Absorbing State Diffusion

Keywords: absorbing state diffusion, generative models

Absorbing State Diffusion for text is a diffusion approach where tokens gradually transition toward a special mask token (absorbing state) — providing a natural discrete diffusion process where the forward process masks tokens with increasing probability and the reverse process learns to unmask, connecting diffusion models to masked language modeling like BERT.

What Is Absorbing State Diffusion?

• Definition: Diffusion process where tokens transition to [MASK] token (absorbing state).
• Forward: Tokens randomly replaced with [MASK] with increasing probability over time.
• Reverse: Model learns to predict original tokens from partially masked sequences.
• Key Insight: Masking is natural discrete corruption process.

Why Absorbing State Diffusion?

• Natural for Discrete Data: Masking is intuitive corruption for text.
• Connection to BERT: Leverages masked language modeling insights.
• Simpler Than Continuous: No embedding/projection complications.
• Interpretable: Easy to understand forward and reverse processes.
• Effective: Competitive with other discrete diffusion approaches.

How It Works

Forward Process (Masking):

• Start: Clean text sequence x_0 = [token_1, token_2, ..., token_n].
• Step t: Each token has probability q(t) of being [MASK].
• Schedule: q(t) increases from 0 to 1 as t goes from 0 to T.
• End: x_T is fully masked [MASK, MASK, ..., MASK].

Transition Probabilities:

P(x_t = [MASK] | x_{t-1} = token) = β_t
P(x_t = token | x_{t-1} = token) = 1 - β_t
P(x_t = token | x_{t-1} = [MASK]) = 0  (absorbing!)

• Absorbing: Once masked, stays masked (can't unmask in forward process).
• Schedule: β_t defines masking rate at each step.

Reverse Process (Unmasking):

• Start: Fully masked sequence x_T.
• Model: Transformer predicts original tokens from masked sequence.
• Input: Partially masked sequence + timestep t.
• Output: Probability distribution over tokens for each [MASK] position.
• Sampling: Sample tokens from predicted distribution, gradually unmask.

Connection to BERT

Similarities:

• Masking: Both use [MASK] token as corruption.
• Prediction: Both predict original tokens from masked context.
• Bidirectional: Both use bidirectional context for prediction.

Differences:

• BERT: Single masking level (15% typically), single prediction step.
• Diffusion: Multiple masking levels, iterative unmasking over T steps.
• BERT: Trained for representation learning.
• Diffusion: Trained for generation.

Insight: Absorbing state diffusion generalizes BERT to iterative generation.

Training

Objective:

• Loss: Cross-entropy between predicted and true tokens at masked positions.
• Sampling: Sample timestep t, mask according to schedule, predict original.
• Optimization: Standard supervised learning, no adversarial training.

Training Algorithm:

1. Sample clean sequence x_0 from dataset
2. Sample timestep t ~ Uniform(1, T)
3. Mask tokens according to schedule q(t)
4. Model predicts original tokens from masked sequence
5. Compute cross-entropy loss on masked positions
6. Backpropagate and update model

Masking Schedule:

• Linear: q(t) = t/T (uniform masking rate increase).
• Cosine: q(t) = cos²(πt/2T) (slower at start, faster at end).
• Tuning: Schedule affects generation quality, requires tuning.

Generation (Sampling)

Iterative Unmasking:

1. Start with fully masked sequence x_T = [MASK, ..., MASK]
2. For t = T down to 1:
   a. Model predicts token probabilities for each [MASK]
   b. Sample tokens from predicted distributions
   c. Unmask some positions (according to schedule)
   d. Keep other positions masked for next iteration
3. Final x_0 is generated text

Unmasking Strategy:

• Confidence-Based: Unmask positions with highest prediction confidence.
• Random: Randomly select positions to unmask.
• Scheduled: Unmask fixed fraction at each step.

Temperature:

• Sampling: Use temperature to control randomness.
• Low Temperature: More deterministic, higher quality.
• High Temperature: More diverse, more creative.

Advantages

Natural Discrete Process:

• No Embedding: No need to embed to continuous space.
• No Projection: No projection back to discrete tokens.
• Interpretable: Masking and unmasking are intuitive.

Leverages BERT Insights:

• Pretrained Models: Can initialize from BERT-like models.
• Masked LM: Builds on well-understood masked language modeling.
• Transfer Learning: Leverage existing masked LM research.

Flexible Generation:

• Infilling: Naturally handles filling masked spans.
• Partial Generation: Can fix some tokens, generate others.
• Iterative Refinement: Multiple passes improve quality.

Controllable:

• Guidance: Easy to apply constraints during unmasking.
• Conditional: Condition on various signals.
• Editing: Modify specific parts while keeping others.

Limitations

Multiple Steps Required:

• Slow: Requires T forward passes (typically T=50-1000).
• Latency: Higher latency than single autoregressive pass.
• Trade-Off: Quality vs. speed.

Unmasking Order:

• Challenge: Optimal unmasking order unclear.
• Heuristics: Confidence-based works but not optimal.
• Impact: Order affects generation quality.

Long-Range Dependencies:

• Challenge: Iterative unmasking may struggle with long-range coherence.
• Autoregressive Advantage: Left-to-right maintains coherence naturally.
• Mitigation: Careful schedule, more steps.

Examples & Implementations

D3PM (Discrete Denoising Diffusion Probabilistic Models):

• Approach: Absorbing state diffusion for discrete data.
• Application: Text, images, graphs.
• Performance: Competitive with autoregressive on some tasks.

MDLM (Masked Diffusion Language Model):

• Approach: Absorbing state diffusion specifically for language.
• Connection: Explicit connection to masked language modeling.
• Performance: Strong results on text generation benchmarks.

Applications

Text Infilling:

• Task: Fill in missing parts of text.
• Advantage: Naturally handles arbitrary masked spans.
• Use Case: Document completion, story writing.

Controlled Generation:

• Task: Generate text with constraints.
• Advantage: Easy to fix certain tokens, generate others.
• Use Case: Template filling, constrained generation.

Text Editing:

• Task: Modify specific parts of text.
• Advantage: Mask regions to edit, unmask with new content.
• Use Case: Paraphrasing, style transfer, improvement.

Tools & Resources

• Research Papers: D3PM, MDLM papers and code.
• Implementations: PyTorch/JAX implementations on GitHub.
• Experimental: Not yet in production frameworks.

Absorbing State Diffusion is a promising approach for discrete diffusion — by using masking as the corruption process, it provides a natural, interpretable way to apply diffusion to text that connects to successful masked language modeling, offering advantages in infilling, editing, and controllable generation while remaining simpler than continuous embedding approaches.

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