Abstract
The shift from rule-based software to modern artificial intelligence represents one of the most consequential paradigm changes in computer science. Large Language Models (LLMs) sit at the center of that transition, showcasing striking advances in natural language understanding, generative reasoning, and—more recently—multimodal synthesis. This post offers a structured, technically grounded tour of the mathematics and architecture behind today’s LLMs. We’ll examine tokenization, embedding spaces, neural network fundamentals, the Transformer and its attention mechanism, scaling laws, alignment methods, and hardware-aware inference optimizations. The goal is to bridge conceptual intuition with the concrete mechanics that make these systems work—at a level suitable for both professional and academic readers.
1. Introduction
Contemporary AI systems are no longer built primarily from handcrafted rules or symbolic decision trees. Instead, they learn statistical structure directly from large datasets using highly parameterized neural networks. Large Language Models are the clearest expression of this trend: they combine representation learning, probabilistic modeling, and massively parallel computation to produce fluent, context-aware text generation.
To understand LLMs well, it helps to move past metaphors like “digital brains” and focus on the operations that actually run—matrix multiplications, attention-weighted mixtures, and gradient-based optimization. What follows is a practical walkthrough of the major components, beginning with how text becomes numbers and ending with the techniques that make inference fast enough to deploy.
2. Tokenization and Embeddings: Where Language Becomes Math
Before any neural computation happens, raw text must be converted into numerical form. Neural networks don’t process words—they process vectors. That conversion typically happens in two steps: tokenization and embedding.
2.1 Subword Tokenization Algorithms
Early NLP systems often tokenized at the word level or character level. Word-level tokenization led to exploding vocabularies and frequent “out-of-vocabulary” failures. Character-level tokenization avoided OOV issues but produced long sequences and weakened semantic cohesion.
Modern LLMs primarily use subword tokenization, which offers a practical balance between vocabulary size and expressive coverage. Common methods include:
Byte Pair Encoding (BPE)
Originally developed for compression, BPE repeatedly merges the most frequent adjacent character pairs until reaching a target vocabulary size. It efficiently encodes common words while still representing rare words by decomposing them. This approach appears in families like GPT-style models and LLaMA.WordPiece
Used in BERT-style encoder models, WordPiece selects merges using likelihood-based criteria rather than raw frequency, and often marks continuation fragments (e.g.,##able) to preserve morphological structure.SentencePiece
Built for multilingual and language-agnostic processing, SentencePiece treats text as a raw character stream and encodes whitespace explicitly. This makes it robust across languages and scripts and is common in models such as T5 and ALBERT.
2.2 Tokenization Artifacts and Failure Modes
Tokenizers reflect the statistics of their training corpora, which means they can introduce quirks. Some rare but recurring substrings may become single tokens with poorly trained embeddings, occasionally producing unstable behavior.
Tokenization also influences numerical reasoning. Because BPE-like approaches treat digits as text fragments, values like 9.11 and 9.9 can be split inconsistently, which can contribute to unreliable magnitude comparisons. Typical mitigations include digit-aware tokenization schemes and structured formatting conventions.
2.3 From Static to Contextual Embeddings
After tokenization, each token ID maps to a dense vector via an embedding matrix.
Earlier approaches like Word2Vec or GloVe produced static embeddings: each word had one vector no matter the context. This struggled with polysemy—“bank” in finance versus “bank” of a river.
Transformers build contextual embeddings. Tokens start as lookup vectors, but each self-attention layer refines them based on neighboring tokens, producing representations that shift meaning depending on context.
3. Neural Networks: The Core Computation
At a foundational level, neural networks are parameterized function approximators. A standard layer applies an affine transformation followed by a nonlinearity:
[
v_j = \sum_{i=1}^{d} w_{ji} x_i + b_j
]
Where (w_{ji}) are learned weights, (b_j) is a bias term, and (x_i) are input activations. Activation functions such as ReLU or GELU introduce nonlinearity—without them, stacking layers collapses into a single linear map.
Training is driven by gradient-based optimization: the model produces outputs, a loss function measures error, and gradients propagate backward through the network to update weights. Over many iterations, the model becomes a strong statistical estimator of patterns present in the data.
4. The Transformer Architecture
4.1 From Recurrence to Parallelism
Older sequence models (RNNs, LSTMs) processed tokens sequentially, limiting parallelization and struggling with long-range dependencies due to vanishing gradients.
Transformers replaced recurrence with full-sequence parallel processing, enabling efficient training at scale across GPUs and distributed clusters.
4.2 Scaled Dot-Product Self-Attention
Self-attention is the key innovation. Each token generates three learned projections:
Query (Q)
Key (K)
Value (V)
Attention is computed as:
[
\text{Attention}(Q,K,V) = \text{Softmax}\left(\frac{QK^T}{\sqrt{d_k}}\right)V
]
The dot product (QK^T) measures similarity, the (\sqrt{d_k}) term stabilizes gradients, and softmax produces normalized weights. Those weights blend value vectors into a context-informed representation.
4.3 Multi-Head Attention
Instead of computing one attention pattern, Transformers compute many in parallel. Each head can learn different relationships (syntax, coreference, sentiment, discourse structure). The results are concatenated and projected, expanding representational capacity.
4.4 Positional Encoding
Because attention alone is permutation-invariant, the model needs an explicit notion of token order. Common strategies include:
Absolute positional embeddings (APE): add position vectors (often sinusoidal).
RoPE (Rotary Position Embeddings): rotate queries and keys to encode relative position geometrically.
ALiBi: add distance-based biases directly into attention scores.
RoPE and ALiBi are especially known for improving behavior on longer contexts.
5. Pretraining Objectives and Scaling Laws
5.1 Masked vs. Causal Language Modeling
Two widely used objectives dominate:
Masked Language Modeling (MLM): predict masked tokens using bidirectional context (BERT-style).
Causal Language Modeling (CLM): predict the next token autoregressively under a causal mask (GPT, LLaMA).
CLM naturally produces fluent generation because it directly trains the model to continue text.
5.2 Compute-Optimal Training and Chinchilla-Style Scaling
Scaling isn’t just “bigger is better.” Performance depends on balancing parameter count (N), dataset size (D), and compute. Chinchilla-style findings suggest that compute-optimal training often requires more data per parameter than earlier scaling strategies assumed—roughly on the order of tens of tokens per parameter in common regimes.
A representative loss approximation is:
[
L(N,D) = 406.4N^{-0.34} + 410.7D^{-0.28} + 1.69
]
The constant term reflects irreducible entropy in the data distribution.
6. Alignment After Pretraining
A pretrained model is fundamentally a next-token predictor—not automatically a helpful or safe assistant. Alignment adapts it for instruction following, safety constraints, and user-facing usefulness.
6.1 Supervised Fine-Tuning (SFT)
SFT trains on curated instruction–response pairs to teach conversational structure and task completion patterns.
6.2 RLHF
Reinforcement Learning from Human Feedback typically trains a reward model from human preferences, then optimizes the policy using methods like PPO to maximize predicted reward. RLHF can be powerful, but it adds complexity and can be sensitive to training instability.
6.3 Direct Preference Optimization (DPO)
DPO reframes alignment as a direct optimization problem using preference pairs, avoiding a separate reward model and simplifying the pipeline while preserving strong outcomes.
7. Inference: Prefill, Decode, and KV Caching
Generation generally happens in two phases:
Prefill: the model processes the full prompt in parallel (often compute-bound).
Decode: the model generates tokens one at a time (often memory-bandwidth-bound).
A major optimization is KV caching, which stores previous keys and values so the model doesn’t recompute attention history at every step. This can significantly reduce compute during decoding, but it increases memory usage and creates VRAM pressure—especially for long contexts and large batch sizes.
8. Escaping the Quadratic Cost of Attention
Standard self-attention scales as (O(N^2)) with sequence length, which becomes expensive for long contexts.
8.1 FlashAttention
FlashAttention is an IO-aware exact attention method that tiles operations into fast on-chip memory, reducing memory traffic and accelerating attention without approximations.
8.2 Sliding Window Attention
Windowed attention limits each token’s attention scope to a local neighborhood, dramatically lowering compute while sacrificing full global access.
8.3 State Space Models (Mamba)
Structured State Space Models maintain fixed-size hidden states, enabling sub-quadratic scaling and constant-memory inference in many settings. The tradeoff is that compressing long histories can reduce exact recall. Hybrid designs increasingly combine attention layers with SSM layers to balance precision and efficiency.
9. Conclusion
Large Language Models aren’t mystical—they’re carefully engineered mathematical systems. Their capabilities emerge from a small set of core ideas executed at massive scale:
efficient subword tokenization,
contextual embedding refinement,
parallel self-attention via Transformers,
compute-aware scaling strategies,
post-training alignment techniques,
and hardware-conscious inference optimizations.
Looking forward, the momentum in AI research continues toward hybrid architectures, better data curation, longer and more reliable context handling, and alignment methods that are both safer and more stable. The most enduring progress will come from treating theory and infrastructure as a single design problem—mathematical rigor paired with scalable systems.
Keywords: Large Language Models, Transformer Architecture, Self-Attention, Tokenization, Scaling Laws, RLHF, FlashAttention, State Space Models
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