Appendix F: Notation reference

The symbols the book's derivations use, grouped by domain and pinned to one meaning each. Where a symbol is genuinely overloaded across domains (the classic offender is , which the book uses for the bootstrap-resample count, the LoRA up-projection matrix, and an NF4 quantization block), the collision is flagged so a reader jumping between chapters is never guessing. Each table gives the symbol, its meaning as the book uses it, and the chapter where it is first introduced.

The book leans on a few standing conventions: vectors and matrices are as written in each chapter (no global bold/roman rule is imposed, because the source chapters set their own); is natural log throughout; expectations are over whatever the subscript names; and "byte" quantities use binary prefixes for capacity (GiB) and decimal for rate (GB/s), as in The hardware baseline and Appendix A.

Reinforcement learning

SymbolMeaningFirst introduced
, state (here, the prompt / partial generation at step )The RL problem
, action (here, the next token or a full completion)The RL problem
policy: the model's probability of action in state , parameters The RL problem
policy (model) parametersThe policy gradient theorem
frozen reference policy for the KL penaltyGRPO
, reward; in RLVR, the verifiable checker's scoreThe RL problem / RLVR
, return: (discounted) sum of future rewards from step Value functions and Bellman equations
discount factor, Value functions and Bellman equations
, state-value: expected return from state under Value functions and Bellman equations
, action-value: expected return taking in , then Value functions and Bellman equations
advantage, Actor-critic and GAE
estimated advantage (GAE, or group-relative in GRPO)Actor-critic and GAE
GAE bias-variance interpolation parameterActor-critic and GAE
temporal-difference residual, Actor-critic and GAE
probability ratio in PPO-clipTrust regions: from TRPO to PPO
PPO clip half-width, ratio clipped to Trust regions: from TRPO to PPO
KL-penalty coefficient in the RL objectiveGRPO
KL divergence from policy to referenceGRPO
GRPO group size: completions sampled per promptGRPO
GRPO group index, GRPO
the -th completion (output) in a groupGRPO
the RL objective (expected return) being maximizedThe policy gradient theorem

Note

is the PPO clip width here, but also names machine epsilon in Tensors, autograd, and number formats and appears as a small constant in RMSNorm. Context disambiguates: an RL objective versus a floating-point rounding bound. Likewise is the KL coefficient in RL but bytes-per-parameter in the quantization/VRAM tables (Appendix A); the RL chapters never mix the two.

Probability and statistics

SymbolMeaningFirst introduced
, expectation, over the distribution named in the subscriptThe policy gradient theorem
varianceREINFORCE and variance reduction
covarianceREINFORCE and variance reduction
model's probability of The language-modeling objective
, an estimator of a mean / parameter (hat = estimated from data)The statistics of evals
eval sample size (number of problems / items)The statistics of evals
in pass@k, the number of sampled completions per problemMetrics and their math
number of bootstrap resamplesThe statistics of evals
95% confidence interval (percentile bootstrap by default)The statistics of evals
significance level (e.g. 0.05)The statistics of evals
effect size (standardized mean difference, Cohen's )The statistics of evals
, standard deviation and its estimateThe statistics of evals
entropy of distribution The language-modeling objective
cross-entropy of relative to The language-modeling objective

Note

is doubly booked and worth watching: it is the statistical effect size here, the head dimension / model dimension in the transformer tables below, and appears as a plain dimension in the roofline matmul argument. The subscript or the sentence resolves it; when a chapter needs both at once it writes for the head dimension explicitly.

Transformer / LLM dimensions

SymbolMeaningFirst introduced
model (hidden) widthThe transformer block
number of transformer layers (num_hidden_layers)Where memory goes / KV cache arithmetic
, number of attention (query) headsAttention from first principles
number of key/value heads (GQA; num_key_value_heads)KV cache arithmetic
head dimension (head_dim, or )Attention from first principles
vocabulary size (context: transformer, not RL value)Tokenization and embeddings
, sequence / context length in tokensPrefill, decode, and the roofline / KV cache arithmetic
token position indexThe language-modeling objective
the token (or activation) at position Tensors, autograd, and number formats
, logits (pre-softmax scores over the vocabulary)The language-modeling objective
query, key, value matrices in attentionAttention from first principles
Jacobian of layer (autograd)Tensors, autograd, and number formats
adjoint of activation , Tensors, autograd, and number formats
total parameter count of a modelAppendix A

Note

is overloaded three ways across the book: the RL state-value function , the attention value matrix , and the vocabulary size . These live in different parts (RL, attention, tokenization) and never appear in the same equation, but it is the collision most likely to trip a reader skimming across parts. is layers in the transformer tables and the scalar loss in the autograd derivation; again, disjoint contexts.

Inference, bandwidth, and the roofline

SymbolMeaningFirst introduced
KV-cache bytes per token, KV cache arithmetic
bytes read from memory per generated token (decode)Prefill, decode, and the roofline
bytes per cached element (2 BF16, 1 FP8)KV cache arithmetic
VRAM pool available for the KV cacheKV cache arithmetic
number of concurrent sequencesKV cache arithmetic
, weight footprint in bytesAppendix A
arithmetic intensity, FLOPs per byte movedPrefill, decode, and the roofline
ridge-point intensity, Prefill, decode, and the roofline
memory bandwidth (~960 GB/s on the baseline card)Prefill, decode, and the roofline
peak compute throughput (FLOP/s)Prefill, decode, and the roofline
Prefill, decode, and the roofline
decode throughput ceiling, Prefill, decode, and the roofline

Note

(lowercase) is bytes-per-cached-element in the KV formula; the quantization table below deliberately writes bit-width as , not , to keep the two apart. (uppercase) is triply booked, the bootstrap-resample count in the statistics table, the LoRA up-projection matrix, and an NF4 quantization block (Parameter-efficient fine-tuning). The KV chapter writes for element bytes and always states "2 for BF16, 1 for FP8" inline so the meaning travels with the symbol.

Number formats and quantization

SymbolMeaningFirst introduced
sign bit (floating point); also the quantization scale (context)Tensors, autograd, and number formats
stored (biased) exponent fieldTensors, autograd, and number formats
exponent bias (127 FP32/BF16, 15 FP16, ...)Tensors, autograd, and number formats
mantissa bit-countTensors, autograd, and number formats
fractional part of the significand, Tensors, autograd, and number formats
unit roundoff, Tensors, autograd, and number formats
machine epsilon, Tensors, autograd, and number formats
spacing between representable values (1 ulp), Tensors, autograd, and number formats
stored integer code in integer quantizationTensors, autograd, and number formats
quantization scale, real Tensors, autograd, and number formats
zero-point (integer offset in asymmetric quantization)Tensors, autograd, and number formats
group/block size for group-wise scales (e.g. 128 AWQ, 32 MXFP4, 64 NF4); chapter 6.3 writes the NF4 block size as Quantization: theory and formats
bit-width of the quantized element (4, 8, 16)Quantization: theory and formats
effective bytes per parameter for a dtype (VRAM tables)Appendix A

Note

This table is the reference the Quantization: theory and formats derivations and the Appendix A budgets both point back to. When those chapters write for a quantization scale, it is the same shown here, not the floating-point sign bit; the surrounding equation always makes clear which. The effective-bytes symbol is what turns a bit-width into the byte-per-parameter figures the VRAM tables multiply by .

Parameter-efficient fine-tuning (LoRA / QLoRA)

The LoRA/QLoRA symbols from LoRA and QLoRA, mathematically and GRPO on 16GB. This group collides hard with the RL and statistics tables above, by inheritance from the source chapters rather than by accident; every clash is flagged below.

SymbolMeaningFirst introduced
LoRA rank: inner dimension of the low-rank update (collides with reward )LoRA and QLoRA, mathematically
LoRA scaling in the update strength (collides with significance level )LoRA and QLoRA, mathematically
LoRA down-projection, (Gaussian-initialized)LoRA and QLoRA, mathematically
LoRA up-projection, (zero-initialized) (collides with bootstrap count )LoRA and QLoRA, mathematically
the low-rank weight update, LoRA and QLoRA, mathematically
(block)NF4 quantization block size in elements (QLoRA uses 64); the quantization table above calls this LoRA and QLoRA, mathematically

Note

This group is a collision minefield with the RL and statistics tables, and every clash is deliberate so the symbols match the source chapters. is the LoRA rank here but the reward in the RL table; is the LoRA scaling factor here but the significance level in the statistics table; and is triply booked, the LoRA up-projection matrix, the bootstrap-resample count (statistics table), and the NF4 block size the quantization table writes as . Chapter 6.3 sets these symbols, so the appendix follows it rather than renaming; the surrounding equation is always what disambiguates.