AstrAI/docs/developer/internals.md

Internals

Mathematical foundations and internal algorithms for AstrAI's training, inference, and preprocessing pipelines. For practical usage guides, see Training, Inference, and Preprocessing.

Contents

• Autoregression & Causal Masking
• Rotary Position Embedding (RoPE)
• Training Loss Formulas
• Training Loop Internals
• Callback Lifecycle
• KV Cache Mathematics
• Mask Algorithm Internals
• Gradient Accumulation Mechanics

Autoregression & Causal Masking

Given a token sequence, the model predicts the probability of the next token. Each generated token is appended to the input and fed back, repeating until an end-of-sequence token or max length.

sequence : [[1, 2, 3, 4, 5, 6]]
input_ids: [[1, 2, 3, 4, 5]]
target_ids: [[2, 3, 4, 5, 6]]

A lower-triangular causal mask prevents attending to future positions:

[[0, -inf, -inf, -inf, -inf],
 [0,    0, -inf, -inf, -inf],
 [0,    0,    0, -inf, -inf],
 [0,    0,    0,    0, -inf],
 [0,    0,    0,    0,    0]]

This ensures position i can only attend to positions \leq i, which is essential for autoregressive generation.

Rotary Position Embedding (RoPE)

RoPE embeds position into Q/K vectors via complex rotation:

q_i = R_i W_q x_i, \quad k_j = R_j W_k x_j, \quad q_i^T k_j = x_i^T W_q^T R_{i-j} W_k x_j

RotaryEmbedding pre-computes a cos/sin table freqs_cis of shape [max_len, dim/2, 2] (f32 — [cos, sin] pairs). forward() returns a [batch, seq_len, dim/2, 2] slice indexed by position_ids. apply_rotary_emb applies the rotation: during training it uses torch complex multiply (autograd-compatible); during inference it auto-dispatches to a fused CUDA kernel when available. The key property is that the dot product q_i^T k_j depends only on the relative position i - j, not the absolute positions.

Critical for inference: RoPE is applied before KV cache write, not after. If applied after caching, position encoding drift occurs because cached K/V would have stale rotation factors.

Training Loss Formulas

SEQ (Pre-training)

Next-token cross-entropy with optional label smoothing:

L_{\text{PT}} = -\frac{1}{T}\sum_{t=1}^{T} \log P(x_t \mid x_{\lt t}; \theta)

SFT (Supervised Fine-Tuning)

Masked cross-entropy (ignore_index=-100) over response tokens only:

L_{\text{SFT}} = -\frac{1}{L}\sum_{t=P+1}^{P+L} \log P(s_t \mid s_{\lt t}; \theta)

Prompt tokens are masked out via loss_mask; only response tokens contribute to the loss.

DPO (Direct Preference Optimization)

Frozen reference model, preference margin via log-ratio:

L_{\text{DPO}} = -\mathbb{E}\left[\log\sigma\left(\beta\log\frac{\pi_\theta(y_w\mid x)}{\pi_{\text{ref}}(y_w\mid x)} - \beta\log\frac{\pi_\theta(y_l\mid x)}{\pi_{\text{ref}}(y_l\mid x)}\right)\right]

Parameters: beta=0.1, reduction="sum".

GRPO (Group Relative Policy Optimization)

Token-level PPO with group-normalized advantages:

\text{Advantage}_i = \frac{r_i - \mu}{\sigma + \epsilon} L_{\text{GRPO}} = -\mathbb{E}_t\left[\min\left(\rho_t A,\; \text{clip}\left(\rho_t, 1-\epsilon, 1+\epsilon\right)A\right)\right] + \lambda \cdot \mathbb{E}_t\left[\frac{\pi_{\text{ref}}}{\pi_\theta} - \log\frac{\pi_{\text{ref}}}{\pi_\theta} - 1\right]

Where \rho_t = \pi_\theta(a_t|s_t) / \pi_{\text{old}}(a_t|s_t) is the per-token importance sampling ratio. Advantages are derived from scalar per-response rewards, group-normalized, and broadcast across all response tokens. Only response tokens contribute to the loss.

Parameters: group_size=4, clip_eps=0.2, kl_coef=0.01.

MoE Load Balancing

MoE layers add a differentiable load-balancing term based on mean router probabilities and top-k expert assignment frequency. The training objective is:

L = L_{\text{task}} + \lambda_{\text{MoE}} L_{\text{aux}}

TrainConfig.moe_aux_loss_coef controls \lambda_{\text{MoE}} (default 0.01). The unweighted and weighted auxiliary losses are logged separately.

Training Loop Internals

Two-level loop: epoch → batch. Optimizer step fires every grad_accum_steps batches.

on_train_begin
  model.train()
  on_epoch_begin
    for batch in dataloader:
      with executor.accumulate(model):
        on_batch_begin
        loss_output = strategy(batch)
        context.loss = loss_output["loss"].item()
        context.metrics = loss_output["metrics"]
        stand_loss = loss_output["loss"] / executor.grad_accum_steps
        executor.backward(stand_loss)
        context.consumed_samples += (
            context.config.batch_per_device * context.world_size
        )
        on_batch_end

        if executor.sync_gradients:
          on_optimizer_step
          optimizer.step()
          strategy.on_optimizer_step()
          optimizer.zero_grad()
          if scheduler:
            scheduler.step()
    on_epoch_end
on_train_end

The loss is divided by grad_accum_steps before backward(), so accumulated gradients sum to the correct mean. Strategy metrics are detached and converted to Python float values before the LossOutput is returned; only LossOutput.loss remains a differentiable tensor.

Callback Lifecycle

Hook	Fires	Default callback
on_train_begin	Before training starts	GradientCheckpointingCallback, CheckpointCallback, MetricCallback
on_epoch_begin	Start of each epoch	ProgressBarCallback
on_batch_begin	Every batch	—
on_optimizer_step	Every accumulation window	MetricCallback, ProgressBarCallback, GradientClippingCallback
on_batch_end	Every batch	CheckpointCallback
on_epoch_end	End of each epoch	MetricCallback, ProgressBarCallback
on_error	On exception during training	CheckpointCallback, MetricCallback
on_train_end	Training exits after on_train_begin completes (via finally)	GradientCheckpointingCallback, CheckpointCallback, MetricCallback

Default callbacks (in order): gradient_checkpointing (activation checkpointing, optional), checkpoint (safetensors, rank-0), metric (JSONL + validation, rank-0), progress_bar (tqdm, rank-0), gradient_clipping. The gradient-clipping callback is always registered and always calls executor.clip_grad_norm() with the numeric max_grad_norm value.

KV Cache Mathematics

At decode time, only the last query token matters. All previous K/V are cached to avoid recomputation:

o_n = \sum_j \text{softmax}\left(\frac{q_n k_j}{\sqrt{d_k}}\right) v_j

The cache stores k_j and v_j for all previous positions. At each decode step, only q_n (the current query) is computed fresh, and attention is computed against the cached K/V.

RoPE ordering: RoPE is applied to Q/K before writing to the KV cache. This is essential because:

1. The cached K values already contain the rotation for their original positions.
2. The new Q is rotated for its current position.
3. The dot product q_n^T k_j then correctly depends on n - j (relative position).

If RoPE were applied after caching, the rotation factors would be inconsistent between cached and new tokens.

Cache Architecture

Three-layer separation (SGLang-inspired):

• KVStorage: Flat token-level buffers [n_layers, size, n_kv_heads, head_dim].
• ReqToTokenPool: Index table [req_idx, pos] → physical token slot, shared across all layers.
• Allocator + PrefixCache: Paged-mode slot allocation with ref-counting, LRU eviction, and hash-based prefix sharing.

PagePool orchestrates all three. In contiguous mode (default), req_to_token is a trivial linear mapping. In paged mode, slots are allocated on demand with prefix caching support. bind_tasks() returns a KVCache dataclass with kv_indptr, a prefix-sum index over sequence lengths computed once per step and shared across layers. Attention layers access buffers directly — no methods, no abstraction.

Attention Backend

Attention computation is decoupled from the model via AttentionBackend ABC (astrai/extension/attention_backend.py):

• TorchNativeBackend (default): writes K/V to cache, gathers via req_to_token indirect indexing, calls F.scaled_dot_product_attention.
• CudaBackend: decode path uses attn_paged_decode with page_size=1 (the req_to_token table serves as the page table, each token slot is a single-token "page"); prefill path uses the ragged-batch attn_paged_prefill (addresses each request via qo_indptr + kv_indptr directly against the flat pool). Falls back to TorchNativeBackend when kernel unavailable.

Rotary embedding is applied via apply_rotary_emb in astrai/extension/rotary_backend.py, which auto-dispatches to the fused CUDA kernel (rotary_emb.cu) during inference or torch complex multiply during training (for autograd compatibility). Both attention backends share the same rotary dispatch.

Backend selection is thread-safe via contextvars, mirroring torch.nn.attention.sdpa_kernel:

from astrai.extension import attn_backend, ATTN_BACKEND

with attn_backend(ATTN_BACKEND.CUDA):
    engine.generate("hello")

Layout convention: all q/k/v are [batch, seq_len, n_heads, head_dim] (blhd). Scale is always 1/sqrt(head_dim).

Mask Algorithm Internals

Template mode (template: true)

1. Prepend BOS token (masked)
2. For each message in the field's array:
   1. Render through chat_template for that single message
   2. Encode rendered text
   3. Apply mask rule for the message's role

Non-template mode

Encode the field value as text. Mask value is 1 (train) or 0 (mask) per the section's action.

Text config detection

When no section uses template and all sections have action: "train", the builder omits loss_mask from the output — all tokens are trained.

Position ID strategies

Mode	Behavior
none	No position IDs generated
doc_reset	Reset position to 0 at each document boundary in packed sequences
continuous	Continuous position IDs across packed documents

Default is doc_reset, which ensures each document in a packed bin starts from position 0, preventing position encoding drift between unrelated documents.

Gradient Accumulation Mechanics

Three cooperating layers enable gradient accumulation:

1. GradientState — tracks the micro-step counter. Fires sync_gradients=True every grad_accum_steps micro-batches. The counter is incremented at the start of accumulate(), before the forward pass.

2. executor._no_sync(model) — suppresses gradient synchronization on non-sync micro-steps:

   • NoneExecutor: nullcontext (nothing to skip)
   • DDPExecutor: model.no_sync() (PyTorch's built-in — skips all-reduce of gradient buckets)
   • FSDPExecutor: set_requires_gradient_sync(False, recurse=True) on each FSDPModule (FSDP2's native mechanism)

3. AccumOptimizer / AccumScheduler — wrap the real optimizer/scheduler. step() and zero_grad() are gated on sync_gradients — they only forward to the inner optimizer when the sync flag is True.

The loss is divided by grad_accum_steps before backward(), so gradients sum to the correct mean across micro-steps. consumed_samples increments by batch_per_device * world_size every micro-batch.

Effective batch size

\text{Effective batch} = \text{nprocs} \times \text{batch\_per\_device} \times \text{grad\_accum\_steps}

Total optimizer steps

samples_per_replica = ceil(dataset_len / nprocs)
batches_per_replica  = ceil(samples_per_replica / batch_per_device)
total_steps          = (batches_per_replica // grad_accum_steps) * n_epoch

This accounts for data-parallel sharding — each rank processes 1/nprocs of the dataset.

Document Update Time: 2026-08-02