Truncated Backpropagation Through Time

Looped transformers reuse one block of layers many times and are often trained by backpropagating through only the last k loops. We measure how much of the weight gradient this throws away, exactly and for every k from one backward pass, on nine looped language models, and report three findings. First, on a trained model this lost gradient is set near the output: once the loop count is well above k, it barely changes with the loop count (within about 0.03 from 32 to 64 loops on the three models we tested, as a pre-registered test predicted for Parcae). Where the backward signal shrinks, $ρ^k$, with $ρ$ the signal's fitted shrink per loop, approximates it at moderate loop counts but overstates it at long ones, because the fitted $ρ$ drifts up as the fit takes in more loops. On Ouro the signal grows, and keeping two of four loops still loses 81% of the final output's gradient. Second, the lost gradient falls during fine-tuning, by 0.07-0.22 within 300 steps on three of four models, on held-out problems too. The fall is not just the read-out's share of the gradient; training only the layers outside the loop reproduced up to half of it, and it also falls under truncated training. So a trained model's value overstates what truncation removes later. Third, we could not show that it predicts the training cost of truncation better than rules that need no measurement: on two new models it tied Parcae's cost at the same k, in a test too small to separate them.

Looped transformers reuse a shared block of layers to increase computational depth without increasing the number of trainable parameters. They are trained with backpropagation through time (BPTT). To save training compute, BPTT is often truncated to the last few loops, but truncation yields biased gradients that may weaken the benefit of looping. In this work, we study the value of truncation in looped transformer training in a systematic way: we jointly vary model size, training tokens, and forward and backward loop counts across nearly 600 training runs and fit a unified scaling law that captures the effects of looping and truncation through effective parameters. Despite the training-compute savings of truncation, our fitted law favors full BPTT when forward and backward loop counts are jointly optimized. In matched-compute runs, full BPTT through fewer loops consistently outperforms truncated settings of the same compute. The law further shows that at fixed model size, the compute-optimal loop count and training-token budget both grow with compute. When model size is also free, the law identifies two fully trained loops as the best looped setting. Experimentally, this setting achieves lower validation loss than the compute-optimal non-looped model with about 27% fewer non-embedding parameters and only 8% more training compute.

Existing online video segmentation methods struggle to track objects in long, complex videos with long-term occlusions. We hypothesize that this limitation is caused by (i) the inability of their temporal propagation mechanism to adaptively select the object information that is propagated across time, and (ii) their inability to be trained on long videos due to memory requirements and vanishing gradients. To address the first limitation, we propose to use a lightweight GRU-based temporal propagation module that can learn to select which information it keeps in memory and propagates across time. Second, to allow training on long videos, we introduce Truncated Query Propagation (TQP), a training strategy in which the model processes a video in chunks of frames, where information about tracked objects is propagated between chunks but backpropagation is only conducted in individual chunks, enabling longer temporal supervision without out-of-memory issues, inference overhead, or vanishing gradients. The resulting model is called the Long-term Video Mask Transformer (LVMT). Extensive experiments on six benchmarks show that LVMT sets a new state of the art across a range of video segmentation tasks, while retaining the speed of the highly efficient model it is based on, making it 10X faster than the prior state of the art. Code: https://github.com/tue-mps/lvmt

When a policy with memory is trained through a differentiable simulator, credit can reach what it stores along two routes: representation credit teaches the network that writes the memory what to summarize, while closed-loop credit travels back through the physical states that earlier actions produced and that the memory later stored. Stopping gradients at stored history, as truncated backpropagation and Transformer-XL do, removes both routes at once. Keeping every forward computation identical and removing one route at a time, we trained Transformer policies for quadrotor tracking to determine when each route matters. Credit through memory improved learning only when velocity observations were noisy or hidden. When velocity was hidden, mildly decaying the gradient between consecutive physical states ($γ= 0.8$) eliminated the truncation cost of removing closed-loop credit while leaving representation credit essential, a separation that held in 14 of 16 new seeds. Two further systems bound this result: in a fixed-wing aircraft, representation credit could replace closed-loop credit on its own, leaving no cost to separate, and in a cart-pole, training with full state-edge credit was too unstable to compare the routes. In both the quadrotor and a fully observed vessel model, static gradient norms failed to predict which cut cost more, so memory truncation choices are best guided by short training runs.

Modern conversational agents condition on an ever-growing dialogue history at each turn, incurring redundant attention and encoding costs that grow with conversation length. Naive truncation or summarization degrades fidelity, while existing context compressors lack cross-turn memory sharing or revision, causing information loss and compounding errors in long dialogues. We revisit the context compression under conversational dynamics and empirically present its fragility. To improve both efficiency and robustness, we introduce Context-Driven Incremental Compression (C-DIC), which treats a conversation as interleaved contextual threads and stores revisable per-thread compression states in a single, compact dialogue memory. At each turn, a lightweight retrieve, revise, and write-back loop shares information across turns and updates stale memories, stabilizing long-horizon behavior. In addition, we adapt truncated backpropagation-through-time (TBPTT) to our multi-turn setting, learning cross-turn dependencies without full-history backpropagation. Extensive experiments on long-form dialogue benchmarks demonstrate superior performance and efficiency of C-DIC; notably, C-DIC shows stable inference latency and perplexity over hundreds of dialogue turns, supporting a scalable path to high-quality dialogue modeling.

Biological and neuromorphic recurrent neural networks (RNNs) are subject to spatial and temporal locality constraints on the information that can plausibly be used during learning. A common strategy to satisfy these constraints is to modify gradient descent by neglecting non-local terms to varying degrees, as in random feedback local online (RFLO) learning and truncated backpropagation through time (tBPTT). However, the learning dynamics of these algorithms, and how they compare with BPTT, remain poorly understood. We apply dynamical systems theory to data-aligned linear RNNs -- whose dynamics can be separated into orthogonal modes -- to compare stationary solutions, stability properties, and convergence rates, finding qualitatively distinct behaviour for RFLO versus BPTT and one-step tBPTT. We further observe that the solutions learned by RFLO are restricted to low-rank perturbations of initial parameters, a result which holds beyond the data-aligned setting. Our work provides analytical insight into how locality constraints shape learning dynamics, with implications for neuroscientific models of learning and alternative optimization approaches for RNNs.

We measure how much one recurrence is worth to a looped (depth-recurrent) transformer, in equivalent unique parameters. From an iso-depth pretraining sweep across recurrence counts $r \in \{1, 2, 4, 8\}$ spanning ${\sim}50\times$ in training compute, we fit a joint scaling law $L = E + A\,(N_\text{once} + r^{\varphi} N_\text{rec})^{-α} + B\,D^{-β}$ and measure a recurrence-equivalence exponent $\varphi = 0.46$. Intuitively, $\varphi$ tells us whether looping a block $r$ times is equivalent in validation loss to $r$ unique blocks of a non-looped model (full equivalence, $\varphi{=}1$) or to a single block run repeatedly with no capacity gain ($\varphi{=}0$). Our $\varphi = 0.46$ sits in between, so replacing unique blocks with shared recurrences increases validation loss at matched training compute. For example, at $r{=}4$ a 410M looped model performs on par with a 580M non-looped model, but incurs the training cost of a 1B non-looped one. We demonstrate the utility of $\varphi$ as a diagnostic tool on two case studies: commonly used truncated backpropagation lowers $\varphi$ to $0.38$, indicating that the loop mechanism is poorly trained under truncation, even though validation loss decreases. Conversely, hyperconnections raise $\varphi$ to $0.65$, a genuine capacity gain. Our method separates true loop improvements from training-side gains, a distinction raw validation loss cannot make.

Recent video diffusion models achieve high-quality generation through recurrent frame processing where each frame generation depends on previous frames. However, this recurrent mechanism means that training such models in the pixel domain incurs prohibitive memory costs, as activations accumulate across the entire video sequence. This fundamental limitation also makes fine-tuning these models with pixel-wise losses computationally intractable for long or high-resolution videos. This paper introduces ChopGrad, a truncated backpropagation scheme for video decoding, limiting gradient computation to local frame windows while maintaining global consistency. We provide a theoretical analysis of this approximation and show that it enables efficient fine-tuning with frame-wise losses. ChopGrad reduces training memory from scaling linearly with the number of video frames (full backpropagation) to constant memory, and compares favorably to existing state-of-the-art video diffusion models across a suite of conditional video generation tasks with pixel-wise losses, including video super-resolution, video inpainting, video enhancement of neural-rendered scenes, and controlled driving video generation.