papers/lora-sb/sau.json

{
  "paper_id": "lora-sb",
  "paper_title": "LoRA-SB: Initialization using Update Approximation for Efficient Low-Rank Fine-Tuning",
  "D1": [
    {
      "id": "lora-sb-D1-001",
      "claim": "LoRA rank configuration: r ∈ {32, 64, 96} for Mistral-7B, Gemma-2 9B, Llama-3.2 3B; r ∈ {8, 16, 24} for RoBERTa-large on GLUE.",
      "source": "Sec 3.1, Sec 3.2, Sec 3.3, Tables 1-3"
    },
    {
      "id": "lora-sb-D1-002",
      "claim": "Scaling factor: s = 1 in LoRA-SB (no α/r tuning); s = α/r in standard LoRA (baseline reference). Parameter count: r² trainable params in LoRA-SB (only R), vs r(m+n) in standard LoRA.",
      "source": "Sec 2.1, Sec 2.5, Sec 2.6, Eq 2, Eq 3, Eq 9"
    },
    {
      "id": "lora-sb-D1-003",
      "claim": "LoRA-SB matrix dimensions: B ∈ ℝ^{m×r} (frozen), A ∈ ℝ^{r×n} (frozen), R ∈ ℝ^{r×r} (trainable). LoRA target modules: key, value, query, attention output, all FC layers (Mistral-7B/Gemma-2 9B/Llama-3.2 3B); self-attention layers only (RoBERTa-large on GLUE).",
      "source": "Sec 2.1, Eq 3, Sec 3.1, Sec 3.2, Sec 3.3"
    },
    {
      "id": "lora-sb-D1-004",
      "claim": "Initialization data subsample fraction: 0.1% (1/1000) of each dataset's total training size. For MetaMathQA (50K total), this yields 50 samples. Ablation range for initialization sample count: n ∈ {1, 5, 25, 50, 100, 200, 500} evaluated on Mistral-7B and Gemma-2 9B.",
      "source": "Sec 3, Sec 4, Table 5, Appendix H"
    },
    {
      "id": "lora-sb-D1-005",
      "claim": "Initialization noise ablation levels: Gaussian noise N(μ=0, σ²) with σ ∈ {10⁻², 10⁻³, 10⁻⁴, 10⁻⁵} added to ΔW_avg. Kaiming random initialization also tested as a control. All variants use truncated SVD for orthonormal B/A, isolating initialization quality as the sole variable.",
      "source": "Sec 4, Table 4"
    },
    {
      "id": "lora-sb-D1-006",
      "claim": "Training hyperparams for Mistral-7B / Gemma-2 9B on MetaMathQA: lr=1e-4, batch_size=1, max_seq_len=512, grad_acc_steps=32, epochs=1, dropout=0, warmup_ratio=0.02, lr_scheduler=Cosine, optimizer=AdamW.",
      "source": "Appendix H, Table 8"
    },
    {
      "id": "lora-sb-D1-007",
      "claim": "Training hyperparams for Llama-3.2 3B on COMMONSENSE170K: lr=2e-3, batch_size=6, max_seq_len=256, grad_acc_steps=24, epochs=2, dropout=0.05, warmup_ratio=0.02, lr_scheduler=Linear, optimizer=AdamW.",
      "source": "Appendix H, Table 8"
    },
    {
      "id": "lora-sb-D1-008",
      "claim": "Training hyperparams for RoBERTa-large on GLUE: lr=1e-3, epochs=30, dropout=0, warmup_ratio=0.06, lr_scheduler=Linear, optimizer=AdamW. Batch size varies per task: 30 (CoLA/STS-B), 128 (RTE/MRPC), 256 (SST-2), 15 (QNLI).",
      "source": "Appendix H, Table 9"
    },
    {
      "id": "lora-sb-D1-009",
      "claim": "LoRA-XS baseline config: α = r for arithmetic/commonsense tasks (Mistral/Gemma/Llama); α = 16 for GLUE (RoBERTa-large). Used as reference PEFT method sharing same architecture as LoRA-SB.",
      "source": "Appendix H"
    },
    {
      "id": "lora-sb-D1-010",
      "claim": "Hardware and infrastructure: 1× NVIDIA A6000 GPU, torch.bfloat16 base model precision, 3 random seeds for result averaging. SVD via torch.svd_lowrank (< 1s per model).",
      "source": "Appendix H, Appendix F"
    },
    {
      "id": "lora-sb-D1-011",
      "claim": "Training datasets: MetaMathQA (50K, math reasoning), COMMONSENSE170K (8-task combined, commonsense reasoning). Evaluation datasets: GSM8K and MATH (math); BoolQ, PIQA, SIQA, HellaSwag, WinoGrande, ARC-e, ARC-c, OBQA (commonsense); CoLA, RTE, MRPC, STS-B, QNLI, SST-2 (GLUE).",
      "source": "Sec 3.1, Sec 3.2, Sec 3.3, Appendix I"
    },
    {
      "id": "lora-sb-D1-012",
      "claim": "AdamW first-step approximation: ε ≈ 0 implies update ≈ −η × sign(g₁) at the first optimizer step. LoRA-SB leverages this to compute ΔW_avg = −η sign(Σ_i ∇_W L(W₀, x_i)) during initialization.",
      "source": "Appendix C, Eq 13"
    }
  ],
  "D2": [
    {
      "id": "lora-sb-D2-001",
      "claim": "LoRA-SB forward pass: W = W₀ + s B R A, where W₀ frozen, B (m×r) and A (r×n) fixed after initialization, R (r×r) sole trainable. s = 1 in LoRA-SB. Only R updated during training.",
      "source": "Sec 2.1, Eq 3"
    },
    {
      "id": "lora-sb-D2-002",
      "claim": "Standard LoRA forward pass (baseline): W = W₀ + s B A, where B (m×r) and A (r×n) both trainable, s = α/r. Total learnable params: r(m+n).",
      "source": "Sec 2.1, Eq 2"
    },
    {
      "id": "lora-sb-D2-003",
      "claim": "LoRA-XS gradient w.r.t. R: g^R_{LoRA-XS} = s Bᵀ g Aᵀ, where g = ∂L/∂W is the full weight gradient, B and A are fixed matrices. Relates the trainable R gradient to the full fine-tuning gradient g.",
      "source": "Sec 2.3, Lemma 2"
    },
    {
      "id": "lora-sb-D2-004",
      "claim": "Equivalent gradient definition: g̃ = s B g^R A, where g̃ is the virtual update to W induced by updates to R. Computed as the matrix product of fixed B, gradient g^R, and fixed A, scaled by s.",
      "source": "Sec 2.3, Definition 1"
    },
    {
      "id": "lora-sb-D2-005",
      "claim": "Optimal gradient computation (general, non-orthonormal B/A): g^R = (1/s²) (BᵀB)⁻¹ g^R_{LoRA-XS} (AAᵀ)⁻¹. Closed-form solution minimizing ||g̃ − g||²_F, making the equivalent gradient optimally approximate the full FT gradient. Requires B, A full-rank.",
      "source": "Sec 2.3, Theorem 3"
    },
    {
      "id": "lora-sb-D2-006",
      "claim": "Simplified optimal gradient (orthonormal B/A): g^R = (1/s²) g^R_{LoRA-XS}, when BᵀB = I and AAᵀ = I. Eliminates matrix inversion at each optimizer step.",
      "source": "Sec 2.6, Eq 8"
    },
    {
      "id": "lora-sb-D2-007",
      "claim": "Final LoRA-SB gradient update rule (s=1, orthonormal B/A): g^R = g^R_{LoRA-XS} = Bᵀ g Aᵀ. Gradient of R equals raw LoRA-XS gradient without scaling or inversion.",
      "source": "Sec 2.6, Eq 9"
    },
    {
      "id": "lora-sb-D2-008",
      "claim": "R parameter update step: R ← R − η g^R, where η is the learning rate (sufficiently small). R is the sole trainable parameter; B and A remain frozen throughout training. The update is applied by the AdamW optimizer each step.",
      "source": "Sec 2.3, Theorem 4; Sec 2.6"
    },
    {
      "id": "lora-sb-D2-009",
      "claim": "Initialization gradient estimation (single sample): ΔW_first_step = −η × sign(∇_W L(W₀, x_i)). Uses sign() to approximate AdamW first-step behavior (moments initialized to zero).",
      "source": "Sec 2.4, Eq 5; Appendix C"
    },
    {
      "id": "lora-sb-D2-010",
      "claim": "Initialization averaged gradient estimation (multi-sample, used in practice): ΔW_avg = −η sign(Σ_{i=0}^{n≤|X|} ∇_W L(W₀, x_i)), where n = 0.1% of training data, x_i randomly sampled, Σ accumulates per-sample gradients, sign() applied element-wise to the sum.",
      "source": "Sec 2.4, Eq 6; Appendix C"
    },
    {
      "id": "lora-sb-D2-011",
      "claim": "Initialization: truncated SVD and factor assignment. U, S, Vᵀ = SVD(ΔW_avg); B_init = U[:, :r]; A_init = (V[:, :r])ᵀ; R_init = (1/s) S[:r, :r]. By Eckart-Young theorem, yields optimal rank-r approximation s B_init R_init A_init ≈ ΔW_avg. With s=1, R_init = S[:r, :r]. Uses torch.svd_lowrank.",
      "source": "Sec 2.6, Eq 7-9; Sec 4; Appendix D, Algorithm 1"
    },
    {
      "id": "lora-sb-D2-012",
      "claim": "Memory-efficient layerwise gradient computation for initialization: accumulate ΔW_avg = -η sign(Σ_{i=0}^{n} ∇_W L(W₀, x_i)) via backward hooks — for each layer l, compute g_l = ∂L/∂W_l(W₀, x_i), accumulate S_l ← S_l + sign(g_l), immediately discard g_l; total memory = O(1) independent of layer count. The sign() approximation matches AdamW first-step behavior (moments initialized to zero: m₀=0, v₀=0 → update ≈ -η · sign(g₁)). Peak memory never exceeds LoRA-SB fine-tuning or standard LoRA training.",
      "source": "Sec 2.6"
    },
    {
      "id": "lora-sb-D2-013",
      "claim": "LoRA-SB complete algorithm: Step 1: Load pre-trained model. Step 2: Estimate ΔW_avg via memory-efficient multi-sample gradient accumulation. Step 3: Compute truncated SVD on ΔW_avg. Step 4: Assign B (U[:,:r]), R (S[:r,:r]/s), A (V[:r,:r]ᵀ). Step 5: Convert model to LoRA-SB (replace linear layers). Step 6: Train with AdamW updating only R via g^R = g^R_{LoRA-XS} (s=1, orthonormal B/A).",
      "source": "Appendix D, Algorithm 1"
    },
    {
      "id": "lora-sb-D2-014",
      "claim": "LoRA-SB ablated variant (good init, no optimal gradient approximation): B_init = U[:r] S[:r,:r], A_init = V[:r]ᵀ, R_init = I. Same SVD source (good initialization) but BᵀB ≠ I, so optimal gradient g^R = (1/s²)(BᵀB)⁻¹ g^R_{LoRA-XS}(AAᵀ)⁻¹ requires explicit inversion. Used to isolate optimal gradient effect from initialization quality.",
      "source": "Sec 4, Eq 11-12"
    }
  ],
  "D3": [
    {
      "id": "lora-sb-D3-001",
      "claim": "Compare LoRA-SB against Full FT and PEFT baselines (LoRA, rsLoRA, PiSSA, DoRA, LoRA-Pro, LoRA-XS) on arithmetic reasoning. Mistral-7B and Gemma-2 9B fine-tuned on MetaMathQA (50K) at ranks r ∈ {32, 64, 96}. LoRA-SB init uses 0.1% training data. Evaluation: GSM8K and MATH (numeric answer accuracy).",
      "source": "Sec 3.1, Table 1, Appendix H, Table 8, Appendix I"
    },
    {
      "id": "lora-sb-D3-002",
      "claim": "Compare LoRA-SB against Full FT and PEFT baselines on commonsense reasoning. Llama-3.2 3B fine-tuned on COMMONSENSE170K (8-task combined) at ranks r ∈ {32, 64, 96}. Evaluation: BoolQ, PIQA, SIQA, HellaSwag, WinoGrande, ARC-e, ARC-c, OBQA, plus average accuracy.",
      "source": "Sec 3.2, Table 2, Appendix H, Table 8, Appendix I"
    },
    {
      "id": "lora-sb-D3-003",
      "claim": "Compare LoRA-SB against Full FT and PEFT baselines on GLUE NLU benchmark. RoBERTa-large (355M) fine-tuned on 6 GLUE tasks at ranks r ∈ {8, 16, 24}. LoRA applied to self-attention layers only. Per-task metrics: Matthew's correlation (CoLA), accuracy (RTE/MRPC/QNLI/SST-2), Pearson correlation (STS-B). All-task average reported.",
      "source": "Sec 3.3, Table 3, Appendix H, Table 9, Appendix I"
    },
    {
      "id": "lora-sb-D3-004",
      "claim": "Ablation: initialization quality impact. Mistral-7B (r=96) with SVD on various matrices: Kaiming random, ΔW_avg + Gaussian noise at σ ∈ {10⁻², 10⁻³, 10⁻⁴, 10⁻⁵}, and clean ΔW_avg (LoRA-SB). All use truncated SVD for orthonormal B/A; differences arise solely from init quality. Comparison metric: accuracy on GSM8K and MATH.",
      "source": "Sec 4, Table 4"
    },
    {
      "id": "lora-sb-D3-005",
      "claim": "Ablation: initialization sample count. Mistral-7B (r=96) and Gemma-2 9B (r=96) initialized with n ∈ {1, 5, 25, 50, 100, 200, 500} random samples from MetaMathQA. Default 0.1% (50 of 50K) selected based on performance plateau after ~25 samples.",
      "source": "Sec 4, Table 5"
    },
    {
      "id": "lora-sb-D3-006",
      "claim": "Ablation: optimal gradient approximation vs initialization quality. Mistral-7B (r=96) trained with two variants: (a) B=US, A=Vᵀ, R=I — good init but no optimal gradient; (b) B=U, A=Vᵀ, R=S — orthonormal B/A enabling closed-form optimal gradient. Training loss curves and final accuracy compared.",
      "source": "Sec 4, Eq 11-12; Appendix E, Fig 3"
    },
    {
      "id": "lora-sb-D3-007",
      "claim": "Benchmark: initialization time overhead vs LoRA-XS (zero overhead, same architecture). Measured on Mistral-7B/MetaMathQA, Gemma-2 9B/MetaMathQA, Llama-3.2 3B/COMMONSENSE170K. One-time overhead: gradient estimation + torch.svd_lowrank (<1s per model). Total overhead 2-4 min vs 3-5 hrs/epoch training (~1.1-1.3%).",
      "source": "Sec 4, Appendix F, Table 6"
    },
    {
      "id": "lora-sb-D3-008",
      "claim": "Benchmark: inference computational cost (MACs and FLOPs). Compare LoRA-SB vs LoRA at minimum rank for matching performance. Models: RoBERTa-large (LoRA r=8 vs LoRA-SB r=16,24), Llama-3.2 3B (r=32 vs r=64,96), Mistral-7B (r=32 vs r=64,96), Gemma-2 9B (r=32 vs r=64,96). Sequence length 256. Extra r×r matrix R adds negligible overhead.",
      "source": "Appendix G, Table 7"
    }
  ],
  "D4": [
    {
      "id": "lora-sb-D4-001",
      "claim": "Phase 1 — Initialization (one-time, before training): Step 1: Estimate per-sample gradient with AdamW sign approximation. Step 2: Accumulate multi-sample ΔW_avg = −η sign(Σ_i ∇_W L(W₀, x_i)) over 0.1% of training data using memory-efficient layerwise hooks. Step 3: Compute truncated SVD on ΔW_avg via torch.svd_lowrank. Step 4: Assign B = U[:, :r], A = V[:r, :]ᵀ, R = S[:r, :r] (with s=1). B and A frozen; only R requires grad.",
      "source": "Appendix D, Algorithm 1; Sec 2.4, Sec 2.6"
    },
    {
      "id": "lora-sb-D4-002",
      "claim": "Phase 2 — Training (only R updated): Step 1: Forward pass W = W₀ + s B R A (B, A frozen). Step 2: Compute optimal gradient g^R = Bᵀ g Aᵀ (simplified: s=1, BᵀB=I, AAᵀ=I). Step 3: Update R ← R − η g^R via AdamW optimizer. Repeat for configured epochs. B and A never updated.",
      "source": "Sec 2.1, Eq 3; Sec 2.6, Eq 9; Sec 2.3, Theorem 4"
    },
    {
      "id": "lora-sb-D4-003",
      "claim": "Phase 3 — Evaluation: Evaluate fine-tuned LoRA-SB model on downstream task test/validation sets. For arithmetic reasoning: GSM8K and MATH (accuracy). For commonsense reasoning: 8 datasets plus average. For GLUE: 6 tasks with task-specific metrics (Matthew's correlation, accuracy, Pearson correlation) plus all-task average.",
      "source": "Sec 3.1, Sec 3.2, Sec 3.3"
    }
  ]
}