The Vercel for Model Training

Distributed Gradient Accumulation

LorientAI implements ring-allreduce with gradient compression using Top-K sparsification where K = ⌈ρ·d⌉ for sparsity ratio ρ ∈ (0,1]. Error feedback accumulation ensures convergence: eₜ₊₁ = eₜ + gₜ - Compress(eₜ + gₜ)

Adaptive Learning Rate Scheduling

Our scheduler combines linear warmup with inverse square root decay, modulated by Adam's bias correction terms. For large batch training, we apply LARS: η̂ₗ = η · ‖wₗ‖/(‖∇L(wₗ)‖ + β‖wₗ‖) per layer l.

Mixed-Precision Numerics

BF16 master weights with FP8 forward pass (E4M3) and backward pass (E5M2). Loss scaling with dynamic exponent: scale = 2^k where k = argmax{2^k · ‖∇L‖∞ < FP16_MAX}.

Tensor Parallelism Sharding

Column-parallel linear layers partition A ∈ ℝᵈˣᵏ into [A₁|A₂|...|Aₚ] across P devices. For attention: Q,K,V heads distributed with Σᵢ softmax(QᵢKᵢᵀ/√dₖ)Vᵢ computed locally before all-gather.

Convergence Guarantees

For L-smooth, μ-strongly convex objectives with SGD:

𝔼[‖θₜ - θ*‖²] ≤ (1 - μη)ᵗ‖θ₀ - θ*‖² + ησ²/μ

Where σ² bounds gradient variance. Our adaptive batching maintains:
Var(ĝ) = σ²/|B| ≤ ε² ⟹ |B| ≥ σ²/ε²

Critical batch size Bₖᵣᵢₜ = tr(H⁻¹Σ)/‖∇L‖² determines scaling efficiency.

Second-Order Approximations

Shampoo preconditioner for matrix parameter W ∈ ℝᵐˣⁿ:

Lₜ = (Σₛ₌₁ᵗ GₛGₛᵀ + εI)^(1/4) ∈ ℝᵐˣᵐ
Rₜ = (Σₛ₌₁ᵗ GₛᵀGₛ + εI)^(1/4) ∈ ℝⁿˣⁿ

Update: Wₜ₊₁ = Wₜ - η·Lₜ⁻¹GₜRₜ⁻¹

Matrix roots computed via coupled Newton iteration:
Xₖ₊₁ = ½(Xₖ + Yₖ⁻¹), Yₖ₊₁ = ½(Yₖ + Xₖ⁻¹)

Attention Complexity Reduction

Standard attention: O(n²d) time, O(n²) memory

Flash Attention tiling with block sizes Bᵣ, Bᶜ:
- Load Qᵢ ∈ ℝ^(Bᵣ×d), Kⱼ,Vⱼ ∈ ℝ^(Bᶜ×d) to SRAM
- Compute Sᵢⱼ = QᵢKⱼᵀ ∈ ℝ^(Bᵣ×Bᶜ)
- Online softmax: mᵢⱼ = max(mᵢ,ⱼ₋₁, rowmax(Sᵢⱼ))
- Rescale: ℓᵢⱼ = e^(mᵢ,ⱼ₋₁-mᵢⱼ)ℓᵢ,ⱼ₋₁ + rowsum(e^(Sᵢⱼ-mᵢⱼ))

IO complexity: O(n²d²/M) for SRAM size M.

Gradient Checkpointing Trade-offs

Memory-compute Pareto frontier for transformer with L layers:

Standard: O(L·n·d) memory, O(1) recomputation
√L checkpointing: O(√L·n·d) memory, O(√L) recomputation
Selective: checkpoint at layers {⌊iL/k⌋ : i ∈ [k]}

Optimal k minimizes: T(k) = T_fwd(1 + (L-k)/k) + T_bwd
Subject to: M(k) = M_act·k + M_param ≤ M_available

Our solver uses DP: V(l,m) = min over checkpoints c ∈ [l]

Rotary Position Embeddings

RoPE encodes position via rotation: (Rθ,mqₘ)ᵀ(Rθ,nkₙ) = qₘᵀRθ,n-mkₙ. Base frequency θᵢ = 10000^(-2i/d). NTK-aware scaling: θ'ᵢ = θᵢ · α^(d/(d-2i)) for context extension factor α.

Grouped Query Attention

GQA with g groups: nₕ heads share nₖᵥ = nₕ/g key-value heads. Memory: O(2·n·d/g) vs O(2·n·d) for MHA. Interpolation from MQA via mean pooling: K_gqa = mean(K_mha[g·i:(g+1)·i]) per group.

SwiGLU Activation

Where Swish(x) = x·σ(βx) and σ is sigmoid. Hidden dim d_ff = ⌊(8/3·d_model·2/3)⌋ rounded to multiple of 256 for tensor core alignment. Gradient: ∂SwiGLU/∂x = σ'(Wx)⊗(Vx)·W + Swish(Wx)·V

RMSNorm Numerical Stability

Backward pass: ∂L/∂x = γ/RMS(x)·(∂L/∂y - ȳ·mean(∂L/∂y ⊙ y)). For mixed precision: compute RMS in FP32, cast to BF16 for multiply. Fused kernel: single pass with Welford's online variance.