---
title: CDD - Constrained Discrete Diffusion
emoji: 🧬
colorFrom: green
colorTo: blue
sdk: docker
app_port: 7860
pinned: false
---

# Constrained Discrete Diffusion (CDD)

Implementation of **"Constrained Language Generation with Discrete Diffusion Models"** (Cardei et al., 2025 — [arXiv:2503.09790v3](https://arxiv.org/abs/2503.09790v3)).

CDD integrates discrete diffusion models with differentiable optimization to enforce constraints on text generation **without retraining**. It achieves **zero constraint violations** across toxicity mitigation, molecular generation, and instruction following.

## Key Idea

Discrete diffusion models (MDLM, UDLM) generate sequences by iteratively denoising from a noise distribution. At each reverse step, CDD inserts an **Augmented Lagrangian Method (ALM) projection** that modifies the denoiser's predicted token distributions to satisfy user-defined constraints:

```
z_T → x_θ(z_T) → [CDD Projection] → z_{T-1} → ... → z_0
```

The projection solves:
```
min_y  D_KL(x_θ || y)   s.t. g(argmax(y)) ∈ C
```

using Gumbel-Softmax relaxation for differentiability and ALM for constraint enforcement.

## Architecture

```
cdd/
├── samplers/
│   └── cdd_sampler.py      # Core ALM projection + sampling loop (§4)
├── constraints/
│   ├── toxicity.py          # Toxicity mitigation constraint (§5.1)
│   ├── molecular.py         # SA + Novelty constraints (§5.2)
│   └── instruction.py       # Counting + Lexical constraints (§5.3)
├── models/
│   └── load_models.py       # MDLM/UDLM model loading utilities
├── experiments/
│   ├── toxicity_mitigation.py     # Full toxicity experiment
│   ├── molecular_generation.py    # Full molecular experiment
│   ├── instruction_following.py   # Full instruction experiment
│   └── train_surrogates.py        # Surrogate model training
└── utils/
    ├── noise_schedule.py    # Diffusion math (schedules, posteriors)
    └── evaluation.py        # Metrics (PPL, entropy, violation rate)
```

## Installation

```bash
pip install -e .

# For GPU models (required for MDLM/UDLM inference):
pip install flash-attn --no-build-isolation

# For molecular evaluation:
pip install rdkit-pypi
```

## Quick Start

### 1. Unconstrained Generation with MDLM

```python
import torch
from cdd.models import load_mdlm
from cdd.samplers import CDDSampler, ALMConfig

# Load pretrained MDLM
model, tokenizer, info = load_mdlm(device="cuda")

# Create sampler (no constraints)
sampler = CDDSampler(
    model=model,
    tokenizer=tokenizer,
    constraint_fn=lambda x: torch.tensor(0.0),  # No constraint
    diffusion_type="mdlm",
    num_timesteps=1000,
    seq_length=128,
    device="cuda",
)

result = sampler.sample(batch_size=1)
print(result["text"][0])
```

### 2. Toxicity-Constrained Generation

```python
from cdd.samplers import CDDSampler, ALMConfig, TOXICITY_ALM_CONFIG

# Setup toxicity constraint (§5.1)
from cdd.constraints import create_toxicity_constraint
constraint = create_toxicity_constraint(threshold=0.5)

sampler = CDDSampler(
    model=model,
    tokenizer=tokenizer,
    constraint_fn=constraint,
    alm_config=TOXICITY_ALM_CONFIG,  # λ=0, μ=1, K=1000, M=10, η=0.2
    diffusion_type="mdlm",
    device="cuda",
)

# Conditional generation from toxic prompt
prefix = tokenizer.encode("The politician was accused of", return_tensors="pt").to("cuda")
result = sampler.sample(batch_size=1, prefix_ids=prefix)
print(result["text"][0])  # Non-toxic completion
```

### 3. Molecular Generation with SA Constraint

```python
from cdd.models import load_udlm
from cdd.samplers import CDDSampler, MOLECULAR_SA_ALM_CONFIG

model, tokenizer, info = load_udlm("kuleshov-group/udlm-qm9", device="cuda")

sampler = CDDSampler(
    model=model,
    tokenizer=tokenizer,
    constraint_fn=sa_constraint,  # SA ≤ 3.5
    alm_config=MOLECULAR_SA_ALM_CONFIG,  # λ=0, μ=1, μ_max=1000, K=1000, M=100, η=1.0
    diffusion_type="udlm",
    num_timesteps=1000,
    seq_length=32,
    device="cuda",
)

result = sampler.sample(batch_size=32)
```

## Reproducing Paper Experiments

### Toxicity Mitigation (§5.1, Table 3)

```bash
python -m cdd.experiments.toxicity_mitigation \
    --threshold 0.5 \
    --num_samples 1000 \
    --device cuda
```

Expected results (CDD τ=0.50): PPL ≈ 59.44, Violation = 0.0%

### Molecular Generation (§5.2, Table 4)

```bash
python -m cdd.experiments.molecular_generation \
    --sa_threshold 3.5 \
    --num_samples 1000 \
    --device cuda
```

### Instruction Following (§5.3, Table 5)

```bash
# Counting task
python -m cdd.experiments.instruction_following \
    --task counting --num_samples 100

# Lexical task
python -m cdd.experiments.instruction_following \
    --task lexical --num_samples 100
```

### Train Surrogate Models

```bash
# Toxicity surrogate (GPT-Neo 1.3B on Jigsaw)
python -m cdd.experiments.train_surrogates --task toxicity

# SA surrogate (GPT-2 on QM9)
python -m cdd.experiments.train_surrogates --task sa
```

## ALM Hyperparameters (Appendix B & C)

| Task | λ₀ | μ₀ | μ_max | K (outer) | M (inner) | η |
|------|-----|-----|-------|-----------|-----------|------|
| Toxicity (NLP) | 0.0 | 1.0 | — | 1000 | 10 | 0.20 |
| Molecular (QM9) | 0.0 | 1.0 | 1000 | 1000 | 100 | 1.0 |

## Base Models

| Model | Type | Params | Vocab | Length | Trained On |
|-------|------|--------|-------|--------|------------|
| [kuleshov-group/mdlm-owt](https://huggingface.co/kuleshov-group/mdlm-owt) | MDLM | 110M | 50,258 | 1024 | OpenWebText |
| [kuleshov-group/udlm-qm9](https://huggingface.co/kuleshov-group/udlm-qm9) | UDLM | 92M | 40 | 32 | QM9 |
| [kuleshov-group/udlm-lm1b](https://huggingface.co/kuleshov-group/udlm-lm1b) | UDLM | 139M | 30,522 | 128 | LM1B |

## Method Details

### Gumbel-Softmax Relaxation (Eq. 6)

Makes `argmax` differentiable:
```
φ̃(x)(v) = exp((log x(v) + ξ_v) / T) / Σ exp((log x(v') + ξ_v') / T)
```

### Augmented Lagrangian (Eq. 7-8)

Optimization at each diffusion step:
```
L(y, λ, μ) = D_KL(x_θ || y) + λ·g(ỹ) + (μ/2)·g(ỹ)²
```

Update rules:
```
λ ← λ + μ·g(ỹ)
μ ← min(2μ, μ_max)
```

### MDLM vs UDLM for CDD

| Property | MDLM | UDLM |
|----------|------|------|
| Noise type | Absorbing ([MASK]) | Uniform |
| Tokens re-sampled each step | Only masked | All |
| CDD paper usage | Text tasks | Molecular |
| Time conditioning | No | Yes |

## Citation

```bibtex
@article{cardei2025constrained,
  title={Constrained Language Generation with Discrete Diffusion Models},
  author={Cardei, Michael and Christopher, Jacob K and Hartvigsen, Thomas 
          and Bartoldson, Brian R. and Kailkhura, Bhavya and Fioretto, Ferdinando},
  journal={arXiv preprint arXiv:2503.09790},
  year={2025}
}
```