SNLP: Layer-Parallel Inference via Structured Newton Corrections

Ligong Han; Kai Xu; Hao Wang; Akash Srivastava

Source-linked AI summary

SNLP: Layer-Parallel Inference via Structured Newton Corrections

Ligong Han, Kai Xu, Hao Wang, Akash Srivastava

arXiv:2605.17842v3May 18, 2026cs.LG

TL;DR

Transformer inference remains bottlenecked by sequential layer dependencies. SNLP replaces costly exact Jacobians with architecture-induced Newton surrogates, reaching up to 2.58× wall-clock speedup and 1.40× without increasing PPL on 0.5B models.

Problem
Transformer inference remains sequential across layers, and existing parallelism techniques do not remove this depthwise dependency.
Method
SNLP uses cheap architecture-induced Newton surrogates and SNLP-aware training to enable finite-iteration layer-parallel inference.
Results
2.58× wall-clock speedup is achieved on 0.5B models, while a less aggressive configuration reaches 1.40× speedup without increasing PPL.
Takeaways & Limitations
SNLP exposes a practical speed-quality frontier for layer-parallel decoding rather than exactly recovering the sequential computation.
Takeaways & Limitations
SNLP’s strongest wall-clock gains appear at 0.5B scale, while larger models and lightweight adaptation of pretrained models remain challenging.

Abstract

from arXiv · show

Autoregressive language models execute Transformer layers sequentially, creating a latency bottleneck that is not removed by conventional tensor or pipeline parallelism. We study whether this layerwise dependency can be relaxed by treating the hidden-state trace across layers as the solution of a nonlinear residual equation and solving it with parallel Newton-style updates. While this view is principled, exact Newton corrections require expensive Jacobian-vector products and naive fixed-point iterations are unstable on trained Transformers. We introduce Structured Newton Layer Parallelism (SNLP), a training and inference framework that replaces exact layer Jacobians with cheap architecture-induced surrogate dynamics. In residual Transformers, this yields Identity Newton (IDN), where the correction reduces to a prefix-sum-like update; in mHC-style architectures, HC Newton (HCN) uses the model's residual mixing matrix. We also study SNLP-aware training, including pretraining regularization and direct SNLP-forward SFT. Experiments on Nanochat-scale Transformers show that SNLP exposes a practical speed-quality frontier: on 0.5B models, it reaches up to 2.58x wall-clock speedup, and a less aggressive configuration reaches 1.40x speedup without increasing PPL. The useful tradeoff comes from the biased finite-iteration computation induced by IDN/HCN rather than exact recovery of the sequential trace. We further show that SNLP-forward SFT can preserve downstream task accuracy, and that SNLP can serve as a drafter for self-speculative decoding while a sequential verifier preserves output correctness.

1 Introduction

SNLP treats the hidden-state sequence across Transformer depth as a nonlinear residual equation and uses cheap architecture-induced Newton surrogates for layer-parallel inference. Its experiments and training co-design show a practical speed-quality frontier, including substantial speedups, preserved quality in selected settings, and solver-induced bias rather than exact sequential-trace recovery.

Motivation: Layer dependencies remain a latency bottleneck despite tensor and pipeline parallelism, kernel fusion, batching, KV caching, and speculative decoding.These methods improve individual layers or token steps but do not remove the depthwise dependency chain.
Method: SNLP formulates layer-parallel inference as structured surrogate Newton solving over the hidden-state trace instead of strict layer-by-layer execution.The approach replaces exact layer Jacobians with cheap structured surrogates; IDN, HCN, and DiagN instantiate the framework across architectures.
Method: Identity Newton uses the residual identity path for additive prefix-style propagation, while HC Newton uses the learned residual mixing matrix in HC/mHC-style models.These architecture-induced surrogates make Newton-style corrections cheaper than exact Jacobian computations.
Training co-design: SNLP-aware training matches finite-iteration SNLP states to sequential traces or directly trains the approximate SNLP-forward computation used at inference.The objectives are designed to make layer dynamics easier to solve with the selected surrogate and support task adaptation under SNLP inference.
Results: 2.58× speedup is achieved at 0.5B scale, while one IDN-regularized configuration reaches 1.40× speedup without increasing PPL.Aggressive settings trade larger speedups for PPL loss, whereas less aggressive settings recover near-sequential quality.
Interpretation: Finite-iteration IDN and HCN induce solver-induced inference bias because practical SNLP differs from sequential execution through approximate corrections, chunking, fusion, and initialization.Exact Newton convergence recovers the sequential trace, but the practical biased computation creates the observed speed-quality tradeoff.

2 Related Work

SNLP connects parallel nonlinear-solver and scan-based computation to depthwise Transformer execution, while differing from prior layer-parallel and efficient-inference methods in targeting cross-layer dependencies during inference.

Parallel nonlinear solvers: SNLP builds on nonlinear-equation views of sequential computation, including DEER’s Newton corrections with parallel scan and later quasi-Newton or Kalman-style approximations.Related work also frames feedforward computation as parallel nonlinear equation solving.
Associative scans and structured recurrences: SNLP applies associative-scan principles to depthwise correction, extending scan-based parallelization from recurrent and state-space models to Transformer depth.The identity, diagonal, or structured surrogate determines the correction dynamics.
Depth mixing and residual architectures: SNLP relates to residual and depth-mixing architectures, including Hyper-Connections, mHC, AttnRes, value residual learning, and x0-style residual connections.These approaches modify information flow or feature persistence across depth.
Efficient language-model inference: Unlike efficient-inference methods that accelerate token decoding or standard layer-stack execution, SNLP targets the dependency chain across layers for a fixed token prefix.The related methods include batching, KV caching, quantization, memory-aware execution, kernel engineering, speculative decoding, early exit, and serving systems.
Layer-parallel computation: SNLP complements multigrid-in-time layer-parallel Transformer training and differs from post-hoc layer-pair parallelization by using Newton corrections with training–inference co-design.The comparison distinguishes SNLP’s inference-side approach from these prior layer-parallel methods.

3 Methods

SNLP reframes the layerwise hidden-state trace as a nonlinear residual system and solves a sequential prefix plus parallel suffix using cheap architecture-induced Newton surrogates. Its practical design combines structured corrections with SNLP-aware training and GPU-oriented suffix execution, yielding finite-iteration inference rather than exact sequential-trace recovery.

Residual formulation: The hidden-state trace h = (h1, . . . , hL) is formulated as G(h) = 0, where Gl(h) = hl − fl(hl−1).The usual sequential forward pass is exactly the zero-residual solution, enabling joint iterative solution across layers.
Motivation: Exact layer Jacobians are infeasible for language-model hidden states, while naive fixed-point updates can be unstable on trained residual networks.SNLP therefore targets a cheap structured approximation that preserves cross-layer information propagation without the full Jacobian cost.
SNLP solver: SNLP evaluates a sequential prefix, computes suffix layers independently, and applies a cheap structured Newton correction that keeps only a lightweight recurrence on the critical path.The expensive nonlinear block forwards are parallelized, while the correction is trivial to compute or architecture-provided.
Correction mechanism: The structured correction propagates information from the prefix through the entire suffix within one solver iteration, unlike naive fixed-point updates that advance only one layer per iteration.After K iterations, naive fixed-point influence reaches only the next K suffix layers, whereas SNLP corrections incorporate all suffix-layer corrections.
Structured surrogates: IDN uses the identity residual path as a prefix-sum-like correction, while HCN uses a small stream-level residual mixing matrix exposed by mHC-style architectures.The HCN construction shows that SNLP can use structured inter-layer sensitivity beyond identity residual paths.
Training and execution: SNLP-aware training matches finite SNLP solves to sequential traces, and inference controls the quality-cost tradeoff through suffix initialization, iteration count K, and batched or fused layer execution.Fusion improves hardware utilization but changes the finite-iteration computation, so practical SNLP inference is not exact sequential-forward recovery.

4 Analysis

SNLP’s practical behavior is governed by solver-induced inference bias from approximate surrogates, finite iterations, initialization, fusion, and chunking rather than exact Newton recovery. Training reduces this bias by stabilizing suffix corrections, while inference trades lower error-compounding variance against evaluation and chunking bias.

Solver-induced bias: Exact Newton convergence recovers the sequential forward pass, whereas practical SNLP introduces solver-induced bias through approximations, finite iterations, initialization, fusion, and chunking.These mechanisms jointly determine the practical inference behavior.
Training-side effects: SNLP-aware training encourages suffix residual branches to match the sequential final state and applies implicit Lipschitz pressure, making suffix features less state-sensitive.For residual blocks, smaller J_g_l makes J_f_l = I + J_g_l closer to the IDN surrogate.
Inference-side effects: Finite-iteration SNLP can remain near the sequential model by reducing layerwise error-compounding variance, but evaluating suffixes at common inputs introduces bias.This tradeoff depends on SNLP training keeping the evaluation bias small.
Inference-side effects: Fusion and aggressive chunking add bias because fused branches receive summed cross-layer signals that can alter feature interactions and degrade performance.The limitation is specifically associated with chunks that are too aggressive.
Layer coupling: SNLP benefits from structured depth coupling: IDN uses identity coupling, HCN uses learned residual mixing, and fused SNLP creates implicit coupling within chunks.This complements HC, mHC, and AttnRes approaches that parameterize or learn interactions across depth.

5 Experiments

Experiments show that SNLP exposes a speed–quality frontier: aggressive layer parallelism reaches 2.58× speedup, while moderate settings achieve 1.40× without increasing PPL. SNLP-aware training preserves downstream accuracy and supports self-speculative drafting with sequential verification.

Inference configurations: SNLP evaluates chunkwise IDN for residual models and HCN for mHC models, varying parallel chunks, fused layers, and prefix-state or one-shot initialization.For example, 8xF2-fwd uses eight fused two-layer chunks with one-shot parallel initialization.
Pretraining evaluation: 1.40× speedup is achieved without increasing PPL, while 2.58× speedup incurs 17.0% PPL loss on the 0.5B w/o x0/VE IDN-regularized model.Additional Newton iterations recover quality at lower speed, exposing a practical speed–quality tradeoff.
Downstream task evaluation: 57.03% average accuracy is obtained with sequential SFT on a 3B IDN-regularized model followed by IDN inference, versus 57.85% for base sequential SFT and inference.This exceeds the 54.60% accuracy of base-model sequential SFT with IDN inference.
Self-speculative decoding: SNLP drafting preserves output correctness because sequential verification guarantees identical output to sequential greedy decoding, while SNLP changes only proposal cost.Ideal speedup depends on block size, drafter speedup, and greedy acceptance rate.
Pretraining evaluation: Moderate IDN/HCN regularization changes sequential PPL by -0.5% to +2.4%, while IDN-regularized models retain parallel behavior after standard SFT.The diagnostics use a 0.5B IDN checkpoint with PPL 16.81.
Diagnostics and ablations: IDN regularization reduces relative per-layer substitution error to 0.03%–0.15% of ∥hS∥, compared with 2%–24% without regularization.Correction lets every active suffix layer affect the final output in one iteration; without correction, influence advances only one layer per iteration.

6 Conclusion … C Analysis Details

SNLP relaxes Transformer layerwise dependencies through structured Newton corrections, exposing a speed-quality frontier with limitations concentrated at larger scales and off-the-shelf adaptation. The paper also documents algorithmic variants, appendix analyses, and changes toward representative long-context evaluation.

6 Conclusion: SNLP replaces exact layer Jacobians with cheap architecture- or objective-induced surrogates, yielding IDN for residual Transformers and HCN for mHC-style models.The framework treats the hidden-state trace across depth as a residual equation and uses structured corrections for training and inference.
6 Conclusion: 1.20× speedup with 3.9% PPL increase is reported for the 3B IDN-regularized model, while stronger gains currently appear at the 0.5B scale.The conclusion frames SNLP as a tradeoff rather than a guarantee of improved quality and notes that wider sequential blocks complicate larger-scale systems implementation.
6 Conclusion: Future work targets off-the-shelf finetuning, post-training and SFT, larger-model kernels and hardware co-design, practical DiagN, and composition with iterative decoding or diffusion language models.These directions are presented as implementation, training, architectural, and decoding extensions of SNLP.
Appendix: The appendix covers algorithmic details, variance reduction, Hyper-Connections and mHC, fused cross-layer coupling, training ablations, inference ablations, and additional correction analyses.Its listed sections also include AR match rate, Jacobi decoding, ELK tempering, off-the-shelf models, diagonal-Jacobian corrections, TinyLlama finetuning, correction ordering, and propagation.
C Analysis Details: The supplied material identifies C Analysis Details and its variance-reduction, Hyper-Connections/mHC, and fused cross-layer-coupling subsections as part of the paper’s analytical appendix.These entries are listed in the appendix contents without further analytical results in the supplied passages.
A Note on Changes from the Previous Version: The revised experiments use sequence length 2048 instead of 128, aligning evaluation with training and reframing SNLP as a practical speed-quality frontier.The previous version emphasized short-context PPL behavior, whereas the revision adopts a more representative long-context evaluation.
B Algorithm: Algorithm 1 executes Transformer blocks sequentially, whereas Algorithm 2 batches IDN/HCN structured corrections after a sequential prefix using suffix length N and K iterations.For HCN, the learned matrix acts on the HC/mHC stream dimension rather than the full hidden dimension.
B Algorithm: DiagN changes only diagonal-estimator acquisition through FD, JVP, or VJP probes; all variants then use the same affine scan recurrence.The algorithm initializes the sequential prefix, estimates diagonal terms in parallel, and solves the affine recurrence before returning the final language-model head output.

C.1 Variance Reduction Derivation · C.2 Connection to Hyper-Connections and mHC · C.3 Fused Cross-Layer Coupling

The appendices explain why IDN can reduce variance from chain-wise error compounding, how HCN generalizes the architecture-induced surrogate to mHC, and why fused SNLP introduces nonzero cross-layer coupling. Together, these derivations motivate training branches to align with cheap structural transitions while recognizing fusion’s biased interactions.

C.1 Variance Reduction Derivation: In a residual suffix, sequential layers evaluate each branch at different accumulated hidden states, whereas one-step IDN evaluates all branches from the same prefix state.The resulting difference is attributed entirely to the evaluation points.
C.1 Variance Reduction Derivation: Sequential execution couples later-layer deviations to earlier deviations through branch Jacobians, producing a variance scale of order ρ^2 P_l(l − S)σ^2 under a first-order independence approximation.IDN removes this compounding term by evaluating all branches at the same prefix state.
C.1 Variance Reduction Derivation: SNLP-aware training reduces IDN’s evaluation-point bias by making branch functions less sensitive over the suffix trajectory, supporting a useful finite-iteration speed-quality tradeoff when the bias is small.The variance reduction helps explain why finite-iteration SNLP can remain close enough to the sequential path.
C.2 Connection to Hyper-Connections and mHC: HCN approximates each mHC block Jacobian with the architecture-derived residual mixing matrix acting only on stream dimension, avoiding full hidden-state Jacobian estimation.The surrogate is available directly from the architecture when nonlinear branch sensitivity is locally small.
C.2 Connection to Hyper-Connections and mHC: IDN and HCN share a template: retain the cheap architecture-induced residual transition and train nonlinear branches to be compatible with it.For residual blocks, IDN uses A_l = I and encourages J_g_l to remain small so J_f_l = I + J_g_l ≈ I.
C.3 Fused Cross-Layer Coupling: Fused attention computes branch projections for all layers in a chunk from shared input h and sums their output projections, followed by a fused MLP.The fused MLP uses concatenated expansion and down-projection weights to sum per-layer MLP branches.
C.3 Fused Cross-Layer Coupling: Fused SNLP is not equivalent to independently summing layer branches because every MLP branch receives the aggregate post-attention state, creating generally nonzero cross-layer coupling.Layer normalization, nonlinear activations, and non-canceling projections can make branches respond to other layers’ attention evidence.

D Training Configuration and Ablations

Training ablations motivated SNLP-aware regularization because vanilla Jacobi diverged on trained models, while spectral regularization did not sufficiently reduce iteration or Jacobian-estimation costs. The preferred regularization settings vary by architecture and scale, and detaching the MSE target can hurt IDN PPL.

Training-side ablations: Vanilla Jacobi diverged on trained models because measured layer Jacobian norms were far from contractive.Spectral regularization substantially reduced the Jacobian norm but only modestly reduced Newton iterations.
Training-side ablations: Spectral regularization did not address the dominant cost of Jacobian estimation, motivating cheap structured IDN/HCN corrections instead of exact Newton acceleration.The training objective therefore shifts toward making structured surrogate corrections useful.
Training configurations: The best loss configuration depends on architecture and scale: 3B prefers a smaller parallel suffix, 0.5B standard is sensitive to λ and stride, and no-x0/VE works best with stride 6.These settings come from trained-from-scratch model ablations.
Training configurations: Detaching the MSE target hurts IDN PPL, whereas HCN on the mHC model uses a detached target by default.PPL is evaluated at sequence length 2048, and ∆PPL is measured relative to each model group's No Reg. baseline.

E Additional Inference Ablations · E.1 Interpreting AR Match Rate

The appendix evaluates correction-ordering and propagation ablations on a 0.5B IDN checkpoint, then interprets autoregressive match rates through a local agreement model. Because mismatches alter later prefixes, modest AR match can still indicate high conditional token agreement.

E Additional Inference Ablations: Correction-ordering and correction-propagation ablations use the 0.5B IDN checkpoint with N = 24, λ = 0.5, stride 3, no detach, and PPL 16.81.This checkpoint is described as more parallelizable and is taken from Table 9.
E Additional Inference Ablations: The ablation checkpoint is selected specifically for its greater parallelizability within the 0.5B IDN setting.
E.1 Interpreting AR Match Rate: AR Match evaluates each sample over T = 32 generated tokens against the sequential baseline.The reported rate is token-level agreement over N samples.
E.1 Interpreting AR Match Rate: The reported AR match rate α counts matched generated tokens relative to the sequential baseline across evaluated samples.The passage defines α as token-level agreement against the sequential baseline over N samples.
E.1 Interpreting AR Match Rate: Under an absorbing-divergence model, β is the probability of matching the next token conditioned on all previous generated tokens matching.A single mismatch changes the prefix for all later autoregressive positions.
E.1 Interpreting AR Match Rate: For T = 32, α = 0.30 corresponds to local conditional agreement of approximately β ≈ 0.905 before prefix divergence.The model assigns token i a matching probability of β^i, so modest AR match can coexist with high local agreement.

E.2 Coupling SNLP with Jacobi Decoding · E.3 ELK Tempering and Preheat Initialization

SNLP and Jacobi decoding parallelize different axes, but coupling them is limited by hidden-state initialization: reuse can lose token agreement, while full reinitialization recovers the nested baseline. ELK tempering and preheat initialization offer auxiliary tradeoff controls, but their benefits are configuration-dependent and therefore excluded from the main search.

E.2 Coupling SNLP with Jacobi Decoding: SNLP parallelizes layers while Jacobi decoding iteratively refines future-token blocks, motivating a joint solver over the layer-token lattice.The proposed coupling could avoid a full inner SNLP solve for every token-level Jacobi iteration.
E.2 Coupling SNLP with Jacobi Decoding: The coupled design aims to jointly update hidden states and token guesses instead of using a nested loop.This formulation treats the computation as a two-dimensional layer-token solve rather than separate sequential axes.
E.2 Coupling SNLP with Jacobi Decoding: Initialization is the main coupling difficulty: full reinitialization reduces to the nested baseline, whereas state reuse carries features computed for obsolete draft tokens.JD changes draft tokens between iterations, so reused hidden states may no longer correspond to the current token sequence.
E.2 Coupling SNLP with Jacobi Decoding: 100% token match is achieved by naive K = 1 across all tested configurations, leaving no inner-iteration gap for coupling to close.These results use a 0.5B IDN stride-0 checkpoint, 8 prompts, 32 generated tokens, and lookahead window 5.
E.2 Coupling SNLP with Jacobi Decoding: Increasing naive K to 2 adds cost without improving match, while h0-JD warm-starting degrades token match to roughly 21–37%.h0-JD recomputes the first parallel input but reuses deeper parallel states from the previous JD iteration.
E.3 ELK Tempering and Preheat Initialization: ELK-style tempering scales the Newton correction without changing block-forward count, so it should not affect speed.It is evaluated as an auxiliary inference knob rather than part of the main configuration search.
E.3 ELK Tempering and Preheat Initialization: Preheat initializes each layer’s parallel suffix using a low-rank affine predictor fitted from calibration embeddings and sequential hidden states.The predictor uses a truncated SVD basis and linear regression; random-token calibration was used in the reported table.
E.3 ELK Tempering and Preheat Initialization: ELK can improve the PPL–iteration tradeoff for some configurations, but preheat is inconsistent, so neither knob is tuned in the main experiments.Preheat may be close to h0 or much worse, and later validation-set calibration did not consistently improve results.

E.4 Off-the-Shelf Model Results

Off-the-shelf model results compare selected post-hoc SNLP configurations across fast relaxed-PPL settings and the lowest-PPL configuration. Matching sequential perplexity requires multiple Newton iterations without speedup, motivating SNLP-aware training.

E.4 Off-the-Shelf Model Results: Table 13 summarizes selected post-hoc SNLP configurations for off-the-shelf models from the L1 timing sweep.The comparison includes fast configurations under relaxed PPL thresholds and the lowest-PPL configuration.
E.4 Off-the-Shelf Model Results: Fast configurations are evaluated under relaxed PPL thresholds alongside the configuration achieving the lowest PPL.
E.4 Off-the-Shelf Model Results: Matching sequential perplexity requires multiple Newton iterations and does not produce speedup.This result supports the need for SNLP-aware training.

E.5 Diagonal-Jacobian Correction Results … E.8 Correction Propagation

Diagonal-Jacobian correction is computationally motivated but performs poorly for asymmetric mHC routing, while post-hoc TinyLlama adaptation remains difficult without SNLP-aware pretraining. Forward correction ordering is most stable, and Newton correction enables immediate influence from all active suffix layers.

E.5 Diagonal-Jacobian Correction Results: Finite-difference Jacobian-vector estimates reuse the already computed f(x), requiring only an additional f(x + ϵv) evaluation.Exact forward-mode JVP returns f(x) and Jv together, whereas VJP computes J⊤v.
E.5 Diagonal-Jacobian Correction Results: 104.2 PPL was the best diagonal result, 17.5% above sequential PPL 88.67, and no configuration reached the 10% fast threshold.On x0+VE mHC models, diagonal correction could match PPL but remained much slower because asymmetric routing makes VJP directions unsuitable for Jv.
E.6 Pretrained TinyLlama Fine-Tuning: TinyLlama-1.1B-Chat-v1.0 was finetuned with IDN regularization on ClimbMix using AdamW and cosine learning-rate decay.The model has 22 layers, hidden dimension 2048, 32 attention heads, and 4 KV heads; Table 16 reports a small grid with sequence length 128 evaluation.
E.6 Pretrained TinyLlama Fine-Tuning: Post-hoc finetuning was difficult for changing pretrained layer-parallel behavior when the model lacked SNLP regularization, supporting a co-design interpretation.Stronger pretrained-model adaptation was left for future investigation.
E.7 Correction Ordering: At K = 1, several non-forward correction orders remained usable, but at K = 4 forward ordering consistently produced the best stable result.Most non-forward orders degraded or diverged; AR-ness averages local- and global-AR-ness scores as defined in DiffuCoder.
E.8 Correction Propagation: Without correction, information advanced only one layer per iteration, whereas Newton correction let all active suffix layers influence the output immediately.The subset ablation compared progressively larger active suffixes with and without correction.