papers/adjoint-matching/sau.json

{
  "paper_id": "adjoint-matching",
  "paper_title": "Adjoint Matching: Fine-tuning Flow and Diffusion Models with Memoryless SOC",
  "D1": [
    {
      "id": "adjoint-matching-D1-001",
      "claim": "Image resolution for autoencoder pre-training and generation: 512×512",
      "source": "Section 7"
    },
    {
      "id": "adjoint-matching-D1-002",
      "claim": "Number of discretization timesteps K=40 for fine-tuning (step size h=1/K=0.025)",
      "source": "Appendix G, Section G.1"
    },
    {
      "id": "adjoint-matching-D1-003",
      "claim": "Practical noise schedule with offsets: σ(t)=√(2(1-t+h)/(t+h)), h=0.025",
      "source": "Appendix G.1, Eq. 236"
    },
    {
      "id": "adjoint-matching-D1-004",
      "claim": "Number of gradient evaluation timesteps per iteration: 20 (10 uniform from [0,0.725] + last 10 from [0.75,0.975])",
      "source": "Appendix G.2"
    },
    {
      "id": "adjoint-matching-D1-005",
      "claim": "Adam optimizer: lr=2e-5 (default), lr=1e-5 (Discrete Adjoint only), β₁=0.95, β₂=0.999, ε=1e-8, weight_decay=0.01, gradient clip=1.0",
      "source": "Appendix G"
    },
    {
      "id": "adjoint-matching-D1-006",
      "claim": "Alternative optimizer config tested: lr=3e-5, β₁=0.97",
      "source": "Appendix G, Table 6"
    },
    {
      "id": "adjoint-matching-D1-007",
      "claim": "Adjoint matching method training precision: bfloat16 (all experiments)",
      "source": "Appendix G"
    },
    {
      "id": "adjoint-matching-D1-008",
      "claim": "Effective batch size: 40 (2× 80GB A100 GPUs, batch 20 each)",
      "source": "Appendix G"
    },
    {
      "id": "adjoint-matching-D1-009",
      "claim": "Fine-tuning prompts per run: 40k (sampled from 100k total), test prompts per run: 1k, 3 independent runs per config",
      "source": "Appendix G, Section 7"
    },
    {
      "id": "adjoint-matching-D1-010",
      "claim": "Iterations per epoch: 1000 (40k prompts / batch 40); Adjoint Matching runs: 1000 iters (1 epoch)",
      "source": "Appendix G"
    },
    {
      "id": "adjoint-matching-D1-011",
      "claim": "Baseline iteration counts: DRaFT-1=4000, DRaFT-40=4000, DPO=1500, ReFL=6000, Continuous Adjoint=750, Discrete Adjoint=1000",
      "source": "Appendix G, Table 3"
    },
    {
      "id": "adjoint-matching-D1-012",
      "claim": "Reward scaling factor λ values: [1000, 2500, 12500] for Adjoint Matching tradeoff",
      "source": "Section 7, Table 2"
    },
    {
      "id": "adjoint-matching-D1-013",
      "claim": "Loss Clipping Threshold (Adjoint Matching): LCT=1.6×λ²; LCT (Continuous Adjoint): 1600×λ²",
      "source": "Appendix G.3"
    },
    {
      "id": "adjoint-matching-D1-014",
      "claim": "Classifier-free guidance weights evaluated: w ∈ [0.0, 1.0, 4.0]",
      "source": "Section 7, Table 5"
    },
    {
      "id": "adjoint-matching-D1-015",
      "claim": "Diversity metric: 40 generations per prompt across 25 prompts; DreamSim features for pairwise distance",
      "source": "Appendix G.4, Eq. 238"
    },
    {
      "id": "adjoint-matching-D1-016",
      "claim": "Inference timesteps ablated: [10, 20, 40, 100, 200] (fine-tuning always at K=40)",
      "source": "Table 8"
    },
    {
      "id": "adjoint-matching-D1-017",
      "claim": "Wall-clock time per iteration: Adjoint Matching=156s, Discrete Adjoint=152s, Continuous Adjoint=204s",
      "source": "Table 3, Appendix G.5"
    }
  ],
  "D2": [
    {
      "id": "adjoint-matching-D2-001",
      "claim": "Unified SDE for Flow Matching and Diffusion Models: dX_t = b(X_t,t)dt + σ(t)dB_t, where b(x,t)=κ_t·x + (σ(t)²/2 + η_t)·𝔰(x,t), κ_t=α̇_t/α_t, η_t=β_t(α̇_t/α_t·β_t − β̇_t)",
      "source": "Section 3, Eq. 10-11"
    },
    {
      "id": "adjoint-matching-D2-002",
      "claim": "Continuous-time DDIM SDE: dX_t = (α̇̄_t/(2ᾱ_t)·X_t − (α̇̄_t/(2ᾱ_t) + σ(t)²/2)·ε^base(X_t,t)/√(1−ᾱ_t))dt + σ(t)dB_t, X_0~N(0,I)",
      "source": "Section 3, Eq. 6"
    },
    {
      "id": "adjoint-matching-D2-003",
      "claim": "Flow Matching velocity to score function: v^base(x,t) = α̇_t/α_t·x + β_t(α̇_t/α_t·β_t − β̇_t)·𝔰(x,t)",
      "source": "Section 3, Eq. 8"
    },
    {
      "id": "adjoint-matching-D2-004",
      "claim": "Noise predictor to score function: 𝔰(x,t) = −ε^base(x,t)/√(1−ᾱ_t)",
      "source": "Section 3, Eq. 9"
    },
    {
      "id": "adjoint-matching-D2-005",
      "claim": "Memoryless noise schedule condition (Proposition 1): σ(t)² = 2η_t. This ensures X_0 and X_1 are independent, removing initial value function bias.",
      "source": "Section 4.3, Proposition 1, Eq. 25"
    },
    {
      "id": "adjoint-matching-D2-006",
      "claim": "DDIM fine-tuning drift+control (memoryless σ=√(2η_t)): b+σu = α̇̄_t/(2ᾱ_t)·x − α̇̄_t/ᾱ_t · ε^finetune(x,t)/√(1−ᾱ_t); control u = −√(α̇̄_t/(ᾱ_t(1−ᾱ_t))) · (ε^finetune − ε^base)",
      "source": "Section 4.3, Eq. 26"
    },
    {
      "id": "adjoint-matching-D2-007",
      "claim": "Flow Matching fine-tuning drift+control (memoryless σ=√(2η_t)): b+σu = 2·v^finetune(x,t) − α̇_t/α_t·x; control u = √(2/(β_t(α̇_t/α_t·β_t−β̇_t))) · (v^finetune − v^base)",
      "source": "Section 4.3, Eq. 27"
    },
    {
      "id": "adjoint-matching-D2-008",
      "claim": "Flow Matching Euler-Maruyama sampling step during fine-tuning: X_{t+h} = X_t + h·(2·v^finetune_θ(X_t,t) − α̇_t/α_t·X_t) + √h·σ(t)·ε_t, ε_t~N(0,I), X_0~N(0,I)",
      "source": "Section 5.2, Algorithm 1, Eq. 40"
    },
    {
      "id": "adjoint-matching-D2-009",
      "claim": "Lean Adjoint ODE (backward): d/dt ã(t;X) = −ã(t;X)^T·∇_x b(X_t,t), with terminal condition ã(1;X) = −∇_{X_1} r(X_1). Solved backwards via Euler: ã_{t−h} = ã_t + h·ã_t^T·∇_{X_t}(2·v^base(X_t,t) − α̇_t/α_t·X_t)",
      "source": "Section 5.2, Eq. 38-39, Algorithm 1, Eq. 41"
    },
    {
      "id": "adjoint-matching-D2-010",
      "claim": "Adjoint Matching loss (Flow Matching): L_AdjMatch(θ) = Σ_{t∈{0,...,1−h}} ||(2/σ(t))·(v^finetune_θ(X_t,t) − v^base(X_t,t)) + σ(t)·ã_t||², where X_t and ã_t are stop-grad",
      "source": "Section 5.2, Eq. 37, Algorithm 1, Eq. 42"
    },
    {
      "id": "adjoint-matching-D2-011",
      "claim": "Clipped Adjoint Matching loss (practical): L̂_AdjMatch(θ) = Σ_{t∈κ} min(LCT, ||(2/σ(t))·(v^finetune − v^base) + σ(t)·ã_t||²), where κ is random timestep subset (20 steps)",
      "source": "Appendix G.3, Eq. 237"
    },
    {
      "id": "adjoint-matching-D2-012",
      "claim": "Reward function for fine-tuning: r(x) = λ × ImageReward(x), where ImageReward from Xu et al. 2023",
      "source": "Section 7, Eq. 40"
    },
    {
      "id": "adjoint-matching-D2-013",
      "claim": "Target tilted distribution: p*(X_1) ∝ p^base(X_1) · exp(r(X_1))",
      "source": "Section 1, Eq. 1; Section 5.2"
    },
    {
      "id": "adjoint-matching-D2-014",
      "claim": "SOC objective for fine-tuning (KL-regularized): min_u E[∫₀¹(½||u(X_t^u,t)||² + f(X_t^u,t))dt + g(X_1^u)], s.t. dX_t^u = (b(X_t^u,t)+σ(t)u(X_t^u,t))dt + σ(t)dB_t, with f=0, g=−r",
      "source": "Section 4.1, Eq. 12-13; Section 4.2"
    },
    {
      "id": "adjoint-matching-D2-015",
      "claim": "SOC optimal control via value function: u*(x,t) = −σ(t)^T·∇_x V(x,t), where V(x,t) = −log E_{X~p^base}[exp(−∫f ds − g(X_1)) | X_t=x]",
      "source": "Section 4.1, Eq. 16-17"
    },
    {
      "id": "adjoint-matching-D2-016",
      "claim": "Classifier-free guidance formula: v_guided(x,t) = (1+w)·v(x,t|y) − w·v(x,t), where w is guidance weight, v(x,t|y) is conditional model, v(x,t) is unconditional",
      "source": "Section 7, Eq. 41; Appendix Table 5"
    },
    {
      "id": "adjoint-matching-D2-017",
      "claim": "DDPM SDE (special case of DDIM with memoryless σ): dX_t = (α̇̄_t/(2ᾱ_t)·X_t − α̇̄_t/ᾱ_t · ε^base(X_t,t)/√(1−ᾱ_t))dt + √(α̇̄_t/ᾱ_t)dB_t",
      "source": "Section 3, Eq. 7"
    },
    {
      "id": "adjoint-matching-D2-018",
      "claim": "U-Net architecture for text-conditional Flow Matching model on latent variables (similar to Rombach et al. 2022 LDM setup): encoder-decoder with skip connections at each resolution level; encoder: x_enc^{(l)} = DownBlock(ResNet(Conv(x_enc^{(l-1)}))) with channel multiplier [C, 2C, 4C, 8C]; decoder: x_dec^{(l)} = ResNet(Conv(Concat(Up(x_dec^{(l-1)}), x_enc^{(l)}))); mid-block with self-attention at lowest resolution (4 heads); text conditioning via cross-attention injected at each resolution level",
      "source": "Section 7"
    },
    {
      "id": "adjoint-matching-D2-019",
      "claim": "DDIM discrete update rule: X_{k+1} = √(ᾱ_{k+1})·(X_k − √(1−ᾱ_k)·ε(X_k,k))/√(ᾱ_k) + √(1−ᾱ_{k+1}−σ_k²)·ε(X_k,k) + σ_k·ε_k",
      "source": "Section 2.2, Eq. 5"
    },
    {
      "id": "adjoint-matching-D2-020",
      "claim": "Continuous Adjoint gradient: dL/dθ = ½∫₀¹ ∂/∂θ||u(X_t,t)||²dt + ∫₀¹ (∂u(X_t,t)/∂θ)^T·σ(t)^T·a(t;X,u)dt, where a(t) is the full adjoint state solving da/dt = −[a^T·∇_x(b+σu) + ∇_x(f+½||u||²)], a(1)=∇g(X_1)",
      "source": "Section 5.1.1, Eq. 30-32"
    }
  ],
  "D3": [
    {
      "id": "adjoint-matching-D3-001",
      "claim": "Main experiment: Fine-tune Flow Matching text-to-image model (512×512, latent space) with ImageReward. Compare Adjoint Matching (λ=1000/2500/12500) against DRaFT-1, DRaFT-40, DPO, ReFL, Continuous Adjoint, and Discrete Adjoint. Evaluate on text-to-image consistency (ClipScore, PickScore), unseen human preference (HPSv2), and sample diversity (DreamSim Diversity).",
      "source": "Section 7, Table 2, Table 3"
    },
    {
      "id": "adjoint-matching-D3-002",
      "claim": "Reward tradeoff ablation: Vary λ=[1000, 2500, 12500] for Adjoint Matching to study KL-regularization vs reward optimization tradeoff. Higher λ increases consistency/human preference but reduces diversity. Compared against DRaFT-1 with varying iterations (early stopping) as alternative tradeoff mechanism.",
      "source": "Section 7, Figure 3, Figure 5, Table 2"
    },
    {
      "id": "adjoint-matching-D3-003",
      "claim": "Classifier-free guidance ablation: Apply CFG weights w=[0.0, 1.0, 4.0] after fine-tuning across Adjoint Matching and DRaFT-1. Higher w improves text-to-image consistency at cost of diversity. Note: only conditional model is fine-tuned; unconditional model is base.",
      "source": "Section 7, Figure 4, Table 5"
    },
    {
      "id": "adjoint-matching-D3-004",
      "claim": "Noise schedule ablation: Compare fine-tuning with memoryless σ(t)=√(2η_t) vs constant σ(t)=1 for Adjoint Matching (λ=12500). Constant σ(t) suffers from initial value function bias — reward optimization underperforms. Also compare sampling with σ(t)=√(2η_t) vs σ(t)=0 after fine-tuning.",
      "source": "Section 7, Table 2, Table 7"
    },
    {
      "id": "adjoint-matching-D3-005",
      "claim": "Inference timesteps ablation: Vary number of sampling steps [10, 20, 40, 100, 200] while keeping fine-tuning at 40 steps. Evaluate Adjoint Matching (λ=12500) and DRaFT-1 against base model. Fine-tuned models robustly improve over base even at low step counts.",
      "source": "Section 7, Table 8"
    },
    {
      "id": "adjoint-matching-D3-006",
      "claim": "Optimizer hyperparameters ablation: Compare lr=3e-5, β₁=0.97 vs default lr=2e-5, β₁=0.95 for DRaFT-1 and Adjoint Matching (λ=1200). Also test Discrete Adjoint stability — default hyperparams cause instability, requiring lr reduction to 1e-5.",
      "source": "Appendix G, Table 6"
    },
    {
      "id": "adjoint-matching-D3-007",
      "claim": "Diversity evaluation protocol: For each prompt, generate 40 samples, compute pairwise DreamSim feature distances across 25 prompts to quantify sample diversity. Metrics: ClipScore diversity (variance of Clip embeddings) and DreamSim diversity (pairwise feature distance).",
      "source": "Appendix G.4, Eq. 238"
    }
  ],
  "D4": [
    {
      "id": "adjoint-matching-D4-001",
      "claim": "[Phase 3/4 — Adjoint Matching Training Loop (Algorithm 1)] ENTRY: (from Phase 2) Memoryless fine-tuning recipe established with σ(t)=√(2η_t); control u parameterized via fine-tuned velocity v^finetune (Eq. 27); base model v^base pre-trained; reward model r(x)=λ×ImageReward(x) provided; hyperparameters configured (N iterations, step size h, step count K=40, m trajectories, optimizer). EXECUTION: Step 1 — Initialize fine-tuned vector field v^finetune = v^base with parameters θ. Step 2 — Sample m trajectories forward from t=0 to t=1 via Euler-Maruyama with memoryless schedule: X_{t+h} = X_t + h·(2·v^finetune_θ(X_t,t) − α̇_t/α_t·X_t) + √h·σ(t)·ε_t, ε_t~N(0,I), X_0~N(0,I). Step 3 — For each trajectory, solve lean adjoint ODE backwards from t=1 to t=0: ã_{t−h} = ã_t + h·ã_t^T·∇_{X_t}(2·v^base(X_t,t) − α̇_t/α_t·X_t), terminal condition ã_1 = −∇_{X_1}r(X_1); X_t and ã_t use stop_grad. Step 4 — Compute clipped Adjoint Matching loss over random timestep subset κ (20 steps): L̂_AdjMatch(θ) = Σ_{t∈κ} min(LCT, ||(2/σ(t))·(v^finetune − v^base) + σ(t)·ã_t||²). Step 5 — Compute gradient ∇_θ L̂ and update θ via Adam. Repeat Steps 1-5 for N=1000 iterations (1 epoch). EXIT: (output to Phase 4) Fine-tuned vector field v^finetune with optimized parameters θ; ready for sampling with any noise schedule to produce tilted-distribution samples.",
      "source": "Section 5.2, Algorithm 1"
    },
    {
      "id": "adjoint-matching-D4-002",
      "claim": "[Phase 2→3→4 Boundary — Fine-tuning/Sampling Separation (Theorem 1)] ENTRY: (from Phase 1) Unified SDE framework dX_t = b(X_t,t)dt + σ(t)dB_t (Eq. 10-11) established; base model trained (FM or diffusion); reward model r(x) available; target tilted distribution p*(X_1) ∝ p^base(X_1)·exp(r(X_1)) defined. CORE PRINCIPLE (Phase 2→3): Fine-tuning MUST enforce memoryless noise schedule σ(t)=√(2η_t) during the SOC optimization in Phase 3 — this is the unique choice (Proposition 1) that makes X_0 ⟂ X_1, thereby removing the initial value function bias V(X_0,0) from the optimal distribution (converting Eq. 23 to Eq. 24). Concretely: fine-tuning uses σ(t)=√(2η_t) which, for DDIM, recovers the continuous-time DDPM process (Eq. 7); for Flow Matching, defines Memoryless Flow Matching (Eq. 27). SEPARATION (Phase 3→4): After fine-tuning completes (Algorithm 1 converges), the fine-tuned vector field v^finetune (or ε^finetune for diffusion) can be plugged into the original generative process with ANY noise schedule for sampling — commonly σ(t)=0 (noiseless ODE sampling, Eq. 3) or the original DDIM noise schedule (Eq. 6). The tilted distribution guarantee holds regardless of the sampling σ(t). EXIT: (feeds Phase 4) Fine-tuned model is schedule-agnostic for sampling; same trained weights support σ(t) ∈ {0, √(α̇̄_t/ᾱ_t), arbitrary} to generate samples from p*(X_1).",
      "source": "Section 4.3, Theorem 1"
    },
    {
      "id": "adjoint-matching-D4-003",
      "claim": "[Phase 1→2 — Memoryless Schedule Derivation Pipeline (Sections 3→4→5)] ENTRY: (starting condition) Existing pre-trained base generative model (Flow Matching or diffusion); known reference flow coefficients α_t, β_t satisfying α_0=β_1=0, α_1=β_0=1 (Eq. 2); base velocity v^base (FM) or noise predictor ε^base (diffusion) pre-trained. EXECUTION: Step 1 (Section 3, Eqs. 6-11) — Unify FM and DDIM into common SDE parameterization: express v^base and ε^base in terms of the same score function 𝔰(x,t) (Eqs. 8-9), then derive the unified SDE dX_t = b(X_t,t)dt + σ(t)dB_t where b(x,t)=κ_t·x + (σ(t)²/2 + η_t)·𝔰(x,t) (Eqs. 10-11). Output: single framework covering all dynamical generative models. Step 2 (Section 4.1-4.2, Eqs. 12-23) — Formulate reward fine-tuning as SOC control-affine problem (Eqs. 12-13) and derive optimal distribution p*(X) = p^base(X)·exp(r(X_1) + V(X_0,0)) (Eq. 23). Diagnose: the V(X_0,0) bias term originates from X_0↔X_1 dependence — when X_0 determines X_1 (e.g., noiseless ODE), V(X_0,0) cannot be factored out, preventing convergence to the tilted distribution. Output: problem statement identifying the bias source. Step 3 (Section 4.3, Proposition 1, Definition 1) — Prove memoryless condition: σ(t)² = 2η_t is necessary and sufficient for X_0 ⟂ X_1 (Definition 1 + Proposition 1). Under this condition, p^base(X_0,X_1) = p^base(X_0)·p^base(X_1), so V(X_0,0) integrates out: p*(X_1) ∝ p^base(X_1)·exp(r(X_1)) (Eq. 24). Theorem 1 further proves this is the only noise schedule that enables arbitrary-schedule sampling after fine-tuning. Output: memoryless noise schedule σ(t)=√(2η_t). Step 4 (Section 4.3 Eqs. 26-27, Section 5.2 Algorithm 1) — Express fine-tuning drift+control in DDIM form (Eq. 26) and Flow Matching form (Eq. 27); parameterize control u in terms of ε^finetune or v^finetune; apply Adjoint Matching algorithm to solve the resulting SOC as a least-squares regression (Eqs. 37-39, Algorithm 1). EXIT: (output to Phase 3 execution) Complete fine-tuning recipe: memoryless schedule σ(t)=√(2η_t); control u parameterized; Adjoint Matching objective and lean adjoint ODE defined; ready to execute Algorithm 1 training loop.",
      "source": "Section 3, Section 4.3, Section 5.2"
    }
  ]
}