| Algorithm 2 Reward-Guided Post-Training with N1 |
| Require: Pre-trained model p1|t pre, proposal distribution h, reward function R, temperature β |
| Require: Model parameters θ, learning rate η, sequence length L |
| 1: Sample diffusion time t ~ ω(t) | ▷ Sample diffusion time and generate noisy sequence |
| 2: Sample clean sequence x1 ~ h(·|xt) | |
| 3: Generate noisy sequence xt ~ p(·|xt) | |
| 4: for i = 1 to L do | ▷ Compute reward-modulated target distribution |
| 5: p1|tR(x1i|x1≠i, xt) ← p1|t(x1i|x1≠i, xt) exp(R(x1i, xt≠i)/β) | |
| 6: end for | |
| 7: L ← L distrib(θ; N1, D, h) | ▷ Compute loss and update parameters |
| 8: θ ← θ - η∇θL | ▷ Gradient descent step |
| Algorithm 3 Reward-Guided Training with Nfull |
| Require: Pre-trained model p1|t pre, proposal distribution h, reward function R, temperature β |
| Require: Model parameters θ, learning rate η |
| 1: t ~ ω(t) | ▷ Sample diffusion time |
| 2: xt ~ p(xt) | ▷ Sample noise |
| 3: Sample mini-batch {x1,b}Bb=1 ~ h(x1|xt) | ▷ Draw samples from proposal |
| 4: Z ← ∑b=1B exp(R(x1,b)/β) | ▷ Compute normalization |
| 5: wb ← exp(R(x1,b)/β)/Z for b = 1,..., B | ▷ Importance weights |
| 6: L ← ∑b=1B wb log p1|t(x1,b|xt)/pθt(x1,b|xt) | ▷ Weighted objective |
| 7: θ ← θ - η∇θL | ▷ Gradient update |
",
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+ "type": "text",
+ "text": "F.2. Experimental Details and Results",
+ "text_level": 1,
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+ "type": "text",
+ "text": "Synthetic Experiments To assess the effectiveness of our reward function tuning methodology, we conducted experiments using a synthetic dataset. This dataset is structured as a 2D discrete grid, specifically a $128 \\times 128$ grid. Initially, we pre-train a discrete diffusion model, denoted as $p^{\\mathrm{pre}}$ , on this grid using the $\\mathcal{L}_{\\mathrm{distrib}}$ objective with a uniform source distribution. Subsequently, we define a reward function $R$ designed to eliminate modes located in the right half of the grid. Concretely, we assign $R(x) = 0$ for all points $x$ in the left half, and $R(x) = -10^{5}$ for those in the right half. Following this setup, we fine-tune the model using the $\\mathcal{L}_{\\mathrm{distrib}}$ objective with $\\mathcal{N}^{\\mathrm{full}}$ , adhering to the procedure detailed in Alg. 3.",
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+ {
+ "type": "text",
+ "text": "The results of this process are illustrated in Figure 5, which displays the intermediate samples generated by the model both before and after fine-tuning. Initially, during the pre-training phase, the model successfully captures all modes present in the data distribution. However, after applying reward-guided fine-tuning, the model effectively suppresses the modes in the right half of the grid, resulting in final samples that exclusively generate the left half of the grid.",
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+ "type": "text",
+ "text": "Toxicity Mitigation A critical challenge in deploying language models is effectively controlling and mitigating toxic content in their outputs. Although toxic generations occur relatively infrequently, their potential negative impact on users and downstream applications makes this an essential area of research (Singhal et al., 2025). Even a small proportion of toxic outputs can significantly undermine the safety, reliability, and trustworthiness of language models in real-world scenarios.",
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+ {
+ "type": "text",
+ "text": "Our experimental methodology builds upon recent advances in controlled text generation (Zhao et al., 2024a; Rector-Brooks et al., 2024; Singhal et al., 2025). To ensure reproducibility, we conduct our experiments using a standardized story-beginning prompt: \"Once upon a time, there was a\". The foundation of our experimental framework is a pre-trained diffusion model developed in Sec. 4.1, which implements $\\mathcal{L}_{\\mathrm{distrib}}$ with absorbing discrete diffusion. To further enhance the model's capabilities and robustness, we perform comprehensive fine-tuning on the Tinystories dataset (Eldan & Li, 2023). This fine-tuning process utilizes the Adam optimizer with $(\\beta_{1} = 0.9, \\beta_{2} = 0.95)$ and a learning rate of $1 \\times 10^{-4}$ , continuing for 100,000 training steps.",
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+ {
+ "type": "text",
+ "text": "For measuring and controlling toxicity, we implement a sophisticated reward function based on a pre-trained RoBERTa classifier (Logacheva et al., 2022). During our evaluation phase, we employ this classifier as our primary metric for assessing",
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+ {
+ "type": "header",
+ "text": "Target Concrete Score Matching: A Holistic Framework for Discrete Diffusion",
+ "bbox": [
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+ "page_idx": 29
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+ {
+ "type": "page_number",
+ "text": "30",
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+ {
+ "type": "text",
+ "text": "content safety, with outputs scored on a continuous scale from 0 (completely non-toxic) to 1 (highly toxic). This granular scoring system allows for precise measurement of our mitigation strategies' effectiveness.",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "The results of our comprehensive evaluation are presented in Fig. 3, where we analyze two critical metrics: the toxicity score and the generative perplexity of the samples. To assess the quality and coherence of the generated text, we measure perplexity using GPT-2 Large (Radford et al., 2019) as an independent evaluator.",
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+ {
+ "type": "text",
+ "text": "We fine-tune the model using the $\\mathcal{L}_{\\mathrm{distrib}}$ objective with $\\mathcal{N}^{\\mathrm{full}}$ , following the procedure outlined in Alg. 3. To investigate the impact of sampling density, we conduct experiments with varying numbers of Monte Carlo samples $N \\in \\{2,4,8,16\\}$ for estimating the importance weights, with results displayed in Fig. 3. For comparative analysis, we include benchmark results from the pre-trained MDLM (Sahoo et al., 2024) model using Best-of-N sampling with $N \\in \\{4,8\\}$ , as reported in (Singhal et al., 2025).",
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+ {
+ "type": "text",
+ "text": "Our experimental results demonstrate several key findings. First, our approach exhibits superior scaling properties with respect to the number of Monte Carlo samples used for importance weight estimation. Second, our fine-tuning methodology achieves more effective toxicity mitigation compared to the pre-trained MDLM model, even when the latter employs Best-of-N sampling techniques. Notably, since our approach is based on fine-tuning rather than inference-time scaling, it eliminates the need for multiple reward function evaluations during inference, resulting in reduced computational overhead and improved efficiency in practical applications.",
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+ "type": "text",
+ "text": "G. TCSM Post-training with Preference Optimization",
+ "text_level": 1,
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+ "type": "text",
+ "text": "G.1. Detailed Algorithm",
+ "text_level": 1,
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+ "type": "text",
+ "text": "Problem Setting We introduce a methodology for fine-tuning pre-trained diffusion models using pairwise preference data, denoted as $\\{(\\mathbf{q},\\mathbf{x}_1^w,\\mathbf{x}_1^l)\\}$ . In this formulation, $\\mathbf{q}$ represents a query or instruction, while $\\mathbf{x}_1^w$ and $\\mathbf{x}_1^l$ represent the preferred (winning) and non-preferred (losing) responses, respectively.",
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+ "text": "The underlying preferences are assumed to emerge from a latent reward model that is not directly observable. Among various approaches for modeling such preferences, we adopt the widely-recognized Bradley-Terry (BT) model (Bradley & Terry, 1952). This model provides an elegant framework for capturing human preference distributions. Specifically, the BT model expresses the probability of one response being preferred over another as:",
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+ "type": "equation",
+ "text": "\n$$\np ^ {*} \\left(\\mathbf {x} _ {1} ^ {w} \\succ \\mathbf {x} _ {1} ^ {l} \\mid \\mathbf {q}\\right) = \\frac {\\exp \\left(R ^ {*} \\left(\\mathbf {q} , \\mathbf {x} _ {1} ^ {w}\\right)\\right)}{\\exp \\left(R ^ {*} \\left(\\mathbf {q} , \\mathbf {x} _ {1} ^ {w}\\right)\\right) + \\exp \\left(R ^ {*} \\left(\\mathbf {q} , \\mathbf {x} _ {1} ^ {l}\\right)\\right)} \\tag {63}\n$$\n",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "where $R^{*}(\\mathbf{q},\\mathbf{x})$ represents the underlying reward function that quantifies the quality of response $\\mathbf{x}$ given query $\\mathbf{q}$ .",
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+ "type": "text",
+ "text": "Building on this foundation, we define our target distribution to emphasize preferred responses. This distribution can be formally expressed as:",
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+ "type": "equation",
+ "text": "\n$$\np _ {\\text {t a r g e t}} \\left(\\mathbf {x} _ {1} \\mid \\mathbf {q}\\right) := p _ {1} \\left(\\mathbf {x} _ {1} ^ {w} \\mid \\mathbf {q}\\right) := p _ {1} \\left(\\mathbf {x} _ {1} \\text {i s w i n n e r} \\mid \\mathbf {q}\\right) = p _ {1} \\left(\\mathbf {x} _ {1} \\mid \\mathbf {q}\\right) \\sum_ {\\mathbf {y} _ {1}} p _ {1} \\left(\\mathbf {y} _ {1} \\mid \\mathbf {q}\\right) p ^ {*} \\left(\\mathbf {x} _ {1} \\succ \\mathbf {y} _ {1} \\mid \\mathbf {q}\\right), \\tag {64}\n$$\n",
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+ "type": "text",
+ "text": "For practical implementation, we leverage a pre-trained diffusion model $p_{1|t}^{\\mathrm{pre}}(\\mathbf{x}_1|\\mathbf{q})$ as our reference distribution, which serves as the starting point for our fine-tuning process.",
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+ "type": "text",
+ "text": "Based on the TCSM with density ratio estimation approach in Sec. 5.1, we learn a new diffusion model $p_{1|t}^{\\theta}$ relative to the pre-trained reference. The detailed algorithm is shown in Alg. 4, where we use BCE loss to estimate the density ratio as an example.",
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+ "type": "header",
+ "text": "Target Concrete Score Matching: A Holistic Framework for Discrete Diffusion",
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+ "type": "page_number",
+ "text": "31",
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+ "type": "code",
+ "sub_type": "algorithm",
+ "code_caption": [],
+ "code_body": "Algorithm 4 Preference Optimization with TCSM using BCE loss \nRequire: Pre-trained diffusion model $p_{1|t}^{\\mathrm{pre}}$ \nRequire: Preference dataset $\\mathcal{D} = \\{(c,\\mathbf{x}^w,\\mathbf{x}^l)\\}$ \nRequire: Model parameters $\\theta$ , learning rate $\\eta$ , time distribution $\\omega (t)$ , coefficient $\\beta$ \n1: for each training iteration do \n2: $t\\sim \\omega (t)$ ▷ Sample diffusion time \n3: $(\\mathbf{c},\\mathbf{x}^{w},\\mathbf{x}^{l})\\sim \\mathcal{D}$ ▷ Sample preference triplet \n4: $\\mathbf{x}_t^w\\sim p_{t|1}(\\cdot |\\mathbf{x}_1^w)$ ▷ Sample noisy sequence for preferred response \n5: $\\mathbf{x}_t^l\\sim p_{t|1}(\\cdot |\\mathbf{x}_1^l)$ ▷ Sample noisy sequence for non-preferred response \n6: ▷ Compute density ratios for preferred and non-preferred responses \n7: $r_{1|t}^{w}(\\mathbf{c})\\gets \\frac{p_{1|t}^{\\theta}(\\mathbf{x}^{w}|\\mathbf{c})}{\\beta p_{1|t}^{\\mathrm{pre}}(\\mathbf{x}^{w}|\\mathbf{c})}$ \n8: $r_{1|t}^{l}(\\mathbf{c})\\gets \\frac{p_{1|t}^{\\theta}(\\mathbf{x}^{l}|\\mathbf{c})}{\\beta p_{1|t}^{\\mathrm{pre}}(\\mathbf{x}^{l}|\\mathbf{c})}$ \n9: ▷ Compute loss \n10: $\\mathcal{L}\\gets -\\log \\frac{r_{1|t}^{w}(\\mathbf{c})}{1 + r_{1|t}^{w}(\\mathbf{c})} -\\log \\frac{1}{1 + r_{1|t}^{l}(\\mathbf{c})}$ \n11: $\\theta \\leftarrow \\theta -\\eta \\nabla_{\\theta}\\mathcal{L}$ ▷ Update model parameters \n12: end for",
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+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "G.2. Experimental Details and Results",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "To evaluate the effectiveness of preference optimization, we employed the IMDB-sentiment dataset (Maas et al., 2011) as our primary evaluation benchmark, with the SiEBERT model (Hartmann et al., 2023) serving as our reward function. For training data, we utilized a carefully curated preference dataset constructed in prior work (Rafailov et al., 2023; Wang et al., 2023). As our foundation model, we used the pre-trained model from Sec. 4.1, which had been extensively trained on the OPENWEBTEXT dataset.",
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+ "type": "text",
+ "text": "The fine-tuning process implemented our density ratio estimation framework, as detailed in Sec. 5.1, with Binary Cross-Entropy (BCE) loss serving as our optimization objective. We adopted parameterization strategy (i) from Sec. 5.1, which defines the density ratio as:",
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+ "type": "equation",
+ "text": "\n$$\nr _ {1 | t} ^ {\\phi : = \\theta} \\left(x _ {1} ^ {i} \\mid \\mathbf {x} _ {1} ^ {\\neq i}, \\mathbf {x} _ {t}\\right) = \\frac {p _ {1 | t} ^ {\\theta} \\left(x _ {1} ^ {i} \\mid \\mathbf {x} _ {1} ^ {\\neq i} , \\mathbf {x} _ {t}\\right)}{\\beta p _ {1 | t} ^ {\\mathrm {r e f}} \\left(x _ {1} ^ {i} \\mid \\mathbf {x} _ {1} ^ {\\neq i} , \\mathbf {x} _ {t}\\right)} \\tag {65}\n$$\n",
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+ "type": "text",
+ "text": "Here, the coefficient $\\beta$ plays a crucial role in balancing two competing objectives: maximizing preference reward optimization while maintaining fidelity to the original pre-trained model. The complete training procedure is outlined in Alg. 4.",
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+ "type": "text",
+ "text": "Our training protocol consisted of 10 full epochs with a batch size of 256. We employed the Adam optimizer with a learning rate of $1 \\times 10^{-5}$ and weight decay of $1 \\times 10^{-5}$ . To ensure stable training, we implemented a linear learning rate warmup for the first $10\\%$ of training steps, with momentum parameters $\\beta_{1} = 0.9$ and $\\beta_{2} = 0.95$ . The noise schedule remained consistent with that of the pre-trained model to maintain continuity in the diffusion process.",
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+ "type": "text",
+ "text": "To thoroughly investigate the effects of preference optimization, we conducted experiments across a range of $\\beta$ values: $\\{0.1, 0.5, 1, 5\\}$ . Our evaluation focused on two key metrics: the mean reward achieved by the fine-tuned model and the entropy of generated samples. As shown in Fig. 2, we observed that models with stronger preference optimization (higher $\\beta$ values) achieved both higher mean rewards and lower sample entropy. This suggests that our approach improves alignment with desired preferences but also leads to less diverse generation of preferred samples.",
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+ "type": "text",
+ "text": "H. TCSM Post-training with AR $\\rightarrow$ Diffusion Distillation",
+ "text_level": 1,
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+ "type": "text",
+ "text": "Problem setting In this case, we assume we have a pre-trained autoregressive model $p_1^{\\mathrm{AR}}(\\mathbf{x}_1)$ trained on the target distribution $p_1(\\mathbf{x}_1)$ , and we show that we can use TCSM to distill it to a diffusion model $p_1^\\theta (\\mathbf{x}_1)$ . Note that this deviates from the regular diffusion models setting, that we have the knowledge of the target distribution $p_1(\\mathbf{x}_1)\\approx p^{\\mathrm{AR}}(\\mathbf{x}_1)$ , and we can use it as a teacher model. In this section, we set the target distribution to be the AR teacher model distributoin",
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+ "type": "header",
+ "text": "Target Concrete Score Matching: A Holistic Framework for Discrete Diffusion",
+ "bbox": [
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+ {
+ "type": "page_number",
+ "text": "32",
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+ {
+ "type": "text",
+ "text": "$p_1(\\mathbf{x}_1) \\coloneqq p_1^{\\mathrm{AR}}(\\mathbf{x}_1)$ . And akin to classical knowledge distillation, we are interested in how to distill the knowledge from the AR teacher model to the diffusion student model.",
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+ "type": "text",
+ "text": "TCSM objectives for distillation We show that our TCSM objectives can naturally integrate the knowledge of the AR teacher model into the training objective.",
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+ "type": "text",
+ "text": "We have",
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+ {
+ "type": "equation",
+ "text": "\n$$\np _ {1 \\mid t} (\\mathbf {x} _ {1} | \\mathbf {x} _ {t}) = \\frac {p _ {1} ^ {\\mathrm {A R}} (\\mathbf {x} _ {1}) p _ {t \\mid 1} (\\mathbf {x} _ {t} | \\mathbf {x} _ {1})}{\\sum_ {\\mathbf {x} _ {1}} p _ {1} ^ {\\mathrm {A R}} (\\mathbf {x} _ {1}) p _ {t \\mid 1} (\\mathbf {x} _ {t} | \\mathbf {x} _ {1})}. \\tag {66}\n$$\n",
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+ "type": "text",
+ "text": "We can also use $p_1^{\\mathsf{AR}}(\\mathbf{x}_1)$ to estimate",
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+ "type": "equation",
+ "text": "\n$$\np _ {1 \\mid t} \\left(x _ {1} ^ {i} \\mid \\mathbf {x} _ {1} ^ {\\neq i}, \\mathbf {x} _ {t}\\right) = \\frac {p _ {1} ^ {\\mathrm {A R}} \\left(x _ {1} ^ {i} , \\mathbf {x} _ {1} ^ {\\neq i}\\right) p _ {t \\mid 1} \\left(\\mathbf {x} _ {t} \\mid x _ {1} ^ {i} , \\mathbf {x} _ {1} ^ {\\neq i}\\right)}{\\sum_ {y _ {1} ^ {i}} p _ {1} ^ {\\mathrm {A R}} \\left(y _ {1} ^ {i} , \\mathbf {x} _ {1} ^ {\\neq i}\\right) p _ {t \\mid 1} \\left(\\mathbf {x} _ {t} \\mid y _ {1} ^ {i} , \\mathbf {x} _ {1} ^ {\\neq i}\\right)}. \\tag {67}\n$$\n",
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+ "type": "text",
+ "text": "Both score-based and distribution-based TCSM objectives can be used to distill the AR teacher model to the diffusion student model, we use the distribution-based TCSM objective in our experiments and assume it is the default setting in following discussions.",
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+ "type": "text",
+ "text": "Efficient estimation of distillation target To optimize the TCSM objective, we need to compute the distillation target $p_1^{\\mathrm{AR}}(\\mathbf{x}_1)$ . Naively, this requires $(V - 1) \\times L + 1$ likelihood evaluations of the teacher autoregressive model for each sequence $\\mathbf{y} \\in \\mathcal{N}^1(\\mathbf{x})$ . Even though that the likelihood evaluation can be done in parallel for the autoregressive model, this procedure is still computationally prohibitive. To address this challenge, we introduce two approaches to efficiently estimate the target concrete score, Top-K estimation and First-order Taylor estimation.",
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+ "type": "text",
+ "text": "Top- $K$ approximation Our empirical analysis reveals that distribution $p_{1|t}(x_1^i | \\mathbf{x}_1^{\\neq i}, \\mathbf{x}_t)$ are naturally sparse. As illustrated in Fig. 6, tokens with high density ratios closely resemble the one-hot encoding of original tokens in the simplex space, but enriched with distributional information. This observation motivates approximating the score vector with only the top- $K$ items, treating the rest as zero, for efficient computation. We leverage this property to propose an efficient top- $K$ approximation that reduces computational complexity from $O(VL)$ to $O(KL)$ by considering only the $K$ most probable tokens at each position. This approximation can be efficiently implemented using batched forward passes and proves effective even with $K \\leq 128$ - for detailed implementation and the complete algorithm, we refer readers to Alg. 5 in the appendix.",
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+ "text": "First-order Taylor approximation We leverage the fact that autoregressive language models, despite operating on discrete tokens, are differentiable functions that can be approximated using Taylor expansion. For sequences that differ by only one position, we can efficiently estimate the likelihood ratio using first-order Taylor approximation: $\\log p_{1|t}(y_1^i,\\mathbf{x}_1^{\\neq i}|\\mathbf{x}_t)\\approx$ $\\log p_{1|t}(x_1^i,\\mathbf{x}_1^{\\neq i}|\\mathbf{x}_t) + \\nabla_{\\mathbf{e}_{\\mathbf{x}_1}}\\log p_{1|t}(\\mathbf{x}_1|\\mathbf{x}_t)^\\top (\\mathbf{e}_{\\mathbf{y}_1} - \\mathbf{e}_{\\mathbf{x}_1})$ . This gradient-based estimation requires just one forward and backward pass through the teacher model; for detailed derivations and implementation, please refer to Alg. 7.",
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+ "type": "text",
+ "text": "Experimental To validate our distillation approach, we conducted comprehensive experiments focusing on language modeling capabilities using the OPENWEBTEXT dataset. Our experimental setup involved two key components: a teacher model and a student model. For the teacher, we pre-trained a transformer-based autoregressive model following the architectural configurations described in (Sahoo et al., 2024). As our student model, we employed an absorbing discrete diffusion model.",
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+ "type": "text",
+ "text": "The training process utilized our Top-K estimation strategy with $K = 128$ , training the student model from scratch. To assess performance, we tracked the validation negative log-likelihood (NLL) loss on the OPENWEBTEXT dataset, which we visualize in Figure Fig. 4. The empirical results demonstrate two significant findings: First, our distillation approach substantially accelerates the student model's learning trajectory compared to standard training. Second, and perhaps more importantly, models trained with our distillation loss consistently achieve lower perplexity scores than baseline approaches throughout the entire training process, indicating improved model quality.",
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+ {
+ "type": "header",
+ "text": "Target Concrete Score Matching: A Holistic Framework for Discrete Diffusion",
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+ "text": "33",
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+ {
+ "type": "image",
+ "img_path": "images/a68ba04584894f14c65f98eb4d577ff6005fb246526d898dd1d0d3f743a7519f.jpg",
+ "image_caption": [
+ "Figure 6: Visualization of the concrete score for sequence \"I traveled to South Carolina last summer\". The x-axis represents the position in the sequence, and the y-axis represents the log-probability ratio. The red line represents the original token, and the blue lines represent the top-K tokens with the highest log-probability ratios. The concrete score is highly sparse, with most of the probability mass concentrated on a few tokens."
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ "page_idx": 33
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+ {
+ "type": "text",
+ "text": "I. Connection to Continuous Target Score Matching",
+ "text_level": 1,
+ "bbox": [
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+ },
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+ "type": "text",
+ "text": "In this section, we elaborate on the relationship between the proposed Target Concrete Score Matching (TCSM) framework for discrete data and the established Target Score Matching (TSM) objective (Bortoli et al., 2024) used in continuous diffusion models. We first briefly review TSM in the context of language modeling via continuous diffusion and then demonstrate how TCSM can be viewed as its discrete analogue under certain approximations.",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "Continuous diffusion models for language often operate in a continuous embedding space. Let $\\mathbf{x}_1 = [x_1^1, \\ldots, x_1^L]$ be a discrete sequence from the vocabulary $\\mathcal{X} = \\{1, \\ldots, V\\}$ . Let $\\mathbf{E} \\in \\mathbb{R}^{d \\times V}$ be a word embedding matrix, where $d$ is the embedding dimension. The one-hot vector for token $k$ is $\\mathbf{e}_k \\in \\{0, 1\\}^V$ . The embedding for token $x_1^l$ is $\\mathbf{E}^\\top \\mathbf{e}_{x_1^l}$ . The forward noisng process typically acts independently on these embeddings:",
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+ {
+ "type": "equation",
+ "text": "\n$$\nq _ {t \\mid 1} (\\mathbf {z} _ {t} | \\mathbf {x} _ {1}) = \\prod_ {l = 1} ^ {L} q _ {t \\mid 1} \\left(\\mathbf {z} _ {t} ^ {l} \\mid x _ {1} ^ {l}\\right) = \\prod_ {l = 1} ^ {L} \\mathcal {N} \\left(\\mathbf {z} _ {t} ^ {l}; \\alpha_ {t} \\mathbf {E} ^ {\\top} \\mathbf {e} _ {x _ {1} ^ {l}}, \\sigma_ {t} ^ {2} \\mathbf {I} _ {d}\\right), \\tag {68}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "type": "text",
+ "text": "where $(\\mathbf{z}_t^l)_{l=1}^L$ forms the sequence of noisy embeddings $\\mathbf{z}_t \\in \\mathbb{R}^{L \\times d}$ , and $\\alpha_t, \\sigma_t$ are schedule parameters. The goal is to learn the score function $\\nabla_{\\mathbf{z}_t} \\log q_t(\\mathbf{z}_t)$ of the marginal distribution $q_t(\\mathbf{z}_t) = \\int q_{t|1}(\\mathbf{z}_t | \\mathbf{x}_1) q_1(\\mathbf{x}_1) d\\mathbf{x}_1$ .",
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+ {
+ "type": "text",
+ "text": "Target Score Matching (TSM) provides an objective when the score of the clean data distribution, $\\nabla_{\\mathbf{z}_1}\\log p_1(\\mathbf{z}_1)$ (where $\\mathbf{z}_1$ represents the clean embeddings and $p_1$ is a density over them), is known or can be estimated. The following identity connects the noisy score to the clean score:",
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+ {
+ "type": "text",
+ "text": "Lemma I.1 (Target Score Matching Identity, adapted from (Bortoli et al., 2024)). Let $q_{t|1}(\\mathbf{z}_t|\\mathbf{z}_1) = \\mathcal{N}(\\mathbf{z}_t; \\alpha_t\\mathbf{z}_1, \\sigma_t^2\\mathbf{I})$ define the forward process conditioned on clean continuous data $\\mathbf{z}_1$ , and let $p_1(\\mathbf{z}_1)$ be a differentiable distribution over $\\mathbf{z}_1$ . Then, the score of the noisy marginal $q_{t}(\\mathbf{z}_{t}) = \\int q_{t|1}(\\mathbf{z}_{t}|\\mathbf{z}_{1})p_{1}(\\mathbf{z}_{1})d\\mathbf{z}_{1}$ is given by:",
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+ {
+ "type": "equation",
+ "text": "\n$$\n\\nabla_ {\\mathbf {z} _ {t}} \\log q _ {t} (\\mathbf {z} _ {t}) = \\frac {1}{\\alpha_ {t}} \\mathbb {E} _ {q _ {1 | t} \\left(\\mathbf {z} _ {1} \\mid \\mathbf {z} _ {t}\\right)} \\left[ \\nabla_ {\\mathbf {z} _ {1}} \\log p _ {1} (\\mathbf {z} _ {1}) \\right], \\tag {69}\n$$\n",
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+ {
+ "type": "header",
+ "text": "Target Concrete Score Matching: A Holistic Framework for Discrete Diffusion",
+ "bbox": [
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+ "page_idx": 33
+ },
+ {
+ "type": "page_number",
+ "text": "34",
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+ "type": "code",
+ "sub_type": "algorithm",
+ "code_caption": [],
+ "code_body": "Algorithm 5 Top-K Estimation \n1: procedure tcs Estimate(xo, teacher_model, L, V, K, tcs) \n2: $\\triangleright x_0$ Input tokens; $L$ : Sequence length; $V$ : Vocabulary size; $K$ : Top- $K$ tokens to select; tcs: list \n3: logits $\\leftarrow$ teacher_model(xo) $\\in \\mathbb{R}^{V\\times L}$ ; original_log_prob $\\leftarrow$ teacher_model_log_prob(xo) \n4: for $l = 1$ to $L$ do \n5: Get top- $K$ tokens: top_tokens $\\leftarrow$ TopK(logits[:, l], K) \n6: If $\\mathbf{x}_0[l]\\notin$ top_tokens, add it to top_tokens \n7: Construct a batch of new sequences $\\widehat{\\mathbf{x}}_0\\gets [\\mathbf{x}_0^{< l},\\mathrm{top\\_tokens},\\mathbf{x}_0^{>l}]$ \n8: Compute log probability of sequences log_prob from new_logs $\\leftarrow$ teacher_model(xo) \n9: Compute log-density ratio: log_density_ratio $\\leftarrow$ log_prob - orig_log_prob \n10: Append log-density ratio to list: tcs $\\leftarrow$ tcs + log_density_ratio \n11: end for \n12: return tcs \n13: end procedure",
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+ "type": "code",
+ "sub_type": "algorithm",
+ "code_caption": [
+ "Algorithm 6 Top-K with N-Gram Estimation"
+ ],
+ "code_body": "1: procedure tcs Estimate $(\\mathbf{x}_1$ , teacher_model, ngram_model, $L,V,K$ , tcs) \n2: $\\triangleright x_{1}$ : Input tokens; $L$ : Sequence length; $V$ : Vocabulary size; $K$ : Top- $K$ tokens to select; tcs: list \n3: logits $\\leftarrow$ teacher_model $(\\mathbf{x}_1)\\in \\mathbb{R}^{V\\times L}$ ; original_log_prob $\\leftarrow$ teacher_model_log_prob $(\\mathbf{x}_1)$ \n4: for $l = 1$ to $L$ do \n5: Get top- $K$ tokens: top_tokens $\\leftarrow$ TopK(logits[,l], $K$ ) \n6: Get N-Gram score for all tokens: n-gram Scores $\\leftarrow$ ngram_model([x1+1,...,x1+N-1]) \n7: Add another top- $K$ tokens: top_tokens $\\leftarrow$ top_tokens + TopK(n-gram Scores, $K$ ) \n8: If $\\mathbf{x}_1[l]\\notin$ top_tokens, add it to top_tokens \n9: Construct a batch of new sequences $\\widehat{\\mathbf{x}}_1\\gets [\\mathbf{x}_1^{< l},\\mathrm{top\\_tokens},\\mathbf{x}_1^{>l}]$ \n10: Compute log probability of sequences log_prob from new_logits $\\leftarrow$ teacher_model(x1) \n11: Compute log-density ratio: log-density_ratio $\\leftarrow$ log_prob - orig_log_prob \n12: Append log-density ratio to list: tcs $\\leftarrow$ tcs + log_density_ratio \n13: end for \n14: return tcs \n15: end procedure",
+ "bbox": [
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+ "page_idx": 34
+ },
+ {
+ "type": "text",
+ "text": "where $q_{1|t}(\\mathbf{z}_1|\\mathbf{z}_t)$ is the posterior distribution.",
+ "bbox": [
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+ "page_idx": 34
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+ {
+ "type": "text",
+ "text": "Proof. The proof follows standard arguments, e.g., in Bortoli et al. (2024), adapted for the scaling factor $\\alpha_{t}$ . Using the property $\\nabla_{\\mathbf{z}_1} \\log q_{t|1}(\\mathbf{z}_t|\\mathbf{z}_1) = -\\alpha_t \\nabla_{\\mathbf{z}_t} \\log q_{t|1}(\\mathbf{z}_t|\\mathbf{z}_1)$ and Bayes' rule $q_{t|1}(\\mathbf{z}_t|\\mathbf{z}_1) = q_{1|t}(\\mathbf{z}_1|\\mathbf{z}_t) q_t(\\mathbf{z}_t) / p_1(\\mathbf{z}_1)$ , we take gradients w.r.t. $\\mathbf{z}_1$ : $\\nabla_{\\mathbf{z}_1} \\log q_{t|1}(\\mathbf{z}_t|\\mathbf{z}_1) = \\nabla_{\\mathbf{z}_1} \\log q_{1|t}(\\mathbf{z}_1|\\mathbf{z}_t) - \\nabla_{\\mathbf{z}_1} \\log p_1(\\mathbf{z}_1)$ . Combining these yields $\\nabla_{\\mathbf{z}_t} \\log q_{t|1}(\\mathbf{z}_t|\\mathbf{z}_1) = -\\frac{1}{\\alpha_t} (\\nabla_{\\mathbf{z}_1} \\log q_{1|t}(\\mathbf{z}_1|\\mathbf{z}_t) - \\nabla_{\\mathbf{z}_1} \\log p_1(\\mathbf{z}_1))$ . Finally, taking the expectation w.r.t. $q_{1|t}(\\mathbf{z}_1|\\mathbf{z}_t)$ : $\\nabla_{\\mathbf{z}_t} \\log q_t(\\mathbf{z}_t) = \\mathbb{E}_{q_{1|t}(\\mathbf{z}_1|\\mathbf{z}_t)}[\\nabla_{\\mathbf{z}_t} \\log q_{t|1}(\\mathbf{z}_t|\\mathbf{z}_1)] = -\\frac{1}{\\alpha_t} \\mathbb{E}_{q_{1|t}}[\\nabla_{\\mathbf{z}_1} \\log q_{1|t}] + \\frac{1}{\\alpha_t} \\mathbb{E}_{q_{1|t}}[\\nabla_{\\mathbf{z}_1} \\log p_1(\\mathbf{z}_1)]$ . Since $\\mathbb{E}_{q_{1|t}}[\\nabla_{\\mathbf{z}_1} \\log q_{1|t}] = \\int \\nabla_{\\mathbf{z}_1} q_{1|t}(\\mathbf{z}_1|\\mathbf{z}_t) d\\mathbf{z}_1 = 0$ (assuming boundary conditions), the identity holds.",
+ "bbox": [
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+ "page_idx": 34
+ },
+ {
+ "type": "text",
+ "text": "Using Lemma I.1, a score network $\\mathbf{s}_{\\theta}(\\mathbf{z}_t,t)$ can be trained by minimizing the TSM loss:",
+ "bbox": [
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+ "page_idx": 34
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+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathcal {L} _ {\\mathrm {T S M}} (\\theta) = \\mathbb {E} _ {t \\sim U (0, 1)} \\mathbb {E} _ {p _ {1} (\\mathbf {z} _ {1}) q _ {t | 1} (\\mathbf {z} _ {t} | \\mathbf {z} _ {1})} \\left\\| \\mathbf {s} _ {\\theta} (\\mathbf {z} _ {t}, t) - \\frac {1}{\\alpha_ {t}} \\nabla_ {\\mathbf {z} _ {1}} \\log p _ {1} (\\mathbf {z} _ {1}) \\right\\| _ {2} ^ {2}. \\tag {70}\n$$\n",
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+ {
+ "type": "text",
+ "text": "Alternatively, using the mean prediction parameterization $\\pmb{\\mu}_{\\theta}(\\mathbf{z}_t,t)\\approx \\mathbb{E}_{q_{1|t}(\\mathbf{z}_1|\\mathbf{z}_t)}[\\mathbf{z}_1]$ , and Tweedie's formula $\\mathbb{E}_{q_{1|t}(\\mathbf{z}_1|\\mathbf{z}_t)}[\\mathbf{z}_1] = \\frac{1}{\\alpha_t} (\\sigma_t^2\\nabla_{\\mathbf{z}_t}\\log q_t(\\mathbf{z}_t) + \\mathbf{z}_t)$ , the TSM objective becomes equivalent to minimizing (up to scaling by $\\lambda_{t} = \\alpha_{t}^{2} / \\sigma_{t}^{2}$ ):",
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+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathcal {L} _ {\\mathrm {T S M}} ^ {\\boldsymbol {\\mu}} (\\theta) = \\mathbb {E} _ {t \\sim U (0, 1)} \\mathbb {E} _ {p _ {1} (\\mathbf {z} _ {1}) q _ {t | 1} (\\mathbf {z} _ {t} | \\mathbf {z} _ {1})} \\left\\| \\boldsymbol {\\mu} _ {\\theta} (\\mathbf {z} _ {t}, t) - \\left(\\frac {\\sigma_ {t} ^ {2}}{\\alpha_ {t}} \\nabla_ {\\mathbf {z} _ {1}} \\log p _ {1} (\\mathbf {z} _ {1}) + \\frac {1}{\\alpha_ {t}} \\mathbf {z} _ {t}\\right) \\right\\| _ {2} ^ {2}. \\tag {71}\n$$\n",
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+ {
+ "type": "header",
+ "text": "Target Concrete Score Matching: A Holistic Framework for Discrete Diffusion",
+ "bbox": [
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+ "page_idx": 34
+ },
+ {
+ "type": "page_number",
+ "text": "35",
+ "bbox": [
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+ "type": "code",
+ "sub_type": "algorithm",
+ "code_caption": [],
+ "code_body": "Algorithm 7 Concrete Score Estimation with first-order Taylor approximation \n1: procedure tcs Estimate(teacher_model, tokens, $V,\\tau$ 2: $\\triangleright$ tokens: Input tokens of shape $(B,L)$ . $V$ : Vocabulary size; $\\tau$ : Temperature \n3: $\\mathbf{x}_1\\gets$ one-hot(tokens, $V$ ) Convert to one-hot vectors \n4: Enable gradient computation for $\\mathbf{x}_1$ \n5: logits $\\leftarrow$ teacher_model(x1) \n6: log_prob $\\leftarrow$ log softmax(logits) \n7: log_prob $\\leftarrow \\sum (\\mathbf{x}_1[:,1:,:\\cdot ]\\cdot \\log\\_prob[:,:-1,:])$ \n8: Compute gradient: grad_log_prob $\\leftarrow \\nabla_{\\mathbf{x}_1}\\log\\_prob$ \n9: $\\triangleright$ Compute log-density ratios \n10: log_prob_ratio $\\leftarrow$ grad_log_prob - $\\sum_{\\mathrm{dim} = -1}(\\mathbf{x}_1\\cdot \\mathrm{grad\\_log\\_prob})$ \n11: Scale by temperature: log_prob_ratio $\\leftarrow$ log_prob_ratio/ \n12: prob_ratio $\\leftarrow$ exp(log_prob_ratio) \n13: return prob_ratio \n14: end procedure",
+ "bbox": [
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+ "page_idx": 35
+ },
+ {
+ "type": "text",
+ "text": "Note: The exact form depends slightly on conventions; here we target a scaled version of the clean score plus noise term. Let $\\mathbf{T}(\\mathbf{z}_1, \\mathbf{z}_t, t) \\coloneqq \\frac{\\sigma_t^2}{\\alpha_t} \\nabla_{\\mathbf{z}_1} \\log p_1(\\mathbf{z}_1) + \\frac{1}{\\alpha_t} \\mathbf{z}_t$ be the target for the mean predictor.",
+ "bbox": [
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+ "page_idx": 35
+ },
+ {
+ "type": "text",
+ "text": "Now, let's connect this to the discrete TCSM objective. Consider the log-probability ratio (concrete score component) for the posterior distribution $q_{1|t}(\\mathbf{x}_1|\\mathbf{z}_t)$ in the continuous setting, where $\\hat{\\mathbf{x}}_1$ differs from $\\mathbf{x}_1$ only at position $i$ (i.e., $\\hat{x}_1^i = j \\neq x_1^i$ , and $\\hat{x}_1^l = x_1^l$ for $l \\neq i$ ):",
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+ "page_idx": 35
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\log \\frac {q _ {1 | t} \\left(\\hat {\\mathbf {x}} _ {1} \\mid \\mathbf {z} _ {t}\\right)}{q _ {1 | t} \\left(\\mathbf {x} _ {1} \\mid \\mathbf {z} _ {t}\\right)} = \\log \\frac {q _ {1} \\left(\\hat {\\mathbf {x}} _ {1}\\right)}{q _ {1} \\left(\\mathbf {x} _ {1}\\right)} + \\log \\frac {q _ {t | 1} \\left(\\mathbf {z} _ {t} \\mid \\hat {\\mathbf {x}} _ {1}\\right)}{q _ {t | 1} \\left(\\mathbf {z} _ {t} \\mid \\mathbf {x} _ {1}\\right)}. \\tag {72}\n$$\n",
+ "text_format": "latex",
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+ "page_idx": 35
+ },
+ {
+ "type": "text",
+ "text": "The second term simplifies due to the product structure of $q_{t|1}$ :",
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+ "page_idx": 35
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\log \\frac {q _ {t | 1} \\left(\\mathbf {z} _ {t} \\mid \\hat {\\mathbf {x}} _ {1}\\right)}{q _ {t | 1} \\left(\\mathbf {z} _ {t} \\mid \\mathbf {x} _ {1}\\right)} = \\log \\frac {q _ {t | 1} \\left(\\mathbf {z} _ {t} ^ {i} \\mid \\hat {x} _ {1} ^ {i}\\right)}{q _ {t | 1} \\left(\\mathbf {z} _ {t} ^ {i} \\mid x _ {1} ^ {i}\\right)} (73) \\\\ \\propto - \\frac {\\left\\| \\mathbf {z} _ {t} ^ {i} - \\alpha_ {t} \\mathbf {E} ^ {\\top} \\mathbf {e} _ {\\hat {x} _ {1} ^ {i}} \\right\\| ^ {2}}{2 \\sigma_ {t} ^ {2}} + \\frac {\\left\\| \\mathbf {z} _ {t} ^ {i} - \\alpha_ {t} \\mathbf {E} ^ {\\top} \\mathbf {e} _ {x _ {1} ^ {i}} \\right\\| ^ {2}}{2 \\sigma_ {t} ^ {2}} (74) \\\\ = \\frac {\\alpha_ {t}}{\\sigma_ {t} ^ {2}} \\left\\langle \\mathbf {z} _ {t} ^ {i}, \\mathbf {E} ^ {\\top} \\left(\\mathbf {e} _ {\\hat {x} _ {1} ^ {i}} - \\mathbf {e} _ {x _ {1} ^ {i}}\\right) \\right\\rangle - \\frac {\\alpha_ {t} ^ {2}}{2 \\sigma_ {t} ^ {2}} \\left(\\| \\mathbf {E} ^ {\\top} \\mathbf {e} _ {\\hat {x} _ {1} ^ {i}} \\| ^ {2} - \\| \\mathbf {E} ^ {\\top} \\mathbf {e} _ {x _ {1} ^ {i}} \\| ^ {2}\\right). (75) \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 35
+ },
+ {
+ "type": "text",
+ "text": "Let's assume embeddings have similar norms, making the last term negligible, or absorb it into the definition.",
+ "bbox": [
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+ "page_idx": 35
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+ "type": "text",
+ "text": "For the first term, $\\log \\frac{q_1(\\hat{\\mathbf{x}}_1)}{q_1(\\mathbf{x}_1)}$ , we use a first-order Taylor approximation in the continuous embedding space $\\mathbf{z}_1 = [\\mathbf{E}^\\top \\mathbf{e}_{x_1^1}, \\dots, \\mathbf{E}^\\top \\mathbf{e}_{x_1^L}]$ corresponding to $\\mathbf{x}_1$ . Let $p_1(\\mathbf{z}_1)$ be the density over these embeddings. Then:",
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+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\log \\frac {p _ {1} \\left(\\mathbf {z} _ {\\hat {\\mathbf {x}} _ {1}}\\right)}{p _ {1} \\left(\\mathbf {z} _ {\\mathbf {x} _ {1}}\\right)} \\approx \\log p _ {1} \\left(\\mathbf {z} _ {\\mathbf {x} _ {1}}\\right) + \\left\\langle \\nabla_ {\\mathbf {z} _ {1}} \\log p _ {1} \\left(\\mathbf {z} _ {1}\\right), \\mathbf {z} _ {\\hat {\\mathbf {x}} _ {1}} - \\mathbf {z} _ {\\mathbf {x} _ {1}} \\right\\rangle - \\log p _ {1} \\left(\\mathbf {z} _ {\\mathbf {x} _ {1}}\\right) (76) \\\\ = \\left\\langle \\nabla_ {\\mathbf {z} _ {1}} \\log p _ {1} \\left(\\mathbf {z} _ {1}\\right), \\mathbf {z} _ {\\dot {\\mathbf {x}} _ {1}} - \\mathbf {z} _ {\\mathbf {x} _ {1}} \\right\\rangle (77) \\\\ = \\left\\langle \\left(\\nabla_ {\\mathbf {z} _ {1}} \\log p _ {1} (\\mathbf {z} _ {1})\\right) _ {i}, \\mathbf {E} ^ {\\top} \\left(\\mathbf {e} _ {\\hat {x} _ {1} ^ {i}} - \\mathbf {e} _ {x _ {1} ^ {i}}\\right) \\right\\rangle , (78) \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
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+ "page_idx": 35
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+ {
+ "type": "text",
+ "text": "where $(\\cdot)_i$ denotes the gradient block corresponding to the $i$ -th position embedding.",
+ "bbox": [
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+ "page_idx": 35
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+ "type": "text",
+ "text": "Combining Eq. (75) (simplified) and Eq. (78), the target concrete score is approximately:",
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+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\mathbf {r} _ {q _ {1 \\mid t}} \\left(\\mathbf {x} _ {1} \\mid \\mathbf {z} _ {t}\\right) _ {i, j} := \\log \\frac {q _ {1 \\mid t} \\left(\\mathbf {x} _ {1} ^ {\\neq i} , x _ {1} ^ {i} \\leftarrow j \\mid \\mathbf {z} _ {t}\\right)}{q _ {1 \\mid t} \\left(\\mathbf {x} _ {1} \\mid \\mathbf {z} _ {t}\\right)} (79) \\\\ \\approx \\left\\langle \\left(\\nabla_ {\\mathbf {z} _ {1}} \\log p _ {1} (\\mathbf {z} _ {1})\\right) _ {i} + \\frac {\\alpha_ {t}}{\\sigma_ {t} ^ {2}} \\mathbf {z} _ {t} ^ {i}, \\mathbf {E} ^ {\\top} \\left(\\mathbf {e} _ {j} - \\mathbf {e} _ {x _ {1} ^ {i}}\\right) \\right\\rangle . (80) \\\\ \\end{array}\n$$\n",
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+ "text": "Target Concrete Score Matching: A Holistic Framework for Discrete Diffusion",
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+ "text": "Now, consider the model prediction $p_{\\theta}(\\mathbf{x}_1|\\mathbf{z}_t)$ , often parameterized via logits $\\pmb{\\mu}_{\\theta}(\\mathbf{z}_t,t)$ such that $p_{\\theta}(x_1^i = j|\\mathbf{z}_t) = \\mathrm{softmax}([ \\pmb{\\mu}_{\\theta}]_{;i})_j$ . The model's concrete score is:",
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+ "type": "equation",
+ "text": "\n$$\n\\mathbf {r} _ {p _ {\\theta}} \\left(\\mathbf {x} _ {1} \\mid \\mathbf {z} _ {t}\\right) _ {i, j} = \\left[ \\boldsymbol {\\mu} _ {\\theta} \\right] _ {j, i} - \\left[ \\boldsymbol {\\mu} _ {\\theta} \\right] _ {x _ {1} ^ {i}, i} = \\langle \\left[ \\boldsymbol {\\mu} _ {\\theta} \\right] _ {:, i}, \\mathbf {e} _ {j} - \\mathbf {e} _ {x _ {1} ^ {i}} \\rangle . \\tag {81}\n$$\n",
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+ "text": "The TCSM objective aims to match $\\mathbf{r}_{p_{\\theta}}$ to $\\mathbf{r}_{q_{1|t}}$ . The TSM objective (Eq. (71)) encourages $\\mu_{\\theta}(\\mathbf{z}_t,t)\\approx \\mathbf{T}'\\coloneqq \\frac{\\sigma_t^2}{\\alpha_t}\\nabla_{\\mathbf{z}_1}\\log p_1(\\mathbf{z}_1) + \\frac{1}{\\alpha_t}\\mathbf{z}_t$ . If this holds, then from Eq. (81):",
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+ "text": "\n$$\n\\mathbf {r} _ {p _ {\\theta}} (\\mathbf {x} _ {1} | \\mathbf {z} _ {t}) _ {i, j} \\approx \\langle [ \\mathbf {T} ^ {\\prime} ] _ {:, i}, \\mathbf {e} _ {j} - \\mathbf {e} _ {x _ {1} ^ {i}} \\rangle = \\left\\langle \\left(\\frac {\\sigma_ {t} ^ {2}}{\\alpha_ {t}} \\nabla_ {\\mathbf {z} _ {1}} \\log p _ {1} (\\mathbf {z} _ {1})\\right) _ {i} + \\frac {1}{\\alpha_ {t}} \\mathbf {z} _ {t} ^ {i}, \\mathbf {e} _ {j} - \\mathbf {e} _ {x _ {1} ^ {i}} \\right\\rangle . \\tag {82}\n$$\n",
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+ "text": "Comparing this to the target approximation in Eq. (80), we see they align (up to scaling factors and potential embedding norm terms) if $\\mathbf{E} = \\mathbf{I}$ . When $\\mathbf{E} \\neq \\mathbf{I}$ , the alignment is approximate.",
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+ "text": "In summary, under the first-order Taylor approximation for the marginal discrete probability ratio and assuming word embeddings $\\mathbf{E}$ behave similarly to an identity mapping (or have negligible impact on the inner products compared to the main terms), minimizing the TCSM objective, which matches discrete concrete scores, serves as an approximation to minimizing the continuous TSM objective. This provides a conceptual link between the two frameworks, highlighting how TCSM adapts score-matching principles to the discrete domain.",
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+ "type": "text",
+ "text": "J. Detailed Model Configurations",
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+ "text": "To enhance clarity and facilitate reproducibility, this section provides a comprehensive summary of the specific models, parameterizations, and training objectives used for each experimental result presented throughout the paper. App. J details the configuration for each key experiment, linking the reported results (identified by their table or figure number) to the underlying methodological choices, including the prior distribution (source distribution for diffusion), the structure of the denoising model $p_{1|t}^{\\theta}$ , the proposal distribution $h_{1|t}(\\mathbf{x}_1|\\mathbf{x}_t)$ used within the loss computation (if applicable), and the specific TCSM training objective function employed.",
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+ "type": "text",
+ "text": "K. Related Works",
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+ "text": "Generative modeling (Goodfellow et al., 2014; Ho et al., 2020; Austin et al., 2021; Song et al., 2021; Song & Ermon, 2019; Zhai et al., 2024) has seen significant advances through diffusion models, initially developed for continuous data like images. Applying these principles effectively to discrete data, such as text or graphs, presents unique challenges due to the non-differentiable nature of discrete spaces and has spurred several distinct lines of research.",
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+ "text": "Score Matching and Continuous Diffusion Foundations The theoretical underpinning for many modern diffusion models is Score Matching (Hyvärinen et al., 2009). This method estimates parameters $\\theta$ for models $p(\\mathbf{x};\\theta)\\propto q(\\mathbf{x};\\theta)$ with intractable normalization constants by minimizing the difference between the model's score function $\\nabla_{\\mathbf{x}}\\log q(\\mathbf{x};\\theta)$ and the data score $\\nabla_{\\mathbf{x}}\\log p_x(\\mathbf{x})$ . A key insight by Hyvärinen et al. (2009) showed that this objective can be computed using only the model score and its derivatives on data samples, avoiding the need for the true data density or normalization constant. A crucial practical development was Denoising Score Matching (DSM) (Vincent, 2011), which established an equivalence between score matching on noise-perturbed data and training specific denoising autoencoders (DAEs). DSM matches the model's score at a noisy point $\\tilde{\\mathbf{x}}$ to the score of the conditional denoising distribution, avoiding the second derivatives required by original score matching and making score estimation more tractable.",
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+ "text": "These principles were central to the development of diffusion models. Early work framed diffusion via forward (noising) and reverse (denoising) Markov processes trained with a variational lower bound (VLB) (Sohl-Dickstein et al., 2015). Subsequently, score-based generative models (Song & Ermon, 2019) directly applied DSM by training a single Noise Conditional Score Network (NCSN) $s_{\\theta}(\\mathbf{x}, \\sigma)$ to estimate scores $\\nabla_{\\mathbf{x}} \\log q_{\\sigma_i}(\\mathbf{x})$ across multiple noise levels $\\{\\sigma_i\\}$ , using annealed Langevin dynamics for sampling. Denoising Diffusion Probabilistic Models (DDPM) (Ho et al., 2020) refined this, particularly for images, by parameterizing the reverse process to predict the added noise $\\epsilon$ and using a simplified VLB-derived objective shown to be equivalent to DSM over multiple noise scales. While highly successful, standard DSM can suffer from high variance at low noise levels. Target Score Matching (TSM) (Bortoli et al., 2024) addresses this by incorporating knowledge of the clean target score $\\nabla \\log p(\\mathbf{x})$ when available, leading to lower variance estimators in the low-noise regime.",
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+ "text": "Target Concrete Score Matching: A Holistic Framework for Discrete Diffusion",
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+ "text": "Continuous Diffusion for Discrete Data One approach to handle discrete data involves operating within continuous embedding spaces, adapting standard continuous diffusion techniques. This allows leveraging powerful continuous models but requires mapping back to the discrete space. Diffusion-LM (Li et al., 2022) applied continuous diffusion to word embeddings, enabling controllable text generation via gradient guidance during sampling. Plaid (Gulrajani & Hashimoto, 2023) focused on likelihood-based training for text, jointly optimizing embeddings and model parameters using the VLB, categorical reparameterization, an output prior, a learned conditional likelihood $p(x|z_0)$ , and self-conditioning. CDCD (Dieleman et al., 2022) employed a probability flow ODE on embeddings, using score interpolation to jointly train embeddings and a denoising Transformer with a cross-entropy loss, along with time warping. Bit Diffusion (Chen et al., 2023) treated the binary representation of discrete data as continuous \"analog bits,\" enhanced by self-conditioning and asymmetric time intervals. While effective, these methods rely on continuous approximations or embeddings, motivating research into models operating directly on discrete domains. Furthermore, many of these works explore non-autoregressive approaches enabling parallel generation (Bowman et al., 2016; Gu et al., 2018; Li et al., 2022; Hoogeboom et al., 2021; Savinov et al., 2022; Che et al., 2017; Zhang et al., 2020; Yu et al., 2017; de Masson d'Autume et al., 2019; Deng et al., 2020), contrasting with sequential autoregressive models.",
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+ "text": "Discrete Diffusion Models A parallel line of research develops diffusion processes inherently designed for discrete state spaces, often using Markov chains. Building on early foundations (Sohl-Dickstein et al., 2015; Hoogeboom et al., 2021), D3PM (Austin et al., 2021) generalized discrete diffusion using various structured transition matrices (e.g., uniform, absorbing, Gaussian-like) and trained via a hybrid VLB/cross-entropy loss. Campbell et al. (2022) extended this to Continuous-Time Markov Chains (CTMCs), deriving a continuous-time ELBO and proposing efficient sampling methods like tau-leaping and predictor-corrector schemes, leveraging factorization for high-dimensional data.",
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+ "text": "Score-like Analogues and Masking Mechanisms for Discrete Diffusion Instead of direct Markov chain simulation, other works define score-like quantities for discrete diffusion. The concrete score, defined as the ratio of marginal probabilities $p_t(\\mathbf{y}) / p_t(\\mathbf{x})$ , acts as a discrete analogue to the continuous score (Meng et al., 2022; Lou et al., 2024). SEDD (Lou et al., 2024) trained models using a score entropy objective ( $L_{DSE}$ ) derived from this ratio, connecting it to the ELBO and using Tweedie $\\tau$ -leaping for sampling. Sun et al. (2023) developed categorical ratio matching within a CTMC framework, learning singleton conditionals $p_t(x^d | \\mathbf{x}^{\\backslash d})$ with a tractable loss and an analytical reverse sampler. Building on this, Ou et al. (2024) showed that for absorbing diffusion, the concrete score factorizes into a time-independent conditional and a time-dependent scalar, simplifying the model (RADD) and yielding the Denoising Cross-Entropy (DCE) loss.",
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+ "type": "text",
+ "text": "Masked (or absorbing) diffusion, which replaces tokens with a special [MASK] token during the forward process, has proven particularly effective. MDLM (Sahoo et al., 2024) introduced a substitution-based parameterization (SUBS) and derived a simplified Rao-Blackwellized ELBO equivalent to weighted Masked Language Modeling (MLM) losses, enabling generative training of encoder-only models. Shi et al. (2024) (MD4) further unified this framework, deriving a simple ELBO with SNR invariance properties similar to continuous diffusion and generalizing to state-dependent masking schedules.",
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+ "type": "text",
+ "text": "Further research has refined the parameterization and mechanisms of discrete diffusion. Reparameterized Discrete diffusion Models (RDM) (Zheng et al., 2023) identified an underlying route-and-denoise mechanism, simplifying the objective to cross-entropy on noisy tokens and enabling adaptive routing during sampling. Liu et al. (2024b) proposed Discrete Diffusion with Planned Denoising (DDPD), factorizing the reverse process into a planner (predicting corruption) and a denoiser, allowing adaptive sampling via the Gillespie algorithm guided by the planner.",
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+ "text": "Discrete Flow Matching offers another generalization pathway. Gat et al. (2024) defined probability paths interpolating discrete distributions and derived corresponding probability velocities, analogous to continuous flow matching, providing a unified sampling theory. (Campbell et al., 2024) formulated discrete flows using CTMCs, learning scores via cross-entropy and enabling inference-time flexibility by adjusting the rate matrix family without retraining, also unifying multimodal generation. Discrete diffusion principles have also been applied to structured data, such as graphs in DiGress (Vignac et al., 2023), using specific noise transitions, auxiliary features, and classifier guidance.",
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+ "text": "Scaling and Adapting Pre-trained Models for Diffusion Language Modeling Significant recent effort has focused on scaling diffusion models for language generation, often by adapting large pre-trained autoregressive (AR) or masked language models (MLMs). DiffusionBERT (He et al., 2023) integrated BERT into an absorbing-state diffusion framework, leveraging pre-trained weights and exploring novel noise schedules and time conditioning. Ye et al. (2023) adapted pretrained MLMs (like XLM-R) for generative tasks by finetuning with an RDM objective, enabling instruction-following",
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+ {
+ "type": "text",
+ "text": "capabilities. AR2Diff (Han et al., 2024) proposed converting pre-trained AR models to diffusion models by enabling bidirectional attention and continuing training with a diffusion objective. DiffuLLaMA (Gong et al., 2024) presented a continual pre-training method to adapt AR models (like LLaMA) into time-embedding-free diffusion models using attention mask annealing. LLaDA (Nie et al., 2025) developed a large masked diffusion model trained with a masking objective, adapting standard pre-training and SFT pipelines for this non-autoregressive paradigm. These works demonstrate the potential of leveraging existing large model architectures and weights to build capable diffusion language models.",
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+ "text": "Guidance and Control in Discrete Diffusion Controlling the generation process of discrete diffusion models is vital for their application. Several approaches modify the sampling procedure or the model itself. Nisonoff et al. (2024) introduced Discrete Guidance (DG), a principled framework for guidance in CTMC-based models, offering exact predictor guidance (PG), predictor-free guidance (PFG), and an efficient Taylor-Approximated Guidance (TAG) variant by exploiting tractable normalization constants during inference. FK-steering (Singhal et al., 2025) provides a general inference-time steering approach using Feynman-Kac interacting particle systems, applicable even with non-differentiable rewards via parallel simulation and resampling. An alternative strategy involves finetuning the model itself to incorporate guidance. Rector-Brooks et al. (2024) proposed Discrete Denoising Posterior Prediction (DDPP), a framework for steering pre-trained Masked Diffusion Models (MDMs) according to a reward function $R(\\mathbf{x}_1)$ . DDPP reframes steering as learning an amortized sampler (via finetuning the MDM) for a target posterior distribution proportional to $p_{\\theta}^{\\mathrm{pre}}(\\mathbf{x}_1)R(\\mathbf{x}_1)$ . By exploiting the relationship between the target denoising posterior, the pre-trained model's posterior, and the reward, DDPP derives several simulation-free training objectives, offering a scalable approach to bake reward-based control into the model. Other methods include informed corrector steps based on confidence scores combined with architectural changes and novel training objectives for masked diffusion (Zhao et al., 2024b), and adaptations of standard classifier-free or classifier-based guidance for discrete domains, sometimes coupled with improved ELBO formulations suitable for guidance (Schiff et al., 2024).",
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+ "text": "LLM Distillation Our work also relates to LLM distillation (Xu et al., 2024b), which focuses on transferring capabilities from large teacher models to smaller student models. Common techniques involve distribution matching, specialized loss functions (e.g., MiniLLM (Gu et al., 2024), DistiLLM (Ko et al., 2024)), using rationales (Hsieh et al., 2023), or dynamic data selection (Liu et al., 2024a). While most existing methods distil knowledge between autoregressive models, our research explores knowledge transfer from powerful AR teachers to bidirectional diffusion students. This presents distinct challenges, particularly regarding the mismatch between the teacher's sequential generation process and the student's non-autoregressive, iterative refinement process, but potentially benefits from similar underlying principles aimed at effective knowledge transfer and mitigating distribution discrepancies.",
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+ "table_body": "| Model Variant / Name(Defining Section/Eq.) | Experiment(Table/Figure) | Prior(SourceDist.) | Denoising Model Parameterization pθ1|t | Proposal distribution h(x1|xt) | Training Objective(Equation / Description) |
| Experiments on TEXT8 (Table 4) |
| TCSM Uniform Lscore(Sec. 4.2) | Table 4 | Uniform | Factorized: pθ1|t(x1|xt) = ∏i=1L pθ1|t(x1xtx1) | p1|t(x1|xt) | Lscore with Gen KL(Monte Carlo version:Eq. (10)) |
| TCSM Uniform Ldistrib(Sec. 4.2) | Table 4 | Uniform | Factorized (as above) | p1|t(x1|xt) | Ldistrib with KL(Cross-Entropy: Factor-ized version of Eq. (9)) |
| TCSM Absorb Lscore(Sec. 4.2) | Table 4 | Mask(Absorb-ing) | Factorized (as above) | p1|t(x1|xt) | Lscore with Gen KL(Monte Carlo version:Eq. (10)) |
| TCSM Absorb Ldistrib(Sec. 4.2) | Table 4 | Mask(Absorb-ing) | Factorized (as above) | p1|t(x1|xt) | Ldistrib with KL(Cross-Entropy: Factor-ized version of Eq. (9)) |
| TCSM Absorb Ldistrib(Sec. 5.1) | Table 4 | Mask(Absorb-ing) | Density Ratio (Strategy ii): pθ1|t(x1|xt) × pθ1|t(x1|xt) exp(fθ(x1|xt)) (Ref = Pre-trained TCSM Absorb Ldistrib) | pref1|t = pre1|t | Post-training phase:DRE objective using Gen KL (Table 5, column 3) |
| Experiments on OPENWEBTEXT (Table 3, Fig. 1, Fig. 4) |
| TCSM Uniform Lscore(Sec. 4.2) | Table 3 | Uniform | Factorized (as above) | p1|t(x1|xt) | Lscore with Gen KL(Eq. (10)) |
| TCSM Uniform Ldistrib(Sec. 4.2) | Table 3 | Uniform | Factorized (as above) | p1|t(x1|xt) | Ldistrib with KL(Factorized version of Eq. (9)) |
| TCSM Absorb Ldistrib(Sec. 4.2) | Table 3 | Mask(Absorb-ing) | Factorized (as above) | p1|t(x1|xt) | Ldistrib with KL(Factorized version of Eq. (9)) |
| TCSM Absorb Ldistrib(Sec. 5.1) | Table 3 | Mask(Absorb-ing) | Density Ratio (Strategy ii, as above) | pref1|t = pre1|t | Post-training phase:DRE objective using Gen KL (Table 5, column 3) |
| TCSM-Bert(Sec. 4.2) | Fig. 1 | Mask(Absorb-ing) | Factorized (as above) | p1|t(x1|xt) | Ldistrib with KL(Target p1|t uses BERT approx. for p1) |
| TCSM-AR(Sec. 4.2) | Fig. 1 | Mask(Absorb-ing) | Factorized (as above) | p1|t(x1|xt) | Ldistrib with KL(Target p1|t uses AR approx. for p1) |
| TCSM-Hollow(Sec. 4.2) | Fig. 1 | Mask(Absorb-ing) | Factorized (as above) | p1|t(x1|xt) | Ldistrib with KL(Target p1|t uses Hollow approx. for p1) |
| TCSM Distillation(Sec. 5.4) | Fig. 4 | Mask(Absorb-ing) | Factorized (Student Model) | p1|t(x1|xt) | Ldistrib with KL(Target p1|t uses AR Teacher via Top-K approx.) |
| Density Ratio Estimation Bregman Comparison (Table 6) |
| TCSM BCE (Reimpl.) (Sec. 5.1) | Table 6 | Mask(Absorb-ing) | Density Ratio (Strategy ii) | pref1|t = pre1|t | DRE objective using BCE(Table 5, column 3) |
| TCSM LSIF(Sec. 5.1) | Table 6 | Mask(Absorb-ing) | Density Ratio (Strategy ii, as above) | pref1|t = pre1|t | DRE objective using LSIF(Table 5, column 3) |
| TCSM Gen KL(Sec. 5.1) | Table 6 | Mask(Absorb-ing) | Density Ratio (Strategy ii, as above) | pref1|t = pre1|t | DRE objective using Gen KL(Table 5, column 3) |
| Post-training Fine-tuning Experiments |
| TCSM Reward Tuning(Sec. 5.2) | Fig. 5 (Synthetic) | Uniform | Standard denoising model pθ1|t(Factorized assumed) | ppre1|t | Weighted KL objective for pR1twith Nfull(Alg. 3, Line 7) |
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+ "text": "Table 9: Detailed summary of model configurations for experiments reported in the paper.",
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+ "content": "Abstract"
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+ "type": "text",
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+ "content": "Discrete diffusion is a promising framework for modeling and generating discrete data. In this work, we present Target Concrete Score Matching (TCSM), a novel and versatile objective for training and fine-tuning discrete diffusion models. TCSM provides a general framework with broad applicability. It supports pre-training discrete diffusion models directly from data samples, and many existing discrete diffusion approaches naturally emerge as special cases of our more general TCSM framework. Furthermore, the same TCSM objective extends to post-training of discrete diffusion models, including fine-tuning using reward functions or preference data, and distillation of knowledge from pre-trained autoregressive models. These new capabilities stem from the core idea of TCSM, estimating the concrete score of the target distribution, which resides in the original (clean) data space. This allows seamless integration with reward functions and pre-trained models, which inherently only operate in the clean data space rather than the noisy intermediate spaces of diffusion processes. Our experiments on language modeling tasks demonstrate that TCSM matches or surpasses current methods. Additionally, TCSM is versatile, applicable to both pre-training and post-training scenarios, offering greater flexibility and sample efficiency."
+ },
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+ "type": "title",
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+ "content": "1. Introduction"
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+ "content": "Discrete diffusion models have emerged as a transformative paradigm in generative modeling, achieving remarkable success across diverse domains. Despite their advancements in closing the performance gap with autoregressive (AR) models through innovative training techniques, these models still face fundamental limitations that impede their broader adoption and practical use."
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+ "content": "The current landscape of discrete diffusion models reveals two critical shortcomings. First, existing approaches are fragmented in their theoretical foundations and training methodologies. Methods such as SEDD (Lou et al., 2024) employ denoising score entropy, while CTMC (Campbell et al., 2022) derives objectives from continuous-time Markov chains, and approaches like those in (Shi et al., 2024; Sahoo et al., 2024; Xu et al., 2024a) specialize in absorbing state diffusion models with specific assumptions. This fragmentation creates a barrier to developing unified and theoretically grounded approaches."
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+ "content": "Second, and perhaps more significantly, current discrete diffusion models predominantly focus on pre-training, largely neglecting the crucial post-training phase that has proven essential for downstream task optimization in autoregressive models. While AR models benefit from well-established post-training techniques such as reinforcement learning with human feedback (Ziegler et al., 2019; Ouyang et al., 2022; Bai et al., 2022), direct preference optimization (Rafailov et al., 2023), and knowledge distillation (Gu et al., 2024), discrete diffusion models lack comparable capabilities. This limitation significantly restricts their practical applicability and prevents them from achieving performance parity with AR counterparts in many real-world scenarios."
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+ "content": "Contributions We introduce Target Concrete Score Matching (TCSM), a novel framework for discrete diffusion models based on the concrete score (Meng et al., 2022). By operating in the clean data space, TCSM seamlessly integrates reward functions and pre-trained models while integrating pre-training and post-training. Our key contributions are:"
+ },
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+ "content": "- We develop the general TCSM framework for discrete diffusion models (Sec. 3), which provides flexibility across various"
+ },
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+ "content": "Correspondence to: Ruixiang Zhang