a predictive state objective--retaining only information from the past necessary to predict the future. By decoupling what to remember from how to update memory, SMT enables time-parallel RNN training with a stable $O(1)$ length gradient path between any two tokens--without ever unrolling the RNN. We find that SMT outperforms BPTT when pretraining various RNN architectures on tasks like language modeling and pixel sequence modeling. SMT enables nonlinear RNNs to better capture long-range dependencies and train in parallel, potentially unlocking the scaling of models that build temporal abstractions of past experience.

Transformers have become the dominant architecture for sequence modeling by using self-attention to enable expressive and highly parallel processing. However, the resulting quadratic time and memory costs limit efficiency in long-context settings. Recurrent models such as LSTMs provide explicit nonlinear state updates and strong state-tracking capabilities, yet their strictly sequential computation limits parallelism. We introduce the Parallel Recursive LSTM (PR-LSTM), a hierarchical recurrent architecture that replaces left-to-right recurrence with recursive nonlinear state composition over a balanced computation tree. Tokens are first mapped independently to latent states, which are then recursively merged by a learned gated composition block. This structure uses the reduction pattern underlying parallel scans as a fixed execution schedule, rather than assuming an associative recurrence. As a result, PR-LSTM retains nonlinear gated state representations while reducing recurrent parallel depth from linear to logarithmic. Empirically, PR-LSTM achieves strong sequence-length generalization on formal-language benchmarks, solving more tasks than standard RNN, LSTM, and Transformer baselines, while avoiding the quadratic scaling of attention. These results suggest that recurrent computation can be reorganized hierarchically to expose parallelism without restricting the transition dynamics to linear or associative forms.

Reconstructing nonlinear dynamical systems (DS) from data (DSR) is a fundamental challenge in science and engineering, but it inherently relies on sequential models. Recent breakthroughs for sequential models have produced algorithms that parallelize computation along sequence length $T$, achieving logarithmic time complexity, $\mathcal{O}(\log T)$. Since sequence lengths have been practically limited due to the linear runtime complexity $\mathcal{O}(T)$ of classical backpropagation through time, this opens new avenues for DSR. This paper studies two prominent classes of parallel-in-time algorithms for this task, both of which leverage parallel associative scans as their core computational primitive. The first class comprises models with linear yet non-autonomous dynamics and a nonlinear readout, such as modern State Space Models (SSMs), while the second consists of general nonlinear models which can be parallelized using the DEER framework. We find that the linear training-time recurrence of the first class of models imposes limitations that often hinder learning of accurate nonlinear dynamics. To address this, we augment DEER with Generalized Teacher Forcing (GTF), a novel variant within the more general nonlinear framework that ensures stable and effective learning of nonlinear dynamics across arbitrary sequence lengths. Using GTF-DEER, we investigate the benefits of training on extremely long sequences ($T>10^4$) for DSR. Our results show that access to such long trajectories significantly improves DSR if the data features long time scales. This work establishes GTF-DEER as a robust tool for data-driven discovery and underscores the largely untapped potential of long-sequence learning in modeling complex DS.

Sequence learning is dominated by Transformers and parallelizable recurrent neural networks (RNNs) such as state-space models, yet learning long-term dependencies remains challenging, and state-of-the-art designs trade power consumption for performance. The Bistable Memory Recurrent Unit (BMRU) was introduced to enable hardware-software co-design of ultra-low power RNNs: quantized states with hysteresis provide persistent memory while mapping directly to analog primitives. However, BMRU performance lags behind parallelizable RNNs on complex sequential tasks. In this paper, we identify gradient blocking during state updates as a key limitation and propose a cumulative update formulation that restores gradient flow while preserving persistent memory, creating skip-connections through time. This leads to the Cumulative Memory Recurrent Unit (CMRU) and its relaxed variant, the $α$CMRU. Experiments show that the cumulative formulation dramatically improves convergence stability and reduces initialization sensitivity. The CMRU and $α$CMRU match or outperform Linear Recurrent Units (LRUs) and minimal Gated Recurrent Units (minGRUs) across diverse benchmarks at small model sizes, with particular advantages on tasks requiring discrete long-range retention, while the CMRU retains quantized states, persistent memory, and noise-resilient dynamics essential for analog implementation.

The proliferation of large-scale and structurally complex data has spurred the integration of machine learning methods into statistical modeling. Recurrent neural networks (RNNs), a foundational class of models for time-dependent data, can be viewed as nonlinear extensions of classical autoregressive moving average models. Despite their flexibility and empirical success in machine learning, RNNs often suffer from limited interpretability and slow training, which hinders their use in statistics. This paper proposes the Parallelized RNN (ParaRNN), a novel model composed of multiple small recurrent units. ParaRNN admits an additive representation that decouples recurrent dynamics into interpretable components, whose behavior can be characterized through recurrence features. This interpretability enables its applications in nonparametric regression for time-dependent data, while the design also allows efficient parallelization. The approximation capacity and non-asymptotic prediction error bounds in a nonparametric regression setting are established for ParaRNN. Empirical results on three sequential modeling tasks further demonstrate that ParaRNN achieves performance comparable to vanilla RNNs while offering improved interpretability and efficiency.

Current transformers discard their rich latent residual stream between positions, reconstructing latent reasoning context at each new position and leaving potential reasoning capacity untapped. The State Stream Transformer (SST) V2 enables parameter-efficient reasoning in continuous latent space through an FFN-driven nonlinear recurrence at each decoder layer, where latent states are streamed horizontally across the full sequence via a learned blend. This same mechanism supports continuous latent deliberation per position at inference time, dedicating additional FLOPs to exploring abstract reasoning before committing to a token. A two-pass parallel training procedure approximates the sequential recurrence, making co-training computationally practical. Hidden state analysis shows that the state stream facilitates reasoning through sharp, content-dependent reorganisations in continuous latent space; the LM head exposes the resulting latent belief states through the output distribution, while the state stream carries them forward to influence future positions. A learned probe shows that at the first generated token, the latent state already predicts whether the eventual answer will survive or break under additional latent computation for every subsequent position. Co-trained into an existing 27B backbone using only a small dataset of GSM8K examples and evaluated using an oracle to route each question to a depth of one to four recurrent forward passes per token, the SST achieves an architectural capacity bound of 61.11% on out-of-distribution GPQA-Diamond, a +15.15 point gain over a fine-tuning-matched baseline, and cuts that same baseline's remaining GSM8K errors by 46%. Together, these results provide a method for efficiently training a nonlinear recurrence and show that the state stream provides additional reasoning capacity beyond that of an otherwise training-matched transformer.

The community is increasingly exploring linear RNNs (LRNNs) as language models, motivated by their expressive power and parallelizability. While prior work establishes the expressivity benefits of LRNNs over transformers, it is unclear what makes LRNNs -- but not traditional, nonlinear RNNs -- as easy to parallelize in practice as transformers. We answer this question by providing a tight connection between types of RNNs and standard complexity classes. We show that LRNNs can be viewed as log-depth (bounded fan-in) arithmetic circuits, which represents only a slight depth overhead relative to log-depth boolean circuits that transformers admit. Furthermore, we show that nonlinear RNNs can solve $\mathsf{L}$-complete problems (and even $\mathsf{P}$-complete ones, under polynomial precision), revealing a fundamental barrier to parallelizing them as efficiently as transformers. Our theory also identifies fine-grained expressivity differences between recent popular LRNN variants: permutation-diagonal LRNNs are $\mathsf{NC}^1$-complete whereas diagonal-plus-low-rank LRNNs are more expressive ($\mathsf{PNC}^1$-complete). We provide further insight by associating each type of RNN with a corresponding automata-theoretic model that it can simulate. Together, our results reveal fundamental tradeoffs between nonlinear RNNs and different variants of LRNNs, providing a foundation for designing LLM architectures that achieve an optimal balance between expressivity and parallelism.

With the emergence of massively parallel processing units, parallelization has become a desirable property for new sequence models. The ability to parallelize the processing of sequences with respect to the sequence length during training is one of the main factors behind the uprising of the Transformer architecture. However, Transformers lack efficiency at sequence generation, as they need to reprocess all past timesteps at every generation step. Recently, state-space models (SSMs) emerged as a more efficient alternative. These new kinds of recurrent neural networks (RNNs) keep the efficient update of the RNNs while gaining parallelization by getting rid of nonlinear dynamics (or recurrence). SSMs can reach state-of-the-art performance through the efficient training of potentially very large networks, but still suffer from limited representation capabilities. In particular, SSMs cannot exhibit persistent memory, or the capacity of retaining information for an infinite duration, because of their monostability. In this paper, we introduce a new family of RNNs, the memory recurrent units (MRUs), that combine the persistent memory capabilities of nonlinear RNNs with the parallelizable computations of SSMs. These units leverage multistability as a source of persistent memory, while getting rid of transient dynamics for efficient computations. We then derive a specific implementation as proof-of-concept: the bistable memory recurrent unit (BMRU). This new RNN is compatible with the parallel scan algorithm. We show that BMRU achieves good results in tasks with long-term dependencies, and can be combined with SSMs to create parallelizable hybrid networks that have both transient dynamics and persistent memory.