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Adjoint sharding for very long context training of state space models

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Adjoint sharding computes backpropagation-equivalent gradients for state-space models by decomposing them into independent vector-Jacobian products, cutting training memory by up to 3X at 1M-token context.

desk verdict A clean adjoint-method decomposition for SSM gradients with a real parallelization idea, but the abstract's memory-reduction claims are unsupported by any experiments and the paper's own storage analysis undermines them. read the letter →

arxiv 2501.00692 v1 pith:55GVQLEE submitted 2025-01-01 cs.LG cs.AIcs.CL

Xingzi Xu , Amir Tavanaei , Kavosh Asadi , Karim Bouyarmane This is my paper · ORCID
classification cs.LGcs.AIcs.CL
keywords adjointmethodstate-spacemodelslong-contexttraininggradientshardingvector-Jacobianproductsmemory-efficienttruncatedbackpropagationthroughtimedistributed
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Training large language models on very long contexts is bottlenecked by GPU memory, because backpropagation stores activations for the whole sequence, so memory grows rapidly with context length. This paper proposes adjoint sharding, which replaces backpropagation with the adjoint method for the recurrent state-space layers, decomposing the gradient into many independent vector-Jacobian products (one per token per layer) that can be computed, stored, and discarded separately. The paper proves that summing these products reproduces exactly the backpropagation gradient, and argues that sharding them across time and layers reduces memory enough to train a 1.27B-parameter state-space model on contexts above 100K tokens with a small cluster. A truncated variant cuts the number of vector-Jacobian products from quadratic to linear in context length. If the memory reduction holds, long-context training and fine-tuning of state-space architectures becomes tractable on far fewer GPUs than today.

What carries the argument

The load-bearing object is the adjoint state $\lambda_{t,\tau}^k = C_t^k \prod_{i=1}^{t-\tau} A_{t+1-i}^k$ for layer $k$, which replaces the chained derivatives of backpropagation. Proposition 1 (the adjoint method for recurrences) supplies the recurrence $\lambda_{i-1}=\lambda_i\,\partial f(i,h_{i-1},\theta)/\partial h_{i-1}$ that makes these states computable by a backward pass without storing the forward computation graph. The residual identity $y_t^K = y_t^0 + \sum_k \tilde{y}_t^k$ then lets the gradient separate into a sum over layers. Each term in the resulting sum is a vector-Jacobian product of the form $\mathrm{vjp}(v) = v \cdot \partial \mathrm{Net}_\theta(\mathrm{Input}_i)$, which can be evaluated with reverse-mode automatic differentiation initialized with the vector $v$, and which the paper treats as costing about one forward pass. The independence of these VJPs is what allows memory to be sharded and computation to be parallelized.

What would settle it

Train a 1.27B-parameter state-space model on a 1M-token sequence with adjoint sharding and record peak GPU memory, comparing with backpropagation under activation checkpointing on the same hardware; if adjoint sharding's peak memory is not roughly 3X lower, the abstract's central memory claim is contradicted.

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Extended reading notes

Core claim

The central claim is that for a residual stack of state-space layers, the gradient of the training loss with respect to every parameter can be written as a sum (over tokens $t$ and layers $k$) of independent vector-Jacobian products, with no cross-dependencies between terms. For a single SSM layer, Proposition 2 gives $$\frac{d l_t}{d\$\theta$} = \Big[\sum_{i=1}^t \mathrm{vjp}^{A_i}\big(\tfrac{dl_t}{dy_t}\lambda_{t,i}\otimes h_{i-1}\big)\Big] \oplus \Big[\sum_{i=1}^t \mathrm{vjp}^{B_i}\big(\tfrac{dl_t}{dy_t}\lambda_{t,i}\otimes \hat{x}_i\big)\Big] \oplus \mathrm{vjp}^{C_t}\big(\tfrac{dl_t}{dy_t}\otimes h_t\big),$$ with adjoint state $\lambda_{t,\tau}=C_t\prod_{i=1}^{t-\tau} A_{t+1-i}$; Proposition 3 extends this to $K$ stacked residual layers, so the full gradient is a sum of such terms over $t$ and $k$. Because each vector-Jacobian product depends only on the saved forward states and the adjoint state, the products can be sharded by time and layer across devices, computed in parallel, and discarded immediately. The paper also introduces truncated adjoint sharding, which limits the time sum to the last $\bar{T}$ states, cutting the number of VJPs from $O(T^2)$ to $O(T\bar{T})$ while retaining implicit dependence on all earlier states through the recurrence.

Load-bearing premise

The promised memory savings depend on the assumption that storing the intermediate values the method needs for every token and every layer—the transition and output matrices, hidden states, and normalized inputs—costs less memory than backpropagation's stored activations, so peak memory still drops sharply.

Editorial extensions

If this is right

  • Training or fine-tuning a 1.27B-parameter SSM on contexts above 100K tokens becomes possible on a small cluster of five instances, where backpropagation tops out near 35K tokens.
  • Gradient computation for residual and recurrent models becomes embarrassingly parallel across time and layers, so wall-clock training time can drop if the independent VJPs are scheduled across GPUs or GPU instances.
  • Truncated adjoint sharding computes an approximate gradient in linear time, making very long contexts computationally feasible where the full method's $O(T^2)$ VJP count would be prohibitive.
  • Because the gradient decomposition is exact for the full method, training with adjoint sharding reproduces backpropagation's training results while using less memory.
  • Activations, gradients, and optimizer states can all be sharded across devices, so per-GPU memory falls roughly as $1/\Upsilon$ with $\Upsilon$ devices.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not report measured memory for the 1M-token, 1.27B-parameter case its abstract cites, only smaller-scale curves; whether the stored states keep peak memory below backpropagation at that scale is an open empirical question.
  • If the VJP decomposition is computed in FP16, rounding may accumulate differently than in backpropagation's sequential chain, so 'equivalent gradients' likely holds only up to numerical precision; a mixed-precision gradient-equality check would clarify this.
  • Truncated adjoint sharding is effectively a truncated-backpropagation-through-time variant for SSMs; its convergence and the optimal truncation length $\bar{T}$ are left for future work, and likely depend on how fast the products $\prod A_t$ decay, which is a testable prediction.
  • The paper's parallel-speedup estimate assumes VJPs are compute-bound at roughly one forward pass each; if memory bandwidth is the binding constraint, the practical speedup on GPU instances will fall below the analyzed figure.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes adjoint sharding, a method that computes gradients for state-space model (SSM) language models by decomposing backpropagation into independent vector-Jacobian products via the adjoint method. It claims this reduces training memory by up to 3X (abstract) or "orders of magnitude" (abstract and conclusion), enabling training on 1M-token contexts for a 1.27B parameter model on five AWS P4 instances. The paper derives gradient-equivalence results in Propositions 2 and 3, introduces a truncated variant and a distributed algorithm, and presents an analytical memory/FLOP analysis. The manuscript contains no experimental section, no measured memory numbers, and no code.

Significance. If the memory-reduction claims were established, adjoint sharding would be a practically valuable technique for long-context SSM training. The gradient-equivalence derivations in Propositions 2 and 3 appear internally consistent and provide a useful formalization of the adjoint approach for recurrent SSM layers. However, the practical impact is entirely unverified: the paper's central empirical assertions are unsupported, and its own memory accounting raises serious doubts about the claimed savings. For a cs.LG systems submission, the absence of experiments is a fundamental gap.

major comments (4)
  1. [Abstract and Conclusion] The abstract states "Empirical results show the proposed adjoint sharding algorithm reduces memory usage by up to 3X with a 1.27B parameter large language model on 1M context length training," and the conclusion repeats "orders of magnitude of memory reduction," yet the manuscript contains no experimental section, no measurement protocol, no memory numbers, and no code. Figure 1 is a plot without axes or experimental setup, and Figure 6 is based on assumed speedups. This is load-bearing because the paper's contribution is a training method whose benefit is memory reduction; the gradient-equivalence proofs alone do not establish any memory savings.
  2. [Section 4.5] The memory analysis states that storing {A_t^k, C_t^k, h_t^k, y_t^k} over T tokens and K layers costs T K(2N+P) + T P FP16 numbers. For the claimed scale (T=1M, K=100, N=225, P=128), this is approximately 5.8e10 FP16 values, or about 116 GB, which exceeds the 40 GB memory of a single AWS P4 (A100) instance. Even distributed over five GPUs, this is roughly 23 GB per GPU before vjp workspaces, model weights, and optimizer states are counted. The paper never compares this peak memory with a backpropagation baseline using activation checkpointing, so the "up to 3X" and "orders of magnitude" claims are unsupported.
  3. [Algorithm 1] Algorithm 1, line 10, requires storing per-token, per-layer states {h_t^k, C_t^k, y_hat_t^k, A_t^k} on each device. This storage grows as O(T K N) with context length and layer count and is not a constant-memory scheme. The claimed advantage of adjoint sharding over backpropagation therefore depends on comparing this O(T K N) state storage against the memory cost of the autograd graph in BPTT, a comparison the paper does not provide. The theoretical gradient equivalence does not imply lower memory.
  4. [Section 4.3] Truncated adjoint sharding is introduced with the phrase "empirically showcase its performance," but no empirical evaluation appears anywhere in the manuscript. The same section later states "We leave investigation of bar(T)'s impact on performances for future works," which explicitly acknowledges that the performance-maintenance claim is not established. The statement that truncated adjoint sharding achieves "similar results" is therefore unsupported.
minor comments (4)
  1. [Abstract and Conclusion] The abstract reports "up to 3X" memory reduction, while both the abstract's earlier sentence and the conclusion claim "orders of magnitude" reduction; these are inconsistent quantitative claims.
  2. [Appendix A.2] In the proof of Proposition 3, the vjpBi terms are written with x_hat_i^k and later changed to y_hat_i^{k-1}; the notation should be harmonized to match Proposition 3 and the main text.
  3. [Figures 1 and 6] Figure 1 and Figure 6 lack axis labels, units, and any description of the experimental or simulated setup, so they cannot be interpreted or reproduced.
  4. [References] The bibliography contains duplicate entries: reference [4] is repeated, and references [52] and [53] both cite the ZeRO paper; these should be consolidated.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the gradient derivation is a direct chain-rule expansion, and the only self-citation is non-load-bearing.

full rationale

The derivative chain is self-contained. Proposition 1 is the standard adjoint-sensitivity formula quoted from the independent references [8,32]; Proposition 2 applies it to the linear SSM recurrence ht = At ht-1 + Bt xhat_t, yt = Ct ht by direct substitution (Eq. 8), and Proposition 3 sums over the residual layers. These are rearrangements of the same chain-rule terms that backpropagation computes, so the claimed gradient equivalence is derived, not assumed. No parameter is fitted to a subset of data and then reported as a prediction: the truncation window \(\bar{T}\) is a user-set hyperparameter whose convergence the paper explicitly leaves to future work, and no experimental memory numbers are used in the derivation. The only author-overlapping citation is [66] in the background sentence "The adjoint method is a constant-memory optimization technique for dynamical systems [9, 66]"; it is paired with the independent [9] and is not load-bearing. The abstract's "up to 3X" and "orders of magnitude" memory claims are unsupported by an experimental section, but that is a verifiability/correctness gap, not circularity, and the paper itself flags the missing implementation and convergence analysis in its Limitation paragraph.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central derivation uses standard adjoint equations and the assumed SSM residual structure. No invented entities appear. The only free parameter is the truncation window \bar{T}, whose value is not fitted in the paper.

free parameters (1)
  • Truncation window \bar{T} = not specified
    The truncation window \bar{T} controls the trade-off between speed and approximation quality in truncated adjoint sharding. Its value is chosen by the user and is not derived from theory or fixed by data in the paper.
assumptions (4)
  • domain assumption SSM dynamics h_t = A_t h_{t-1} + B_t x_t and y_t = C_t h_t.
    Section 3.1 defines the model class that the adjoint derivation uses; the gradient formulas depend on this linear recurrent form.
  • domain assumption Residual structure y_t^K = y_t^0 + sum_k \tilde{y}_t^k.
    Section 3.2 introduces the residual network; Proposition 3 uses it to separate gradients across layers.
  • standard math Adjoint equations from Proposition 1 of [8] are correct.
    Section 3.3 states the adjoint method for recurrence relations as a cited proposition and builds on it without proof.
  • domain assumption VJPs can be computed with memory comparable to a forward pass and can be parallelized on modern GPUs.
    Section 4.5 and the Limitations paragraph rely on this implementation assumption to argue for speedups, but provide no measured implementation.

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Cite this review

Pith. "Pith review of Adjoint sharding for very long context training of state space models." pith.science (2026). https://pith.science/paper/55GVQLEE

@misc{pith2026250100692,
  author       = {Pith},
  title        = {Pith review of: Adjoint sharding for very long context training of state space models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/55GVQLEE}},
  note         = {Machine review of arXiv:2501.00692}
}
read the original abstract

Despite very fast progress, efficiently training large language models (LLMs) in very long contexts remains challenging. Existing methods fall back to training LLMs with short contexts (a maximum of a few thousands tokens in training) and use inference time techniques when evaluating on long contexts (above 1M tokens context window at inference). As opposed to long-context-inference, training on very long context input prompts is quickly limited by GPU memory availability and by the prohibitively long training times it requires on state-of-the-art hardware. Meanwhile, many real-life applications require not only inference but also training/fine-tuning with long context on specific tasks. Such applications include, for example, augmenting the context with various sources of raw reference information for fact extraction, fact summarization, or fact reconciliation tasks. We propose adjoint sharding, a novel technique that comprises sharding gradient calculation during training to reduce memory requirements by orders of magnitude, making training on very long context computationally tractable. Adjoint sharding is based on the adjoint method and computes equivalent gradients to backpropagation. We also propose truncated adjoint sharding to speed up the algorithm while maintaining performance. We provide a distributed version, and a paralleled version of adjoint sharding to further speed up training. Empirical results show the proposed adjoint sharding algorithm reduces memory usage by up to 3X with a 1.27B parameter large language model on 1M context length training. This allows to increase the maximum context length during training or fine-tuning of a 1.27B parameter model from 35K tokens to above 100K tokens on a training infrastructure composed of five AWS P4 instances.

Figures

Figures reproduced from arXiv: 2501.00692 by the authors.

Figure 1
Figure 1. Compared to backpropagation (red lines), adjoint sharding (blue lines) significantly reduces memory re [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Adjoint sharding dissembles large models’ gra [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Lines in red are fine-tuning free methods and lines in blue are fine-tuning methods. Fine-tuning methods [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: The adjoint states are computed sequentially backwards. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Computation schematic of dl t/dθAk , dl t/dθBk , and dl t/dθCk . We provide the proof to proposition 3 in section A.2. Define Λ t k = {λ t,τ k } t τ=1, proposition 3 shows that the gradients of each network’s parameters computed with each token only correlate through t…
Figure 6
Figure 6. Figure 6: Training time (/day) per epoch comparison for [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.