The RNN Homunculus: Learning to Gate Long-Term Memory

We develop a statistical theory of temporal learnability in recurrent neural networks, quantifying the maximal temporal horizon $\mathcal{H}_N$ over which gradient-based learning can recover lag-dependent structure at finite sample size $N$. The theory is built on the effective learning rate envelope $f(\ell)$, a function that captures how gating mechanisms and adaptive optimizers jointly shape the coupling between state-space dynamics and parameter updates during Backpropagation Through Time. Under heavy-tailed ($α$-stable) fluctuations, where empirical averages concentrate at rate $N^{-1/κ_α}$ with $κ_α= α/(α-1)$, the interplay between envelope decay and statistical concentration yields explicit scaling laws for the growth of $\mathcal{H}_N$: logarithmic, polynomial, and exponential temporal learning regimes emerge according to the decay law of $f(\ell)$. These results identify envelope decay as the key determinant of temporal learnability. Slower attenuation of $f(\ell)$ enlarges $\mathcal{H}_N$, while heavy-tailed fluctuations compress it by weakening statistical concentration. Moreover, envelope geometry outweighs dataset size: slowing the envelope's decay enlarges $\mathcal{H}_N$ more than adding data, so more complex architectures that realize slower-decaying envelopes can be more data-efficient than simpler ones. Experiments across multiple gated architectures and optimizers corroborate these structural predictions.

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.

Long-term dependencies remain a major challenge for sequential decision-making in the field of AI: RNNs suffer from vanishing gradients and the limited expressivity of vector-based hidden states, whilst Transformer-based models are limited by the quadratic scaling of attention. Recent work has proposed tackling this problem with the Test-Time Training (TTT) framework, which stores episodic memories in the parameters of a neural network through gradient descent at both train and test-time. This approach has seen success in the domain of Natural Language Processing, however, to the best of our knowledge it has not yet been applied to the domain of Reinforcement Learning (RL), nor has there been a study analysing how this memory practically functions. In this paper, we study the potential of the TTT framework for offline RL by augmenting a Decision Transformer with TTT layers, dubbed the Decision Titan. We analyse performance and properties of the model in the X-Maze environment, an extension of T-Maze designed to test sequential memory, and investigate how the memory mechanism learns by visualising gate values over time. Our key findings are that Decision Titan can learn long-term dependencies with ranges 20x longer than the context window, generalises to lengths 1.7x the training data, but crucially temporal generalisation depends on the time embeddings used, and the ability to learn long-term dependencies depends on how the relevant information is encoded.