| { |
| "paper_id": "masked-diffusion-token-ordering", |
| "paper_title": "Train for the Worst, Plan for the Best: Understanding Token Ordering in Masked Diffusions", |
| "D1": [ |
| { |
| "id": "masked-diffusion-token-ordering-D1-001", |
| "claim": "MDM training uses Adam optimizer with beta1=0.9, beta2=0.95 across all experiments", |
| "source": "Section 3.2, Appendix C.1" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-002", |
| "claim": "MDM training weight decay: 0.1 applied to all model parameters", |
| "source": "Section 3.2, Appendix C.1" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-003", |
| "claim": "Learning rate schedule: cosine decay, max = 4e-4, min = 4e-5", |
| "source": "Section 3.2, Appendix C.1" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-004", |
| "claim": "Training sequence length: L=2048 tokens for standard language modeling setup", |
| "source": "Section 3.2, Appendix C.1" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-005", |
| "claim": "IsoFLOP tokens formula: total tokens = C / (6 * N), where C is FLOPs and N is non-embedding parameters", |
| "source": "Section 3.2, Appendix C.1" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-006", |
| "claim": "L&O-NAE-SAT error analysis: latent variables N = 20, observations P = 280", |
| "source": "Section 3.3, Appendix C.2" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-007", |
| "claim": "L&O-NAE-SAT: pad tokens = 212, pad token value = 2", |
| "source": "Section 3.3, Appendix C.2" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-008", |
| "claim": "L&O-NAE-SAT error analysis: model size = 19M, architecture = MDM with RoPE, max sequence length = 512", |
| "source": "Section 3.3, Appendix C.2" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-009", |
| "claim": "L&O-NAE-SAT error analysis: training iterations = 2000, proxy Bayes iterations = 50000", |
| "source": "Section 3.3, Appendix C.2" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-010", |
| "claim": "L&O-NAE-SAT: masked latent tokens ell = 11, repeat trials = 1000", |
| "source": "Section 3.3, Appendix C.2" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-011", |
| "claim": "L&O-NAE-SAT adaptive inference: 5 (N,P) configurations — (25,275), (30,270), (40,260), (50,250), (100,200)", |
| "source": "Section 4.2, Table 1" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-012", |
| "claim": "Text error analysis: model size = 170M, architecture = MDM pretrained on text", |
| "source": "Section 3.2, Section 3.3, Appendix C.1" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-013", |
| "claim": "Text error analysis: num samples for expectation = 1024", |
| "source": "Section 3.3, Appendix C.2" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-014", |
| "claim": "Text adaptive inference: generator model size = 1.1B, evaluator = LLaMA2-7B", |
| "source": "Section 4.2, Appendix D.1, Figure 3" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-015", |
| "claim": "Sudoku experiment: MDM model size=6M parameters, architecture based on GPT-2 with depth=8, width=256", |
| "source": "Section 4.2, Section 4.3, Appendix D.2" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-016", |
| "claim": "Sudoku: learning rate = 0.001, batch size = 128, epochs = 300", |
| "source": "Section 4.2, Appendix D.2" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-017", |
| "claim": "Sudoku: reverse sampling steps = 50, Gumbel noise coefficient = 0.5", |
| "source": "Section 4.2, Appendix D.2" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-018", |
| "claim": "Zebra: MDM model size = 19M, learning rate = 0.001, batch size = 128, epochs = 300", |
| "source": "Section 4.2, Section 4.3, Appendix D.2" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-019", |
| "claim": "Zebra: reverse sampling steps = 50, Gumbel noise coefficient = 0.5", |
| "source": "Section 4.2, Appendix D.2" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-020", |
| "claim": "LLaDA-8B: model size = 8B, task categories = infilling and instruction-answering", |
| "source": "Section 4.4, Table 4" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-021", |
| "claim": "LLaDA-8B inference strategies: vanilla, top probability, top probability margin", |
| "source": "Section 4.4" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-022", |
| "claim": "Pi-learner: closer swaps = L/10 = 204, much-closer swaps = sqrt(L) = 45, uniform random baseline = L*log(L) = 15606 (all for L = 2048)", |
| "source": "Section 3.2, Appendix C.1" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-023", |
| "claim": "MDM model sizes used across experiments: 6M, 19M, 170M, 1.1B, 8B", |
| "source": "Section 1, Section 3.3, Section 4.2, Section 4.4" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-024", |
| "claim": "ARM baseline model sizes: 42M parameters; evaluator: LLaMA2-7B for perplexity assessment", |
| "source": "Section 4.3, Table 2, Table 3" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-025", |
| "claim": "Sudoku dataset: approximately 1,000,000 hard puzzles, 7 training strategies", |
| "source": "Section 4.2, Section 4.5" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-026", |
| "claim": "Planted CSP NAE-SAT: arity k = 3, vocabulary m = 3, D_KS/k = 64, D_cond/k ≈ 50, N = 10000", |
| "source": "Section 3.1, Example 3.2, Appendix B.4" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-027", |
| "claim": "Planted Coloring: arity k = 2, vocabulary m = 5, D_KS/2 = 16, D_cond/2 ≈ 13.23", |
| "source": "Section 3.1, Appendix B.1" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-028", |
| "claim": "SlimPajama dataset: 627B tokens cleaned and deduplicated", |
| "source": "Section 3.2" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D1-029", |
| "claim": "L&O-NAE-SAT error repeats = 1000, text error samples = 1024", |
| "source": "Section 3.3" |
| } |
| ], |
| "D2": [ |
| { |
| "id": "masked-diffusion-token-ordering-D2-001", |
| "claim": "Forward masking per token: q_{t|0}(x_t^i | x_0^i) = Cat(alpha_t * e_{x_0^i} + (1-alpha_t) * e_0). Each token independently masked to 0 with probability 1-alpha_t.", |
| "source": "Section 2, Eq 1" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-002", |
| "claim": "Reverse transition probability: q_{s|t}(x_s^i | x_t, x_0) = Cat(e_{x_t^i}) if x_t^i != 0; Cat((1-alpha_s)/(1-alpha_t)*e_0 + (alpha_s-alpha_t)/(1-alpha_t)*e_{x_0^i}) if x_t^i = 0. Unmasked tokens stay unchanged; masked tokens sample from mixture.", |
| "source": "Section 2, Eq 2" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-003", |
| "claim": "Denoising network approximation: g_theta(x_s^i | x_t) approximates q_{s|t} by replacing true x_0^i with denoising network prediction p_theta(·|x_t, t). Time-embedding-free: p_theta(·|x_t) as x_t encodes t via number of masked tokens.", |
| "source": "Section 2" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-004", |
| "claim": "MDM training loss (continuous): L_theta = integral_0^1 (alpha_t'/(1-alpha_t)) * E[sum_{i: x_t^i=0} -log p_theta(x_0^i | x_t, t)] dt. Minimizes negative log-likelihood only on masked positions.", |
| "source": "Section 2, Eq 3" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-005", |
| "claim": "MDM loss (discrete mask sum, Prop E.1): L_theta = -sum_{n=1}^{L} E_{x(n)~q_tilde(·|x_0)} [1/n * sum_{l: x^l(n)=0} log p_theta(x_0^l | x(n))]. Sum over all possible numbers n of masked tokens.", |
| "source": "Section 2.1.1, Appendix E.1" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-006", |
| "claim": "MDM loss (all masks weighted, Prop 2.1): L_theta = -sum_{M subseteq [L], i in M} 1/binomial(L,|M|) * 1/|M| * E[log p_theta(x_0^i | x_0[M])]. Sum over ALL possible mask subsets, weight inversely proportional to subset size and count.", |
| "source": "Section 2.1.1, Proposition 2.1, Eq 4" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-007", |
| "claim": "MDM loss = expected any-order autoregressive loss (Appendix E.1): L_theta = -E_{pi~Unif(S_L)} [sum_{j=0}^{L-1} log p_theta(x_0^{pi(j)} | x_0[pi{j,...,L-1}])]. MDM solves exponentially more subproblems than ARM (all orders vs. one).", |
| "source": "Section 2.1.1, Eq 5, Appendix E.1" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-008", |
| "claim": "ARM training loss (left-to-right): log p_theta(x_0) = sum_{i=0}^{L-1} log p_theta(x_0^i | x_0[{i,...,L-1}]). Autoregressive model predicts each token from left to right given causal prefix.", |
| "source": "Section 2.1.1, Eq 6" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-009", |
| "claim": "Pi-learner likelihood: log p_theta(x_0) = sum_{i=0}^{L-1} log p_theta(x_0^{pi(i)} | x_0[pi{i,...,L-1}]). Generalized AR model predicting tokens in fixed permutation order pi; used to measure hardness of different generation orders.", |
| "source": "Section 3.2, Eq 7" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-010", |
| "claim": "Vanilla MDM inference Step (a) — select positions: S subseteq {i | x_t^i = 0}, P(i in S) = (alpha_s - alpha_t) / (1 - alpha_t). Randomly chooses which masked positions to unmask at each reverse step.", |
| "source": "Section 2.1.2, Section 4" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-011", |
| "claim": "Vanilla MDM inference Step (b) — sample tokens: x_s^i ~ p_theta(x^i | x_t) for all i in S. For each selected position, sample new token from denoising network's predicted categorical distribution.", |
| "source": "Section 2.1.2, Section 4" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-012", |
| "claim": "Full vanilla MDM inference loop: start from fully masked x_1 = (0,...,0), iterate from t=1 to t=0 in T discretized steps, at each step randomly select positions S via Bernoulli and sample tokens from p_theta.", |
| "source": "Section 2.1.2, Section 4" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-013", |
| "claim": "Adaptive MDM inference Step (a) — oracle position selection: S = F(theta, x_t) subseteq {i | x_t^i = 0}. Instead of random selection, use oracle F to strategically choose which masked positions to unmask next.", |
| "source": "Section 4" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-014", |
| "claim": "Full adaptive MDM inference loop: x_1 fully masked; for k=T downto 1: S = F(theta, x_t); x_s^i ~ p_theta(x^i|x_t) for i in S. Uses oracle F at each step instead of random selection.", |
| "source": "Section 4" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-015", |
| "claim": "Top Probability oracle: certainty(i) = max_j p_theta(x^i=j | x_t); S = argTopK_{i: x_t^i=0} certainty(i). Selects K masked positions with highest maximum predicted probability.", |
| "source": "Section 4.1" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-016", |
| "claim": "Top Probability Margin oracle: certainty(i) = |p_theta(x^i=j1|x_t) - p_theta(x^i=j2|x_t)| where j1=argmax, j2=argmax_{j!=j1}; S = argTopK certainty(i). Better uncertainty estimate when multiple tokens compete at high probability.", |
| "source": "Section 4.1" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-017", |
| "claim": "Top Probability Margin with Gaussian noise (temperature variant): F(theta, x_t) = TopK_i (|p_theta(x^i=j1|x_t) - p_theta(x^i=j2|x_t)| + epsilon_i), epsilon_i ~ N(0, sigma^2). Adds noise to prevent greedy deterministic sampling, used for text generation diversity.", |
| "source": "Section 4.1, Appendix D.1.2" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-018", |
| "claim": "Number of tokens to unmask per step (deterministic K): K = N_masked * (alpha_s - alpha_t) / (1 - alpha_t) where N_masked = |{i: x_t^i = 0}|. Keeps noise-level marginal consistent with training distribution.", |
| "source": "Section 4.1" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-019", |
| "claim": "Gumbel noise augmented oracle (for puzzles): F(theta, x_t) = TopK_i (certainty(i) + gamma * g_i), g_i ~ Gumbel(0,1), gamma = 0.5. Adds Gumbel noise to certainty scores for controlled stochasticity in Sudoku/Zebra inference.", |
| "source": "Section 4.2, Appendix D.2" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-020", |
| "claim": "Chain rule sampling path likelihood (ideal MDM): p_theta(x_0) = prod_{i=0}^{L-1} p_theta(x_0^{pi(i)} | x_0[pi{i,...,L-1}]) = p_data(x_0) for any permutation pi. Under perfect model, all generation orders yield correct likelihood, justifying exploration of adaptive orders.", |
| "source": "Section 4" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-021", |
| "claim": "Token entropy of generated sequence: H(x) = -sum_j p_j * log p_j where p_j = (# occurrences of token j in x) / L. Used to verify adaptive inference maintains generation diversity comparable to vanilla inference.", |
| "source": "Section 4.2, Figure 3" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-022", |
| "claim": "Belief Propagation variable-to-factor update (Def B.10): MS_c^{i->S}[t+1] proportional to prod_{T: i in T, T!=S} MS_c^{T->i}[t]. Variable i sends to factor S the product of incoming messages from all other factors.", |
| "source": "Appendix B.4" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-023", |
| "claim": "Belief Propagation factor-to-variable update (Def B.10): MS_c^{S->i}[t+1] proportional to sum_{sigma_bar in [m]^{S\\i}} g(sigma_bar union_i c) * prod_{j in S\\i} MS_{sigma_bar_j}^{j->S}[t]. Complexity O(m^{k-1}) per factor update.", |
| "source": "Appendix B.4" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D2-024", |
| "claim": "Belief Propagation marginal estimation: p_hat(sigma_i = c | observations) proportional to prod_{T: i in T} MS_c^{T->i}[converged]. Product of converged factor-to-variable messages, normalized to probability distribution.", |
| "source": "Appendix B.4" |
| } |
| ], |
| "D3": [ |
| { |
| "id": "masked-diffusion-token-ordering-D3-001", |
| "claim": "Pi-learner scaling laws on text (Section 3.2): Measure how model likelihood degrades as token generation order deviates from left-to-right (natural text order) using pi-learners at different permutation distances — identity (ARM), Closer (L/10 swaps), Much-closer (sqrt(L) swaps), uniform random (MDM). IsoFLOP analysis with AdamW, cosine LR, L=2048.", |
| "source": "Section 3.2, Figure 2 (left)" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D3-002", |
| "claim": "Error imbalance on L&O-NAE-SAT (Section 3.3): For each masking size ell in [1, N-1], randomly mask ell latent tokens and ell*(P/N) observation tokens, measure MSE error per position. Compare Bayes-optimal predictor (MDM 50000 iters) vs MDM 2000 iters. N=20, P=280, max_seq=512, 19M MDM with RoPE, 1000 repeats.", |
| "source": "Section 3.3, Figure 2 (right)" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D3-003", |
| "claim": "Error imbalance on text (Section 3.3): Sample permutations pi from three distributions (Unif(S_L), Closer, identity), compute cumulative validation loss E[sum_i log p_theta(x0^{pi(i)} | x0[pi{i,...,L-1}])] using 170M pretrained MDM on 1024 text samples.", |
| "source": "Section 3.3, Figure 2 (right)" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D3-004", |
| "claim": "Adaptive MDM inference on L&O-NAE-SAT (Section 4.2): Compare Top Probability Margin oracle vs vanilla random selection across 5 synthetic (N,P) configurations. Oracle selects which tokens to unmask based on certainty = |p_theta(x^i=j1|xt) - p_theta(x^i=j2|xt)|. 19M MDM trained per configuration.", |
| "source": "Section 4.2, Table 1" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D3-005", |
| "claim": "Adaptive MDM inference on text (Section 4.2): Compare Top Probability Margin oracle with Gaussian noise (temperature) vs vanilla inference. K set to match expected vanilla count. Evaluated via generative perplexity (LLaMA-7B) and token entropy. 1.1B MDM pretrained on text, unconditional generation.", |
| "source": "Section 4.2, Figure 3" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D3-006", |
| "claim": "Sudoku puzzle solving (Sections 4.2-4.3): Compare adaptive MDM inference strategies (Top probability, Top probability margin, both with Gumbel noise gamma=0.5) vs vanilla MDM and ARM (with/without order info via teacher forcing). 6M GPT-2 MDM, 42M ARM, lr=0.001, batch=128, 300 epochs, 50 inference steps. Dataset from Shah et al. 2024 (7-strategy puzzles).", |
| "source": "Section 4.2, Section 4.3, Table 2" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D3-007", |
| "claim": "Zebra (Einstein) puzzle solving (Sections 4.2-4.3): Same adaptive strategies as Sudoku — Top probability, Top probability margin with Gumbel noise. 19M MDM, 42M ARM, lr=0.001, batch=128, 300 epochs, 50 inference steps. Dataset from Shah et al. 2024.", |
| "source": "Section 4.2, Section 4.3, Table 3" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D3-008", |
| "claim": "LLaDA-8B adaptive inference on coding and math (Section 4.4): Evaluate Top Probability and Top Probability Margin oracles vs vanilla on large-scale 8B MDM. Infilling tasks (HumanEval-Single/Multi/Split, ROCStories) use non-autoregressive sampling; instruction-answering tasks (Math, MMLU) use semi-autoregressive with explicit length. Inference-only modification, no additional training.", |
| "source": "Section 4.4, Table 4" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D3-009", |
| "claim": "Easy-to-hard generalization on hard Sudoku (Section 4.5): Test robustness of adaptive MDM inference on ~1M hard puzzles requiring unseen strategies or backtracking. Models trained only on easy 7-strategy puzzles. Compare MDM (vanilla, Top prob, Top margin) vs ARM with teacher-forced ordering. 6M GPT-2 MDM, 42M ARM, same training config.", |
| "source": "Section 4.5, Table 5" |
| } |
| ], |
| "D4": [ |
| { |
| "id": "masked-diffusion-token-ordering-D4-001", |
| "claim": "Phase 1 (Train MDM): Train time-embedding-free p_θ by minimizing loss over all mask subsets M⊆[L] (Proposition 2.1). Equivalent to expected any-order AR loss, solving Θ(L·2^L) subproblems vs L for ARM. (Section 2, Appendix E.1)", |
| "source": "Section 2, Section 2.1.1, Proposition 2.1, Appendix E.1" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D4-002", |
| "claim": "Phase 2 (Diagnose error imbalance): (i) L&O-NAE-SAT: measure MDM per-position MSE vs Bayes-optimal proxy across masking sizes, 1k repeats. (ii) Text: train π-learners at four permutation distances from identity, measure cumulative validation loss. Confirms MDM performance varies across subproblems. (Section 3.2-3.3)", |
| "source": "Section 3.2, Section 3.3, Figure 2" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D4-003", |
| "claim": "Phase 3 (Design oracle for position selection): Implement oracle F(θ,x_t) selecting K masked positions to unmask per reverse step. (A) Top Probability: certainty = max predicted probability. (B) Top Probability Margin: certainty = gap between top-2 probabilities. Augment with Gumbel noise (γ=0.5) for puzzles, Gaussian noise for text diversity. (Section 4.1)", |
| "source": "Section 4.1" |
| }, |
| { |
| "id": "masked-diffusion-token-ordering-D4-004", |
| "claim": "Phase 4 (Adaptive inference loop): Start from fully masked x_1. Per reverse step: S=F(θ,x_t) selects positions by certainty, then sample x_s^i∼p_θ(x^i|x_t) for i∈S. T=50 steps for puzzles. Evaluate via GenPPL, prediction accuracy, puzzle solve rate, and task-specific metrics across 5 domains. (Section 4.2-4.5)", |
| "source": "Section 4.2, Section 4.3, Section 4.4, Section 4.5" |
| } |
| ] |
| } |
