Transformer-based sequential recommenders with causal self-attention often rely heavily on the most recent interaction at inference time, but how this behavior is structurally expressed in the representation used for prediction remains unclear. We combine prediction-time diagnostics with norm-based analysis of the full attention block. First, we show that SASRec-style models exhibit highly localized last-item reliance. We then find that, although self-attention aggregates contextual information, residual addition sharply shifts the full-block representation toward same-position contributions, which we term residual dominance. To probe this interpretation, we use inference-time residual scaling as a controlled diagnostic intervention. Changing the residual strength induces a monotonic trade-off between structural mixing and last-item reliance, while reducing residual strength recovers a subset of final-position misses for which representations at non-final positions already rank the ground-truth item correctly. Our results provide a structural account linking extreme last-item reliance to residual dominance at inference time. The code is publicly available.
Residual connections rely on a static residual pathway, and are essential for training deep neural networks. Hyper-connections (HC) increase the expressivity of residual routing by incorporating multiple residual streams and learning dynamic information flow, while manifold-constrained (mHC) variants stabilize training through doubly stochastic residual mixing. However, a generator-level bottleneck remains in existing methods: they use dense, unstructured generators for pre-branch aggregation, residual mixing, and post-branch redistribution, which results in parameter count growing rapidly with the number of streams. To address this issue, we propose \underline{\textbf{T}}ensorized \underline{\textbf{E}}fficient \underline{\textbf{M}}anifold-constrained \underline{\textbf{P}}arameterization for \underline{\textbf{E}}xpressive Residual \underline{\textbf{R}}outing (\textbf{TEMPER}), which represents these generators as multi-way tensors over the input-stream, feature, and output-stream modes, and parameterizes them using tensor networks. Such a structured low-rank formulation is shown to preserve token-dependent manifold-constrained routing interface while substantially reducing parameter growth. It also promotes interpretability and intuition, as: i) tensor ranks control the dimensionality of the learned routing subspace, with full ranks recovering dense routing; while ii) the generator approximation errors bound differences in routing logits and, consequently, in the routed-block outputs. Comprehensive experiments show that TEMPER matches or outperforms existing methods across language modeling and commonsense reasoning tasks, while requiring substantially fewer additional parameters. At eight residual streams, TEMPER achieves the best CORE score while using about $84\%$ fewer additional parameters than mHC, thus showing a stronger performance-parameter efficiency trade-off.
Residual connections are a fundamental component of transformer architectures, yet the roles of the attention and feed-forward residual pathways remain poorly understood when considered independently. This paper presents a reproducibility study of partial residual ablations in Pre-LN GPT-style transformers trained at two scales (10M and 124M parameters). I compare four architectural configurations by selectively removing the attention residual connection, the feed-forward residual connection, or both. Across all experiments, removing the attention residual (FFNOnly) consistently causes deterministic collapse to the No-Residual performance floor. In contrast, removing the feed-forward residual (AttnOnly) exhibits a reproducible recovery effect at 10M scale under a controlled 8-seed deterministic study, while its behavior at 124M remains unresolved because of substantial seed variance. During the investigation, I identified and corrected an experimental measurement confound in runtime gain scaling and document both the failed intermediate reproduction and the subsequent controlled replication. Based on the empirical results, I propose a cross-position routing hypothesis to explain the observed asymmetry while explicitly distinguishing confirmed findings from unresolved questions. To support reproducibility, I release the complete source code, experiment configurations, checkpoints, training logs, and all experimental results, including intermediate non-reproducing runs.
Jie Zhang, Cheng-Fang Su, Yi-Jui Huang +1cs.LG cs.AI cs.CL
Residual connections are the de facto mechanism for training deep neural networks stably. Geodesic Normalization (GeoNorm) recasts them on a Riemannian manifold, orthogonalizing each layer output against the current hidden state and applying the resulting update through the Riemannian exponential map. Every hidden state thus keeps a constant $\ell_{2}$-norm, confining the residual stream to a hypersphere. The exponential map, however, is only one member of a broad family of retraction maps. We show that on the hypersphere this entire family collapses to a single scalar design choice. What distinguishes one retraction from another is only how the magnitude of an update is converted into a rotation angle within the plane spanned by the hidden state and the update. This view places Euclidean residual connections and GeoNorm in one framework. Instantiating it with the metric projection retraction and the Cayley retraction yields Proj-SpheretNorm and Cay-SpheretNorm, which are exactly norm-preserving yet require only algebraic operations. Both prove to be members of a one-parameter family of angular retractions, $p$-SpheretNorm, whose rotation angle saturates rather than growing without bound. The two methods above are recovered exactly at $p = 1$ and $p = 2$, while the identity map and GeoNorm arise only as limits at either end. On nanoGPT, all three methods outperform existing lightweight deep connection schemes, and the best validation loss is attained at finite $p$, indicating that the exponential map is not the preferred retraction for spherical residual streams but merely one end of a spectrum.
Depth-routing residual architectures allow Transformer layers to retrieve earlier representations instead of inheriting only the immediately preceding state. Existing Block Attention Residuals, however, use a single content-dependent depth mixture to construct the inputs to queries, keys, and values. This design couples two functionally different decisions: queries and keys determine where attention matches, whereas values determine what content is retrieved. We therefore ask whether matching and content retrieval should be forced to read from the same depth. We introduce Role-Decoupled Attention Residuals (RD-AttnRes), a minimal extension that shares one depth route between queries and keys while learning an independent value route over the same residual sources. Tying the two routing queries exactly recovers the parent architecture, while decoupling them adds only one model-width vector per layer and introduces no additional token-to-token attention operation. We evaluate RD-AttnRes using a frozen, paired pretraining protocol on FineWeb-Edu with five matched seeds for both 120M- and 343M-parameter models and a 2.0B-token training budget. RD-AttnRes improves validation negative log-likelihood in all 10 matched comparisons. The mean reductions are 0.0301 and 0.0247, corresponding to perplexity reductions of 2.97 percent and 2.43 percent at 120M and 343M parameters, respectively. Early-budget controls indicate that neither the additional parameter count, duplicated routing execution, nor a fixed value route reproduces the improvement. Routing diagnostics further reveal persistent divergence between the query-key and value depth distributions. These results suggest that, within the evaluated training regime, attention matching and content retrieval benefit from distinct reads over the residual hierarchy.
Recent work extends Transformer residual pathways along two complementary axes: historical retrieval selects information from earlier depths, whereas multi-stream methods maintain multiple residual trajectories. These capabilities have largely been studied in isolation, and assigning an independent retriever to each stream still prevents one trajectory from influencing depth selection in another. We propose Dual Attention Residuals (DAR), which brings multi-stream interaction into historical retrieval through reciprocal cross-stream addressing. For each target stream, DAR computes depth weights from normalized states in the opposite stream and applies them to values from the target stream's own history. The retrieved states are combined for an unchanged Transformer branch and updated through constrained gated writes; a block-form variant operates on block-level histories to control overhead. Across dense models from 0.1B to 1B parameters and a 7B sparse-MoE model, DAR consistently improves validation loss over standard residual Transformers and Attention Residuals. Routing ablations show that the gain cannot be explained by an additional stream or value projection alone. Representation and intervention analyses further show that reciprocal cross-stream selection preserves depth-wise diversity and avoids the redundancy or functional imbalance observed in alternative two-stream designs.
Valentijn Oldenburg, Floris de Kam, Bente Zuijdam +4cs.LG
Most parameter-efficient finetuning (PEFT) methods adapt weights or activations, thus leaving one of the key Transformer components unchanged: residual connections. This paper investigates Manifold-Constrained Hyper-Connections (mHC), a generalisation of residual connections, as a novel PEFT approach, wrapping frozen OLMo-2 backbones with learned residual routing modules. We find that mHC can finetune frozen Transformers, but that its role differs fundamentally from the original pre-training setting: in finetuning, fixing the residual mixing matrix to identity often improves performance. As a standalone PEFT method, mHC does not consistently outperform LoRA. However, at matched trainable parameter budgets, mHC+LoRA combinations improve language-modelling loss and show task-dependent benchmark gains at both 1B and 7B scale. Overall, our results identify residual routing as a distinct and promising novel PEFT axis.
Residual connections add every sublayer's proposed update with a fixed coefficient of one; the network never evaluates whether an update is reliable before committing it. Drawing on the human-factors principle of independent verification, we introduce Review Residuals, which scale each update by a learned, input-dependent gate conditioned on both the current state and the proposed update: h_l = h_{l-1} + r_l * u_l with r_l = sigmoid(W[RMSNorm(h_{l-1}), RMSNorm(u_l)]). Conditioning the gate on the update is the property that distinguishes it from prior gated and scaled residuals. We report two findings. First, a depth-stability result: a convex (Highway-style) form of the gate reintroduces vanishing gradients and fails to train beyond ~20 layers, whereas the additive, identity-preserving form trains stably at all depths we tested. Second, an emergence-with-scale result: trained from scratch across five sizes (60M-1B parameters, multi-seed), Review Residuals show no advantage at small scale but at 590M significantly outperform both a parameter-matched Highway gate and a parameter-matched standard residual (p<0.05), with a larger advantage at 1B. The benefit grows with model size rather than shrinking.
This work proposes an element-based Convolutional Neural Network (CNN) to accelerate density-based Topology Optimization (TO), termed eCNNTO. TO generally undergoes a large number of iterations, where finite element analysis is performed in every iteration, leading to the efficiency bottleneck especially when dense meshes are used to achieve high-resolution designs. To address this limitation, eCNNTO is proposed to build upon Kallioras et al. (2020), where a Deep Belief Network (DBN) was trained for every element to predict its near-optimal density from its early history, thereby skipping the great majority of iterations and significantly accelerating the TO procedure. However, the method lacks spatial correlations among neighboring elements and may lead to disconnected features in the final structure. The proposed method employs CNN with residual connections to address this issue. On top of it, a novel training strategy is introduced to further enhance the optimization efficiency, where the training dataset consists of the final stage density histories rather than early ones. This change can also help reduce the required training data size. eCNNTO requires only a small dataset to train and yet it can be generalized to problems with largely different boundary conditions, loading cases, design domain geometries, mesh resolutions, as well as non-design domains. In the end, the generalization capabilities and efficiency of eCNNTO are demonstrated through a variety of examples in two and three dimensions, achieving up to 90% and 97% reduction of iterations, respectively.
The well known phenomenon of exploding and vanishing gradients in deep neural networks is analyzed using multiplicative ergodic theory. The effect of adding a residual connection is explained in this context. Specifically, a characterization of Liapunov exponents due to Furstenberg and Kifer is exploited in order to make a precise statement about the Liapunov spectrum and the effect of residual connections on it.
Ekaterina Alimaskina, Gleb Molodtsov, Aleksandr Beznosikovcs.LG cs.AI
Hyper-Connections (HC) replace the single Transformer residual stream with multiple streams, introducing a permutation symmetry over stream indices. We study how this symmetry is resolved in practice: whether streams specialize in a balanced way or exhibit dominant-stream usage. Using fine-grained diagnostics for HC-based language models, we trace how multi-stream representations are actually used. We find that after an early seeding stage, residual mixing often remains close to identity, limiting a core HC mechanism for exchanging information between streams. Moreover, both signal and interpretable features concentrate in a dominant stream, and the nominally multi-stream residual connection can underutilize its capacity, behaving closer to a single-stream residual pathway. Finally, we show that breaking symmetry at stream initialization reduces dominant behavior and improves performance across \textit{m}HC variants. Our code is publicly available.
Speech emotion recognition is an important component of modern human-computer interaction systems. However, many state-of-the-art approaches rely on large pretrained models with high computational and memory requirements, limiting their applicability. This paper proposes ResLSTM-SA, a lightweight architecture that integrates residual connections with soft attention within an LSTM-based framework. Evaluated on the RAVDESS dataset under strict speaker-independent partitioning, the proposed model outperforms conventional attention-based LSTM baselines and several previously reported CNN- and hybrid CNN-LSTM architectures in terms of unweighted average recall (UAR). The best-performing variant (ResLSTM-SA-h64) achieves a maximum UAR of 0.6517 with only 46.8k trainable parameters, delivering competitive accuracy with three orders of magnitude fewer parameters than large-scale self-supervised alternatives, thereby enabling efficient deployment on edge devices and real-time voice assistants. The source code is available at https://github.com/Mak-Sim/ResLSTM-SER.
Diffusion Transformers (DiTs) have become a de facto backbone of modern visual generation, and nearly every major axis of their design -- tokenization, attention, conditioning, objectives, and latent autoencoders -- has been extensively revisited. The residual stream that governs how information accumulates across layers, however, has been directly inherited from the original Transformer. In this paper, we present a systematic empirical analysis of cross-layer information flow in DiTs, jointly along depth and denoising timestep, and identify three concrete symptoms of traditional residual addition, namely monotonic forward magnitude inflation, sharp backward gradient decay, and pronounced block-wise redundancy. Motivated by this diagnosis, we propose Diffusion-Adaptive Routing (\textsc{DAR}), a drop-in residual replacement that performs \emph{learnable, timestep-adaptive, and non-incremental} aggregation over the history of sublayer outputs. Moreover, the proposed \textsc{DAR} is compatible with many modern Transformer enhancement methods, such as REPA. On ImageNet $256\times256$, \textsc{DAR} improves SiT-XL/2 by $2.11$ FID ($7.56$ vs.\ $9.67$) and matches the baseline's converged quality with $8.75\times$ fewer training iterations. Stacked on top of REPA, it yields a $2\times$ training acceleration in the early stage, suggesting cross-layer information routing as an underexplored design axis in diffusion modeling, one that operates orthogonally to existing representation-alignment objectives. Beyond pretraining, \textsc{DAR} can also be applied during the fine-tuning stage of large-scale T2I models and preserves high-frequency details during Distribution Matching Distillation.
Weight matrices in deep networks exhibit geometric continuity -- principal singular vectors of adjacent layers point in similar directions. While this property has been widely observed, its origin remains unexplained. Through experiments on toy MLPs and small transformers, we identify two mechanisms: residual connections create cross-layer gradient coherence that aligns weight updates across layers, and symmetry-breaking nonlinearities constrain all layers to a shared coordinate frame, preventing the rotation drift that would otherwise destabilize weight structure. Crucially, a nonlinear but rotation-preserving activation fails to retain continuity, isolating symmetry breaking -- not nonlinearity itself -- as the active ingredient. Activation and normalization play distinct roles: activation concentrates continuity in the leading singular direction, while normalization distributes it across multiple directions. In transformers, continuity is projection-specific: Q, K, Gate, and Up (which read from the residual stream) develop input-space ($\mathbf{v}_1$) continuity; O and Down (which write to it) develop output-space ($\mathbf{u}_1$) continuity; V alone, lacking an adjacent nonlinearity, develops only low continuity.