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REVIEW 5 major objections 5 minor 12 references

LEDOM: Reverse Language Model

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A purely right-to-left language model can verify forward answers: reranking by reverse posterior — how well an answer reconstructs its question — lifts AIME 2024 accuracy by 6.6 points and AMC 2023 by 15 points.

desk verdict Open-source reverse LM is the real news; the verification proof assumes its own central premise, but the paper deserves a serious referee. read the letter →

arxiv 2507.01335 v4 pith:6KNGJJRN submitted 2025-07-02 cs.CL cs.AI

Xunjian Yin , Sitao Cheng , Yuxi Xie , Xinyu Hu , Li Lin , Xinyi Wang , Liangming Pan , William Yang Wang
show 1 more author
Xiaojun Wan
This is my paper · ORCID
classification cs.CLcs.AI
keywords reverselanguagemodelright-to-leftfactorizationnoisychanneldecodingposteriorverificationhallucinationrerankingreversalcurseabductiveinferencemathematicalreasoning
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

The paper's claim is that the left-to-right factorization of language is only half the story. Training a model on the same data, architecture, and tokenizer but predicting tokens right-to-left produces complementary abilities — abductive inference, question synthesis, and a natural grasp of inverse relations — rather than merely a worse forward model. That reverse model then serves a practical purpose: by Bayes' theorem, its posterior $P(\text{question} \mid \text{answer})$ measures whether a candidate answer genuinely reconstructs the original prompt, a verification signal forward likelihood alone cannot supply. Combined with forward likelihood in a noisy-channel score, this 'Reverse Reward' reranks candidate math solutions, and the paper reports that it beats both greedy decoding and random Best-of-N selection across three math-specialized forward models, with gains up to 6.6 points on AIME 2024 and 15 points on AMC 2023. If correct, the contribution is a label-free, parameter-light verifier that transfers across forward models of very different quality.

What carries the argument

The load-bearing object is the reverse autoregressive factorization $P(x) = \prod_{t=1}^{T} P(x_t \mid x_{t+1:T})$, trained by reversing the token sequence and applying a standard causal Transformer; at every position the hidden state encodes what comes after, not what came before. Its downstream instrument is the Reverse Reward score $R(x,y) = P_{\mathrm{FLM}}(y \mid x)^{1-\lambda} \cdot P_{\mathrm{RLM}}(x \mid y)^{\lambda}$, the noisy-channel combination of forward likelihood with the reverse posterior $P_{\mathrm{RLM}}(x \mid y)$, the probability that a candidate response reconstructs the original prompt when read back-to-front. The argument rests on two identities: Bayes' theorem, which reduces the score to $\log P(y \mid x) - \lambda \log P(y) + c$ — forward likelihood regularized by a marginal response-complexity penalty — and the conditional-entropy asymmetry $H(Y \mid X) - H(X \mid Y) = H(Y) - H(X)$, which shows reverse scoring is the tighter, more concentrated evaluation signal whenever responses are longer or more variable than prompts, as in reasoning tasks.

What would settle it

Measure posterior degradation directly at scale: on a math benchmark, sample correct and incorrect candidate solutions from a forward model with matched forward likelihood and record how often the reverse posterior ranks the correct solution higher; then repeat with the reverse reward model applied out-of-domain, for example without the OpenMathInstruct fine-tune or on response distributions unlike its training data. If the ranking rate is near chance, or if the Table 4 gains on AIME 2024 and AMC 2023 disappear under those conditions, posterior verification is not what produced the reported improvement.

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Extended reading notes

Core claim

The paper's central claim is that reverse posterior scoring penalizes hallucination. Proposition 1 states: for a prompt $x$, if a correct response $y^*$ and a hallucinated response $y'$ have comparable forward likelihood $P_{\mathrm{FLM}}(y \mid x)$, and the hallucinated response exhibits 'posterior degradation' — $P_{\mathrm{RLM}}(x \mid y') < P_{\mathrm{RLM}}(x \mid y^*)$ — then the combined bidirectional score $R(x,y) = P_{\mathrm{FLM}}(y \mid x)^{1-\lambda} \cdot P_{\mathrm{RLM}}(x \mid y)^{\lambda}$ ranks the correct response higher for any $\lambda > 0$. Because $\log R(x,y) = \log P(y \mid x) - \lambda \log P(y) + c$, bidirectional scoring is exactly noisy-channel decoding, the information-theoretic scheme that treats a received output as a corrupted rendering of a hidden message and decodes by combining likelihood with a prior; here it is forward likelihood regularized by a marginal-complexity penalty that suppresses generic, prompt-independent responses. The empirical version of the claim is Table 4: Reverse Reward improves accuracy on GSM8K, MATH-500, AIME 2024, and AMC 2023 over greedy decoding and uniform random Best-of-N across DeepSeekMath, QwenMath, and OpenMath2, with the largest gains, up to +6.6 points on AIME 2024 and +15 points on AMC 2023, on competition-level problems. The paper also claims the reverse model itself develops distinct capabilities — abductive inference, question synthesis, backward-from-goal derivation — and resolves the reversal curse naturally, while matching forward models on semantic-understanding tasks at 2B scale.

Load-bearing premise

Everything rests on one empirical premise: that a hallucinated response — one whose reasoning inserts steps absent from the original premises — actually scores lower under the reverse posterior than a correct response with comparable forward likelihood. The paper states this premise behind Proposition 1, points to a single qualitative case (Table 7) as support, and its own Limitations section concedes that the quality of the reverse posterior under distribution shift is not established.

Editorial extensions

If this is right

  • Reverse posterior scoring is a label-free verifier: it adds one reverse-model forward pass per candidate and improves accuracy across forward models whose greedy GSM8K accuracy spans 42.0% to 95.6%, so it complements rather than duplicates forward model quality.
  • Bidirectional scoring is equivalent to forward likelihood with a marginal-complexity penalty $-\lambda \log P(y)$, so the reranker suppresses generic, prompt-independent responses instead of merely favoring fluent or long ones.
  • Finer verification granularity helps on some tasks: step-level beam search with the same score prunes hallucinated derivation paths early and further improves AMC 2023 and GSM8K, though it underperforms on AIME 2024, where pruning long chains can discard partially correct beams.
  • The reverse factorization itself resolves the reversal curse and generates premises from conclusions, making abductive inference, question synthesis, and backward-from-goal derivation available as engineering resources.
  • The two directions fail on different tasks and make different errors on the same tasks; that complementarity, the paper argues, is what makes bidirectional combination fruitful beyond the specific math-verification application demonstrated here.

Reading between the lines

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

  • The noisy-channel argument is not math-specific. A direct extension would test Reverse Reward on other hallucination-prone settings — summarization faithfulness, open-domain factuality, instruction following — where a faithful output should reconstruct its prompt better than a fabricated one; the paper does not run these tests.
  • Posterior degradation is itself a measurable diagnostic: ranking candidate answers by reverse posterior alone, against gold or human labels on a broad suite, would isolate how much of the reranking signal is fidelity to the prompt rather than generic fluency preferences.
  • Because the reverse model is architecture-identical to the forward model, the same GPU pool hosts both directions at roughly equal per-pass cost; whether the verification signal strengthens or saturates as both models scale is an open question the paper explicitly leaves.
  • The safety asymmetry the paper observes — reverse generation completing unsafe prompts that forward safeguards block — implies alignment must be direction-aware; a concrete probe would test whether safety fine-tuning on forward data transfers to a reversed factorization at all.
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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

5 major / 5 minor

Summary. The paper trains LEDOM, a purely reverse autoregressive language model at 2B/7B scale on 435B tokens, with matched forward models on identical architecture and data. It reports behavioral differences (abductive reasoning, question synthesis, reversal-curse resolution) and then proposes Reverse Reward, a bidirectional scoring method that combines forward likelihood P(y|x) with reverse posterior P(x|y) to rerank or beam-search forward generations on math benchmarks. The empirical section reports consistent gains over greedy decoding and Best-of-N random selection on GSM8K, MATH-500, AIME 2024, and AMC 2023, with a formal Proposition 1 claiming that posterior degradation is penalized.

Significance. If the empirical claims survive scrutiny, the contribution is substantial: LEDOM is, to my knowledge, the first open large-scale purely reverse autoregressive model, and the bidirectional scoring idea is a parameter-light verification signal that transfers across forward models of varying quality. The release of models, code, and data is a genuine strength, and the matched FLM/RLM training setup is a clean experimental design for isolating the effect of factorization direction. The consistency of the Table 4 gains across three base models and four benchmarks, including large gains on AIME and AMC, is a meaningful empirical signal. However, the central proof is conditional on an unverified posterior-degradation premise, the SFT data used to train the reverse reward model overlaps with two of the evaluation benchmarks, and key hyperparameters are unreported. These issues currently prevent the paper from establishing the mechanism it claims.

major comments (5)
  1. [Section 5.1, Proposition 1 and Eq. (8)] Proposition 1 is a direct restatement of the posterior-degradation assumption: it proves that if PRLM(x|y') < PRLM(x|y*) at comparable forward likelihood, then R ranks y* above y'. This is formally valid but does not establish that bidirectional scoring penalizes hallucination independently of the assumed premise. Moreover, Eq. (8) shows that at fixed forward likelihood the combined score is log P(y|x) - lambda log P(y) + c, so the ordering is exactly the ordering of -log P(y), i.e., of the reverse posterior through Bayes' rule. The paper should be explicit that the proposition is a conditional guarantee and that the substantive claim is the empirical posterior-degradation hypothesis, not a derived theorem.
  2. [Section 6.4, Remark 1, Table 7] The only direct evidence for posterior degradation is the single qualitative Carla-download example in Table 7. This is insufficient to support the load-bearing premise of Proposition 1 and the interpretation of Table 4. The authors should report a quantitative evaluation on a sample of Best-of-N candidates: e.g., the distribution of PRLM(x|y) for correct versus incorrect responses matched on forward likelihood, with separation statistics and error bars. The Limitations section itself concedes that the posterior approximation under distribution shift is not established, so without such measurements the reported gains cannot be attributed to the reverse-posterior mechanism.
  3. [Section 6.1, Appendix D.2.1, Table 4] The reverse reward model is SFT on 100,000 examples from OpenMathInstruct-2, which is constructed from GSM8K and MATH (the source of MATH-500). This creates a direct overlap between the reward model's fine-tuning distribution and two of the four evaluation benchmarks in Table 4, so the GSM8K and MATH-500 gains may reflect memorization of solution styles or instances rather than a general reconstruction penalty. The authors should quantify the overlap, filter any overlapping instances, or report results on held-out benchmarks (e.g., AIME 2024 and AMC 2023 are less subject to this concern) with the SFT-leakage controlled.
  4. [Section 6.2, Table 4 caption, Appendix D.2.3] Two central reproducibility details are missing or inconsistent. Table 4's caption and Section 6.5 state that Best-of-N samples 64 candidates, while Appendix D.2.3 says N=4 for response-level reranking; these must be reconciled. In addition, the combination weight lambda in Eq. (7) is never reported anywhere in the main text or appendices, even though the method's behavior depends entirely on lambda (lambda=0 reduces R to forward likelihood). The authors should report the exact lambda value(s) used and ideally include a sensitivity analysis over lambda.
  5. [Section 5.2 and Algorithm 1] Step-wise beam search applies R(x,s<=t) to partial sequences, but Eq. (6) defines RRLM for a complete response y. Scoring a partial reasoning chain with a model trained to reconstruct the prompt from a full response is an approximation that is not discussed or validated. Since beam search is presented as a distinct contribution and one table row (OpenMath2 AMC 2023) shows its benefit, the authors should either justify this approximation or provide an ablation comparing step-level and response-level scoring under matched compute.
minor comments (5)
  1. [Table 8 caption] Table 8 is a beam-search case on MATH-500, but its caption says 'A specific case of Best-of-N by Qwen-Math on GSM8K'; the caption should be corrected to match the content.
  2. [Section 2.2, Eq. (4)] The claim that longer or more variable responses imply H(Y)>H(X) is heuristic; entropy is a property of the distribution, not of sequence length alone. The statement should be qualified or supported by an argument about the actual distributions.
  3. [Eq. (6)] The notation in Eq. (6) uses T for the prompt length while conditioning on response tokens y; the index ranges should be made explicit to avoid confusion about what is being summed and conditioned on.
  4. [Appendix D.2.1] The phrase 'precious token prediction' appears to be a typo; the intended term is likely 'prefix token prediction' or 'prior token prediction'.
  5. [Figure 3] Figure 3 reports accuracy of 'FLM' with Reverse Reward beam search, but the base model is not named in the caption or text; it should identify which forward model was used and clarify whether the curve is for MATH-500 or GSM8K, since both appear on different y-axes.

Circularity Check

1 steps flagged · score 6.0 of 10

Proposition 1's 'proof' reduces to its posterior-degradation premise by construction of Eq. (7); benchmark gains are empirical and not themselves circular.

  1. self definitional [Section 5.1, Proposition 1 (Eqs. 7-9)]
    "Proposition 1 (Posterior Verification Penalizes Hallucination). Let y∗ be a correct response and y′ a hallucinated response to prompt x, with comparable forward likelihood: PFLM(y∗|x)≈PFLM(y′|x). If the hallucinated response exhibits posterior degradation, PRLM(x|y′)<PRLM(x|y∗), then for any λ>0: R(x,y∗)>R(x,y′). (9) Proof. The forward terms in Eq. (7) are approximately equal by assumption. The ordering is determined by RRLM(x,y∗)λ > RRLM(x,y′)λ, which follows from posterior degradation and monotonicity of exponentiation for λ>0."

    Equation (7) defines R(x,y) as PFLM(y|x)^(1−λ) · RRLM(x,y)^λ. Under the proposition's assumption PFLM(y∗|x) ≈ PFLM(y′|x), the claimed inequality R(x,y∗)>R(x,y′) is exactly the assumed posterior degradation RRLM(x,y∗)>RRLM(x,y′), transformed by the monotone function z↦z^λ. The proof adds no independent content: the 'prediction' that bidirectional scoring penalizes hallucination is the posterior-degradation premise restated. The paper's Remark 1 defers empirical support to a single qualitative case (Table 7), and the Limitations concede that the posterior approximation under distribution shift 'is not established', so the proposition does not establish that the premise actually holds.

full rationale

The empirical evaluation in Table 4 is not circular: Reverse Reward scores candidates with LEDOM, which was fine-tuned on OpenMathInstruct-2, and the reported gains are measured against greedy decoding and random Best-of-N on held-out math benchmarks; no fitted parameter is renamed as a prediction there. The circularity is confined to the theoretical Proposition 1, whose conclusion is its posterior-degradation assumption restated through the definition of R in Eq. (7). The paper is transparent that the framework's effectiveness 'rests on a testable hypothesis', but the proof supplies no content beyond that definition, and the only offered verification is one qualitative case study. Thus the benchmark contribution stands as an independent empirical result, while the advertised formal guarantee is definitionally forced. This yields partial circularity (score 6) rather than full circularity, because the main empirical claims do not reduce to the assumption.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

Free parameters: the combination weight is the main unreported knob; the candidate count conflicts; beam settings are arbitrary. Axioms: Bayes and the chain rule are standard; the two domain assumptions do the real work, namely posterior fidelity and posterior degradation, both acknowledged as unproven or only anecdotally supported in the paper. The SFT corpus choice is an additional domain assumption because OpenMathInstruct-2 overlaps the evaluated math distributions. No invented entities beyond trained models and a scoring rule, which are methods rather than new ontological posits.

free parameters (3)
  • lambda (posterior weight in Eq. 7) = not reported
    Weight controlling the reverse posterior contribution; no value or tuning protocol is given, so the main results cannot be checked for fitting to evaluation sets.
  • N (number of Best-of-N candidates) = 64 in main text (Table 4 caption); 4 in Appendix D.2.3
    All accuracy numbers in Table 4 depend on N; the contradiction makes the reported gains ambiguous and not reproducible as specified.
  • Beam width k and expansion count n = k=4, n=3 (Appendix D.2.3)
    Step-wise beam search hyperparameters chosen without sensitivity analysis; Figure 3 varies N only.
assumptions (4)
  • standard math The chain rule permits either left-to-right or right-to-left factorization of P(x), and both have the same theoretical optimum.
    Invoked in Sections 2.1 and 2.2 to justify comparing FLM and RLM probabilities.
  • domain assumption The reverse model's posterior approximates the true posterior P(x|y) closely enough for ranking, even under distribution shift.
    Needed for Eq. 7 and Proposition 1; the Limitations section states this approximation quality is not established.
  • domain assumption Hallucinated responses have lower reverse posterior than correct responses at comparable forward likelihood.
    This is the load-bearing premise of Proposition 1; Remark 1 says it is verified empirically, but only a single case is shown in Table 7.
  • domain assumption Fine-tuning the reverse scorer on OpenMathInstruct-2 improves posterior scoring without leaking benchmark answers into the scorer.
    Appendix D.2.1-D.2.2; if the fine-tuning set overlaps the GSM8K/MATH evaluation problems, the gains are inflated. No overlap analysis is given.

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Cite this review

Pith. "Pith review of LEDOM: Reverse Language Model." pith.science (2026). https://pith.science/paper/6KNGJJRN

@misc{pith2026250701335,
  author       = {Pith},
  title        = {Pith review of: LEDOM: Reverse Language Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6KNGJJRN}},
  note         = {Machine review of arXiv:2507.01335}
}
abstract

Autoregressive language models are trained exclusively left-to-right. We explore the complementary factorization, training right-to-left at scale, and ask what reasoning patterns emerge when a model conditions on future context to predict the past. We train LEDOM, an open-source purely reverse autoregressive language model (2B/7B parameters, 435B tokens), and find it develops capabilities distinct from forward models, including abductive inference, question synthesis, and natural resolution of the reversal curse. We then explore one application of the reverse model: combining forward likelihood $P(y \mid x)$ with reverse posterior $P(x \mid y)$ through noisy channel duality. We propose Reverse Reward, which reranks forward outputs using reverse posterior estimates, and prove that bidirectional scoring penalizes hallucinated reasoning chains whose backward reconstruction degrades. Reverse Reward yields gains of up to 6.6\% on AIME 2024 and 15\% on AMC 2023 across multiple strong baselines. We release all models, code, and data here: https://github.com/Arvid-pku/LEDOM.

Figures

Figures reproduced from arXiv: 2507.01335 by the authors.

Figure 1
Figure 1. Forward vs. Reverse Language Modeling. FLMs decompose P(x) left-to-right; our RLM decom￾poses P(x) right-to-left. Both use identical decoder￾only Transformer architectures; only the factorization direction differs. We introduce LEDOM, a purely reverse-trained autoregressive language model pre-trained on 435B tokens at 2B and 7B parameter scales, with matched FLMs sharing identical architecture, tok￾enizer, and data.… view at source ↗
Figure 2
Figure 2. Verification by Inversion. The FLM generates candidate reasoning paths. LEDOM scores each by computing PRLM(query | response). Darker shading = higher posterior score. The final output combines forward likelihood with reverse posterior evaluation. that reverse models can provide unsupervised feed￾back for reranking, we explore using reverse poste￾rior estimates to verify forward outputs, extending the idea with a fo… view at source ↗
Figure 3
Figure 3. Accuracy of FLM with Reverse Reward beam [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Training loss curves comparing LEDOM and FLM. The LEDOM exhibits slower convergence and higher final loss, indicating greater uncertainty during reverse-temporal modeling. multiple-choice questions, given an open book of facts. (The table uses "OpenBookQA", re￾ferring …
Figure 5
Figure 5. Figure 5: An example case of reverse language model evaluation on GSM8K, which includes input and output that [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Full outputs from LEDOM across various NLP tasks. Italicized outputs are partially redacted due to safety concerns. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

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Reference graph

Works this paper leans on

12 extracted references · 9 canonical work pages

  1. [1]

    - 40- Carla downloads at a rate of 2 GB per minute

    **Calculate the time to download 40- The file size is 200 GB. - 40- Carla downloads at a rate of 2 GB per minute. - Time to download 80 GB is 80 2 = 40minutes

    work page

  2. [2]

    **Add the time taken for the restart:** - The restart takes 20 minutes

    work page

  3. [3]

    - Carla downloads at a rate of 2 GB per minute

    **Calculate the time to download the remaining 60- The remaining file size is 200−80 = 120 GB. - Carla downloads at a rate of 2 GB per minute. - Time to download 120 GB is 120 2 = 60minutes

    work page

  4. [4]

    **Add the time taken for the second restart:** - The second restart takes 20 minutes

    work page

  5. [5]

    Therefore, the total time it takes Carla to download the file is140 minutes

    **Sum up all the times:** - Time to download the first 40- Time for the first restart: 20 minutes - Time to download the remaining 60- Time for the second restart: 20 minutes Total time is40 + 20 + 60 + 20 = 140minutes. Therefore, the total time it takes Carla to download the file is140 minutes. Best-of-N by Reverse Reward: To determine the total time it ...

    work page

  6. [10]

    - 40- Carla downloads at a rate of 2 GB per minute

    **First Segment: Downloading 40- The file size is 200 GB. - 40- Carla downloads at a rate of 2 GB per minute. - Time to download 80 GB: 80 2 = 40minutes

    work page

  7. [11]

    **Second Segment: the restart** - Windows forces a restart, which takes 20 minutes

    work page

  8. [12]

    - Carla downloads at a rate of 2 GB per minute

    **Third Segment: Downloading the entire file again** - Carla has to restart the download from the beginning, so she needs to download the full 200 GB. - Carla downloads at a rate of 2 GB per minute. - Time to download 200 GB: 200 2 = 100minutes. Now, we sum up the time for all segments: 40minutes+ 20minutes+ 100minutes= 160minutes Thus, the total time it ...

    work page

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    arXiv preprint arXiv:2110.14168

    Training verifiers to solve math word prob- lems. arXiv preprint arXiv:2110.14168. OpenCompass Contributors. 2023. Opencompass: A universal evaluation platform for foundation models. https://github.com/open-compass/ opencompass. Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kr...

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    arXiv preprint arXiv:2305.20050

    Let’s verify step by step. arXiv preprint arXiv:2305.20050. 10 Todor Mihaylov, Peter Clark, Tushar Khot, and Ashish Sabharwal. 2018. Can a suit of armor conduct elec- tricity? a new dataset for open book question answer- ing. arXiv preprint arXiv:1809.02789. John X. Morris, We...

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    and MAP-Neo (Zhang et al., 2024). Our sam- pling strategy and the specifics of each component are as follows: General-Domain Texts (DGeneral)This compo- nent comprises 284.16 billion tokens randomly sampled from the DCLM-Baseline dataset (Li et al., 2024). DCLM is a benchmark ...

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    In The Thirteenth International Conference on Learning Representations, ICLR 2025, Singapore, April 24-28, 2025

    Openmathinstruct-2: Accelerating AI for math with massive open-source instruction data. In The Thirteenth International Conference on Learning Representations, ICLR 2025, Singapore, April 24-28, 2025. OpenReview.net. Hugo Touvron, Thibaut Lavril, Gautier Izacard, Xavier Martin...

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