1 Residual Context Diffusion Language Models

Method

  • Traditional dLLMs discard the [MASK] at unpredicted positions in each round and reset them as new blank [MASK]s. To avoid wasting computational power, this method retains these discarded semantics and transforms them into vectors for the next round of prediction.

Algorithm

  1. Generate the residual vector by performing a weighted sum of the model’s predicted probability and the word embeddings:

$\Delta_i^{(t_k)} = \sum_{j=1}^V p_{i,j}^{(t_k)} E_{j,:}$

$E_{j}$ is the embedding for that word.

  1. Weight of the residual vector to determine if the vector is worth retaining:

$\alpha_i^{(t_k)} = \frac{-\sum_{j=1}^V p_{i,j}^{(t_k)} \log p_{i,j}^{(t_k)}}{\log V}$

The larger $\alpha$ is, the less certain the model is about the word, and the more it should be retained. If high-confidence words are not predicted, it indicates they should likely not be retained further.

$\log V$ is used to compress $\alpha$ between 0 and 1.

  1. If the word at position $i$ is not predicted, perform residual addition by adding the residual term to the [MASK].

If it has been predicted, it equals the word embedding.

$\tilde{e}i^{(t_k)} = \begin{cases} (1 - \alpha_i^{(t{k-1})})E(x_i^{(t_k)}) + \alpha_i^{(t_{k-1})}\Delta_i^{(t_{k-1})}, & \text{if } x_i^{(t_k)} = \text{[MASK]} \ E(x_i^{(t_k)}), & \text{otherwise} \end{cases}$

Training

1 Reference Model Initialization

Train a $\mathcal{M}_{ref}$ and freeze all parameters to output $p^{(t)}$.

2 Residual-Aware Target Training

Train a $\mathcal{M}_{target}$, where the input is a sentence with [MASK]s.

  1. Calculate the weight of the residual vector $\alpha$ based on the probabilities calculated by $\mathcal{M}_{ref}$.
  2. Calculate the word vector $ \Delta_i^{(t_k)} = \sum_{j=1}^V p_{i,j}^{(t_k)} E_{j,:} $, where probabilities are from $\mathcal{M}{ref}$ and embeddings are from $\mathcal{M}{target}$.
  3. Calculate the final weighted result as input.
  4. Input into $\mathcal{M}{target}$ to predict the unmasked original sentence. Loss function: $\mathcal{L} = \mathbb{E}\left[\frac{1}{t}\sum{i:m_i=1} -\log P_{\theta_{target}}(x_i^{(0)} | {e_{i’}^{(t)}}_{i’=1}^b)\right]$

Inference

1 Warm-Start

$\mathcal{M}_{ref}$ outputs the probability distribution $p^{(t_0)}$ for the first step.

2 Recursive Decoding

At each time step $t_k$:

  1. Check every position in the sequence and calculate a new word vector as input based on $\alpha$ and $\Delta$ from step $t_{k-1}$.
  2. $\mathcal{M}_{target}$ predicts each [MASK] and retains the original scores.
  3. Select high-confidence positions for generation and remask uncertain ones.
  4. For remasked parts, apply Temperature-Scaled Entropy Alignment to original scores to recalculate $\alpha$ and $\Delta$ for the next round.

Temperature-Scaled Entropy Alignment

In the inference phase, $\Delta$ and $\alpha$ no longer depend on $\mathcal{M}_{ref}$.

$\mathcal{M}_{target}$ tends to be over-confident when relying on its own previous round’s predictions, causing $\alpha$ to trend towards 0, which makes the final result indistinguishable from [MASK].

Before converting scores to probabilities, a Residual temperature scalar is introduced to flatten extreme probability values:

$p_i^{(t_k)}(T_{res}) = \text{softmax}(z_i / T_{res})$

2 Deferred Commitment Decoding for Diffusion Language Models

Method

  • Boundary-Induced Context Truncation: Block decoding leads to inaccuracies at block boundaries because it cannot see the content of the next block, resulting in a loss of context.
  • Use a dynamic sliding window for deferred decoding of low-confidence words.

Algorithm

1 Sliding Window

  1. Left Boundary: $$L^{(t)} = \arg\min_{i \ge l} {i | x_i^{(t)} = \langle \text{MASK} \rangle}$$ The first un-decoded position serves as the left boundary.

  2. Right Boundary: $$R^{(t)} = \arg\max_{i \le r} \left{ i \mid i \le L^{(t)} + s_{max} \text{ and } \sum_{k=L^{(t)}}^{i-1} [x_k^{(t)} = \langle \text{MASK} \rangle] \le s_{init} \right}$$ The length does not exceed $s_{max}$; the number of [MASK]s included in the window does not exceed $s_{init}$.

$r$ is the block’s right boundary (absolute right boundary).

2 Deferred Decoding

$$\mathcal{S}^{(t)} = \mathcal{M}^{(t)} \cap [L^{(t)}, R^{(t)}) \cap {i \mid \mathcal{C}(i) \ge \min(\tau, \max_j \mathcal{C}(j))}$$ Determines which [MASK]s in the window will be decoded.

Confidence must be greater than a threshold or the highest confidence within the window.

3 Dynamic Block Extension (DBE)

$$\min_{i \in \mathcal{S}^{(t)}} \mathcal{C}(i) < \tau_{low} \text{ and } R^{(t)} - L^{(t)} < s_{max} \text{ and } \sum_{k=L^{(t)}}^{R^{(t)}-1} [x_k^{(t)} = \langle \text{MASK} \rangle] < s_{init}$$

For semi-causal DLMs, the sliding window still operates between blocks. If the window reaches the block boundary (conditions 2 and 3) and confidence within the window is very low (condition 1), dynamically extend the block size: $$r’ = \min(r + e_{step}, \text{blocksize} + e_{max})$$ Extend the window right boundary by $e_{step}$ units, not exceeding a maximum limit.

4 DCD’s Combination with KV Cache

$$\mathcal{W}^{(t)} = {x_i^{(t)} \mid i \in [L^{(t-1)} - r, R^{(t)} + r)}$$ Prefix Cache: Temporarily stores KV for all tokens before the active interval.

Dual Cache: Simultaneously stores KV for both prefix and suffix tokens.

Reconstruct the cache every time $B’$ masked words are successfully decoded.

3 WAVEFRONTDIFFUSION: DYNAMIC DECODING SCHEDULE OR IMPROVED REASONING

Method

  • Establish a Wavefront, diffusing outward from already decoded words rather than using fixed block boundaries; finally, perform pruning by setting a capacity.
  • MHCO Metric.

Algorithm

1 Definition

Set a radius $R$ to include context:

$$\mathcal{W}{t}={i \mid dist(i,\mathcal{C}{t})\le R}$$

$C_t$ is the set of confirmed tokens.

2 Decoding Iteration Process

  1. Scoring Perform one forward pass to obtain confidence: $$s_{j}=\max_{v\in\mathcal{V}}p_{\theta}(x_{j}=v \mid x_{t},c)$$ The probability of the most likely word in the vocabulary is its confidence.

  2. Denoising Control the number of denoising actions per round; if not divisible, distribute the remainder across the first $extra$ steps: $$k_{t}=k_{base}+\mathbb{I}[t\le extra]$$

$$kbase = ⌊N/T/⌋, extra =N \pmod T$$

$N$ is the total number of words to decode. $T$ is the number of steps. $\mathbb{I}$ is the indicator function, equal to 1 if the condition is met.

  1. Expansion $$\mathcal{W}{t}=\bigcup{i\in\mathcal{C}{t}}{j \mid dist(j,i)\le R, x{j}=\text{[MASK]}}$$ For each token in the newly decoded set $C_t$, perform an expansion with radius $R$.

  2. Pruning If the maximum capacity $F$ is exceeded, remove tokens with lower confidence.

4 AdaBlock-dLLM: SEMANTIC-AWARE DIFFUSION LLM INFERENCE VIA ADAPTIVE BLOCK SIZE

Method

  • If the block is too small, deferred decoding results in high-confidence tokens not being decoded immediately; if tokens within a block have low confidence but must be decoded, it creates issues.
  • Divide confidence regions and set a Volatility Band.

Algorithm

Adaptive block size adjustment based on semantics. Set a delimiters set $D$. The delimiter with the highest confidence in the window is denoted as $c_{max}$. If $c_{max} \ge \tau_{\mathcal{D}}$: Set the current block size $B$ as the distance from the starting point to this delimiter; otherwise, the current area is semantically ambiguous, and $B$ reverts to the default fixed block size $B_0$ (or the remaining length).

Index Window: Prevents premature decoding of <EOS> by setting a window to mask out positions that are too far away.

  1. Perform initial denoising before entering the next block.
  2. Apply the algorithm above to define a block.
  3. Iteratively denoise within this block.

5 Improving the Throughput of Diffusion-based Large Language Models via a Training-Free Confidence-Aware Calibration

Method

  • Fast-dLLM parameters are fixed and not dynamic.
  • Perform calculations and a series of adaptive operations dynamically based on average confidence.

Algorithm

clip: Used to set upper and lower bounds (min/max).

1 Adaptive Block Size

$$B_t = clip(B_{min} + (B_{max} - B_{min})\overline{c}, B_{min}, B_{max})$$ $\overline{c}$ is the average confidence. The higher the average confidence, the closer the size is to the maximum value.

2 Adaptive steps

$$S_t = clip(S_{base} + (S_{max} - S_{base})(1 - \overline{c}), S_{base}, S_{max})$$ Similar to point 1, lower average confidence leads to larger steps, representing more calculations.

3 Adaptive vocabulary size

$$V_t = clip(V_{phase}(g_t) \cdot f_{conf}(\overline{c}) \cdot f_{rep}(r_t), V_{min}, V_{max})$$ $g_t$ is generation progress. $V_{phase}(g_t)$: Larger vocabulary size at the start of generation. $f_{conf}(\overline{c})$: Larger vocabulary size when confidence is low to expand the search range. $r_t$ is the repetition rate of generated tokens. $f_{rep}(r_t)$: Expand vocabulary size if recently generated tokens are repetitive.

4 Adaptive threshold

$$\tau_t = \tau_{base}(1 - g_t) + \tau_{min}g_t$$ At the start of generation, the threshold is high ($\tau_{base}$), then it decreases to $\tau_{min}$.

6 Next Semantic Scale Prediction via Hierarchical Diffusion Language Models

Method

  • Introduce clustered semantics between tokens and [MASK]. When adding noise, transform to cluster tokens first, then to [MASK]. During denoising, revert to clusters first, then to specific tokens.
  • MDLM is a special case of HDLM, equivalent to having only one cluster [MASK].

Algorithm

1 Forward Noise Addition

Mapping matrix $\Gamma$ to obtain clusters $c(x) = \Gamma x$: $$q_{t}(z_{t}|x)=Cat(z_{t};\alpha_{t}x+\beta_{t,c}c(x)+\beta_{t,m}m)$$ $Cat$ is the Categorical distribution. At time step $t$, there are three possible states:

  1. $\alpha_{t}$ is the probability of remaining as a specific token.
  2. $\beta_{t,c}$ is the probability of becoming cluster $c$.
  3. $\beta_{t,m}$ is the probability of becoming [MASK]. As $t$ increases, $\alpha_t$ decreases.

Stochastic Perturbations

$$\beta_t \pi_t = \xi \beta_{t,c} c(x) + \sum_{c \neq c(x)} \frac{1-\xi}{N_c-1} \beta_{t,c} c + \beta_{t,m} m$$ Puts incorrect words into the set with probability $1-\xi$.

2 Backward Denoising

$$p_{\theta}(z_{s}|z_{t})=q_{t|s}(z_{t}|z_{s})\frac{q_{s}(z_{s}|x_{\theta})}{q_{t}(z_{t}|x_{\theta})}$$

3 Training

$$ELBO = -\mathbb{E}{t,z{t}}\left[\delta_{z_{t},c}w_{t,c}CE\left(x,\frac{x_{\theta}\odot(\Gamma^{\top}\Gamma x)}{x_{\theta}^{\top}\Gamma^{\top}\Gamma x}\right)+\delta_{z_{t},m}w_{t,m}CE(\Gamma x,\Gamma x_{\theta})\right]$$

  1. $\delta_{z_t,c}$ indicates the current token is a cluster; words outside the cluster are zeroed out, and only the correct token is found within this cluster for generation.
  2. $\delta_{z_t,m}$ indicates the current token is a [MASK]; it must first find a suitable cluster.

Example

  1. During forward propagation, after a token becomes a cluster, it maintains this state for duration $t_{m,\tau}$ (uniform distribution over $[\tau, 1]$) before potentially becoming [MASK].
  2. Calculate the token’s state: $$\beta_{t,c} = \int_{0}^{t} \frac{1-t}{1-\tau} (-d\alpha_\tau)$$ Remaining time divided by total span multiplied by decay factor. $$\beta_{t,m} = \int_{0}^{t} \frac{t-\tau}{1-\tau} (-d\alpha_\tau)$$ Similarly: $$\alpha_t = (1-t)^\gamma$$
  3. Calculate loss: $$w_{t,c} = \frac{-\alpha_t’}{\beta_{t,c}}$$ $$w_{t,m} = \frac{-\alpha_t’}{\beta_{t,m}}$$

Optimization

  1. Gradient of cluster-level loss $$-\nabla_{x_{\theta}}(\delta_{z_{t},m}w_{t,m}CE(\Gamma x,\Gamma x_{\theta}(z_{t},t)))=\delta_{z_{t},m}w_{t,m}\Gamma^{\top}\left(\Gamma x\odot\frac{1}{\Gamma x_{\theta}}\right)$$

  2. Auxiliary loss No separate classification head is added to categorize tokens into clusters; cluster probabilities are obtained directly from $x_\theta$ via the posterior.

  3. Flexible loss weights Clipping weights $w_{t,c}$ and $w_{t,m}$ if they are too large or too small.

  4. Force transition in decoding Disallow prediction of words outside of the cluster.