Michael Brenndoerfer | Data & AI, Private Equity, Technology

Michael Brenndoerfer

Weight Tying › Scaling Considerations

Michael BrenndoerferUpdated June 17, 20256 min read

Language AI HandbookAll chapters

Part 16 of 67Transformer Architectures

Chapter 115Weight Tying › Scaling Considerations

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Weight tying becomes more or less impactful depending on how large the vocabulary is relative to the rest of the model. In a tiny model, the embedding matrices might account for the majority of parameters. In a massive model with many deep transformer layers, the embedding fraction shrinks. Understanding this scaling relationship helps you calibrate how much you should care about tying in different contexts.

The fundamental relationship is simple: the number of embedding parameters grows linearly with vocabulary size and embedding dimension, while the number of transformer layer parameters grows quadratically with the hidden dimension (due to the $4d^2$ parameters in attention and feedforward components). As models scale up in depth, the relative importance of the embeddings decreases.

In[16]:

Code

def analyze_weight_tying_impact(
    vocab_size, hidden_dim, num_layers, d_ff_multiplier=4
):
    """Calculate what fraction of parameters weight tying saves."""

    # Embedding parameters
    embedding_params = vocab_size * hidden_dim
    tied_savings = embedding_params  # One matrix instead of two

    # Per-layer transformer parameters (approximate)
    attention_params = 4 * hidden_dim * hidden_dim  # Q, K, V, O projections
    ffn_params = 2 * hidden_dim * (hidden_dim * d_ff_multiplier)  # Up and down
    layer_norm_params = 4 * hidden_dim  # Two layer norms
    layer_params = attention_params + ffn_params + layer_norm_params

    # Total model parameters
    total_layers = num_layers * layer_params
    total_untied = (
        2 * embedding_params + total_layers
    )  # Input + output embeddings
    total_tied = embedding_params + total_layers  # Shared embedding

    return {
        "embedding_params": embedding_params,
        "layer_params": total_layers,
        "total_untied": total_untied,
        "total_tied": total_tied,
        "savings": total_untied - total_tied,
        "savings_pct": 100 * (total_untied - total_tied) / total_untied,
    }


# Analyze different model configurations
configs = [
    ("Small (GPT-2)", 50257, 768, 12),
    ("Medium", 50257, 1024, 24),
    ("Large", 50257, 1280, 36),
    ("XL", 50257, 1600, 48),
    ("Large vocab", 128000, 1024, 24),  # Like newer models
]

Out[17]:

Console

Weight Tying Impact Analysis
=====================================================================================
Config                    Vocab   Hidden  Layers         Savings % of Model
-------------------------------------------------------------------------------------
Small (GPT-2)            50,257      768      12      38,597,376      23.8%
Medium                   50,257     1024      24      51,463,168      12.7%
Large                    50,257     1280      36      64,328,960       7.7%
XL                       50,257     1600      48      80,411,200       4.9%
Large vocab             128,000     1024      24     131,072,000      23.2%

For smaller models, weight tying can save 10 to 20 percent of total parameters. As models grow deeper, the relative savings decrease because transformer layers dominate, but the absolute parameter savings remain substantial. Even in a very large model, eliminating 38 million or 131 million parameters is meaningful from both a memory and computational perspective.

The most striking case in the table is the large vocabulary configuration. Models with 128,000-token vocabularies, used by newer multilingual models and those covering many programming languages, see their embedding fraction rise even at larger hidden dimensions. This means weight tying becomes relatively more valuable as vocabulary size increases, a trend that will continue as models adopt even larger vocabularies.

To understand why weight tying matters more for some models than others, let's visualize how parameters are distributed:

Out[18]:

Visualization

Stacked bar chart showing parameter distribution across embedding and transformer layers for five model configurations. — Parameter distribution in different model configurations. Smaller models allocate a larger fraction of their parameters to embeddings, making weight tying more impactful. Larger and deeper models are dominated by transformer layer parameters, reducing the relative benefit. The large vocabulary model stands out as an exception where embedding costs remain high despite substantial depth.

The visualization makes the trade-off clear. In GPT-2 Small, embeddings consume over 15 percent of parameters, so weight tying provides substantial savings. In the XL configuration, embeddings are less than 5 percent of the model. The "Large vocab" case is instructive: despite having more layers, the 128K vocabulary pushes embedding costs back up, reminding us that vocabulary size is a critical variable in this calculation.

Out[19]:

Visualization

Bar chart showing absolute parameter savings in millions for five model configurations. — Absolute parameter savings from weight tying across different model scales. Larger vocabularies produce more savings in absolute terms, with the large-vocab model saving over 130 million parameters.

Bar chart showing percentage parameter savings for five model configurations. — Relative parameter savings as a percentage of total model parameters. Weight tying provides larger relative benefits for smaller models where embeddings constitute a greater fraction of total capacity.

The "Large vocab" configuration shows an interesting pattern: models with bigger vocabularies, like those using 128K-token vocabularies for multilingual support or code generation, benefit more from weight tying because the embedding matrix is a larger fraction of total parameters. As newer models adopt increasingly large vocabularies to support diverse scripts and domains, weight tying becomes progressively more important as an efficiency measure.