gemma pytorch repo
Thread (29 posts)
Thinking Summary
The user is asking about the organization of the Gemma 2 source code based on the gemma_pytorch repository. Let me analyze the code structure and highlight how Gemma 2 is organized within the codebase. From what I can see: 1. The repository supports multiple Gemma architectures (Gemma 1, Gemma 2, and Gemma 3) 2. Gemma 2 has specific model configurations and architectural features 3. There are dedicated decoder layer classes for different architectures Let me look at the key organizational aspects:
Looking at the gemma_pytorch repository structure, here's how the Gemma 2 source code is organized:
Architecture Integration
Gemma 2 is integrated into a unified codebase that supports multiple Gemma architectures (1, 2, and 3). The code uses an Architecture
enum to differentiate between versions:
class Architecture(enum.Enum):
GEMMA_1 = 1
GEMMA_2 = 2
GEMMA_3 = 3
Configuration System
Gemma 2 has dedicated configuration functions in gemma/config.py
:
get_config_for_2b_v2()
- 2B parameter variantget_config_for_9b()
- 9B parameter variantget_config_for_27b()
- 27B parameter variant
Each configuration specifies Gemma 2's unique features:
- Architecture:
Architecture.GEMMA_2
- Pre/Post FFW Normalization:
use_pre_ffw_norm=True
,use_post_ffw_norm=True
- Logit Softcapping:
final_logit_softcapping=30.0
,attn_logit_softcapping=50.0
- Attention Types: Alternating pattern of
LOCAL_SLIDING
andGLOBAL
attention - Sliding Window:
sliding_window_size=4096
Model Implementation
The model architecture in gemma/model.py
uses conditional logic to handle different architectures:
Decoder Layer
Gemma 2 has its own decoder layer class Gemma2DecoderLayer
with specific features:
- Additional normalization layers (pre/post feedforward)
- Support for both local sliding and global attention types
- Attention logit softcapping
Model Construction
for i in range(config.num_hidden_layers):
if config.architecture == gemma_config.Architecture.GEMMA_1:
self.layers.append(GemmaDecoderLayer(config))
elif config.architecture in (Architecture.GEMMA_2, Architecture.GEMMA_3):
attn_type = config.attn_types[i % len(config.attn_types)]
self.layers.append(Gemma2DecoderLayer(config, attn_type))
Key Gemma 2 Features
- Mixed Attention Pattern: Alternates between local sliding window attention and global attention across layers
- Additional Normalizations: Extra RMSNorm layers before and after the MLP block
- Softcapping: Applied to both attention logits and final output logits
- Query Pre-attention Scaling: Custom scaling factor for certain variants (e.g., 27B model)
Variant Selection
The get_model_config()
function maps variant strings to configurations:
'2b-v2'
→ Gemma 2 2B model'9b'
→ Gemma 2 9B model'27b'
→ Gemma 2 27B model
This organization allows the codebase to support multiple Gemma architectures while sharing common components and maintaining architecture-specific features through configuration and conditional logic.
Looking at the 2b-v2 model, how does data flow from inputs through the network to the output token selection? Explain each step with references to specific functions
Thinking Summary
The user is asking about the data flow through the Gemma 2b-v2 model specifically. I need to trace through the code from input to output, referencing specific functions. Let me walk through this step by step. Looking at the code: 1. The 2b-v2 model is a Gemma 2 architecture variant 2. The main entry point appears to be the `forward` method in `GemmaForCausalLM` 3. I need to trace through embedding, model layers, and sampling Let me trace through the forward pass:
Let me trace the data flow through the Gemma 2b-v2 model from input tokens to output token selection:
1. Input Tokenization and Embedding
Function: GemmaForCausalLM.forward()
# Input: input_token_ids [batch_size, input_len]
hidden_states = self.embedder(input_token_ids)
# Output: [batch_size, input_len, hidden_size=2304]
# Normalize embeddings by sqrt(hidden_size)
normalizer = torch.tensor(self.config.hidden_size**0.5, dtype=hidden_states.dtype)
hidden_states = hidden_states * normalizer
2. Positional Encoding Setup
Function: GemmaForCausalLM.forward()
# Create frequency encodings for rotary embeddings
freqs_cis[AttentionType.LOCAL_SLIDING] = self.freqs_cis.index_select(0, input_positions)
freqs_cis[AttentionType.GLOBAL] = self.freqs_cis.index_select(0, input_positions)
3. Model Layers (26 layers for 2b-v2)
Function: GemmaModel.forward()
→ Gemma2DecoderLayer.forward()
Each layer alternates between LOCAL_SLIDING and GLOBAL attention:
3a. Pre-Attention Processing
residual = hidden_states
hidden_states = self.input_layernorm(hidden_states) # RMSNorm
3b. Self-Attention
Function: GemmaAttention.forward()
# Project to Q, K, V
qkv = self.qkv_proj(hidden_states) # Linear projection
xq, xk, xv = qkv.split([self.q_size, self.kv_size, self.kv_size], dim=-1)
# Reshape for multi-head attention
# 2b-v2: 8 attention heads, 4 KV heads, head_dim=256
xq = xq.view(batch_size, -1, 8, 256)
xk = xk.view(batch_size, -1, 4, 256)
xv = xv.view(batch_size, -1, 4, 256)
# Apply rotary embeddings
xq = apply_rotary_emb(xq, freqs_cis=freqs_cis)
xk = apply_rotary_emb(xk, freqs_cis=freqs_cis)
# Update KV cache
k_cache.index_copy_(1, kv_write_indices, xk)
v_cache.index_copy_(1, kv_write_indices, xv)
# Compute attention scores
q.mul_(self.scaling) # Scale by 1/sqrt(head_dim)
scores = torch.matmul(q, k.transpose(2, 3))
# Apply attention type-specific mask
if self.attn_type == AttentionType.LOCAL_SLIDING:
# Use sliding window of 4096 tokens
scores = scores + local_mask
else:
# Use global causal mask
scores = scores + mask
# Apply attention logit softcapping (50.0 for 2b-v2)
scores = scores / 50.0
scores = torch.tanh(scores)
scores = scores * 50.0
# Softmax and apply to values
scores = F.softmax(scores.float(), dim=-1)
output = torch.matmul(scores, v)
# Project back
output = self.o_proj(output) # Linear projection
3c. Post-Attention Processing
hidden_states = self.post_attention_layernorm(hidden_states)
hidden_states = residual + hidden_states # Residual connection
3d. MLP Block
Function: GemmaMLP.forward()
residual = hidden_states
hidden_states = self.pre_feedforward_layernorm(hidden_states) # Pre-norm
# MLP with GeGLU activation
gate = self.gate_proj(hidden_states) # Project to intermediate_size=9216
gate = F.gelu(gate, approximate="tanh")
up = self.up_proj(hidden_states) # Project to intermediate_size=9216
fuse = gate * up # Element-wise multiply
hidden_states = self.down_proj(fuse) # Project back to hidden_size=2304
hidden_states = self.post_feedforward_layernorm(hidden_states) # Post-norm
hidden_states = residual + hidden_states # Residual connection
4. Final Normalization
Function: GemmaModel.forward()
hidden_states = self.norm(hidden_states) # Final RMSNorm
# Output: [batch_size, input_len, hidden_size=2304]
5. Token Selection
Function: Sampler.forward()
# Select output position
hidden_states = hidden_states.index_select(1, output_positions).squeeze(dim=1)
# Shape: [batch_size, hidden_size=2304]
# Compute logits
logits = torch.matmul(hidden_states, embedding.t())
# Shape: [batch_size, vocab_size=256000]
# Apply final logit softcapping (30.0 for 2b-v2)
logits = logits / 30.0
logits = torch.tanh(logits)
logits = logits * 30.0
# Temperature scaling (if temperature provided)
logits.div_(temperatures.unsqueeze(dim=1))
# Convert to probabilities
probs = torch.softmax(logits, dim=-1, dtype=torch.float)
# Apply top-p and top-k filtering
probs_sort, probs_idx = torch.sort(probs, dim=-1, descending=True)
# ... filtering logic ...
# Sample next token
next_token_ids = torch.multinomial(probs, num_samples=1)
Summary of Data Flow
- Embedding: Tokens → 2304-dim embeddings (normalized)
- 26 Transformer Layers: Each with:
- RMSNorm → Attention (local/global) → RMSNorm → Residual
- RMSNorm → MLP (GeGLU) → RMSNorm → Residual
- Final Norm: RMSNorm on hidden states
- Logit Generation: Project to vocabulary space
- Sampling: Softmax → Top-p/Top-k filtering → Multinomial sampling
The 2b-v2 model's unique features include:
- Alternating local (4096 window) and global attention
- Double normalization (pre/post) around MLP blocks
- Softcapping on both attention (50.0) and final logits (30.0)
- 8 attention heads with 4 KV heads (2:1 ratio)
Thinking Summary
The user is asking about the "cis" in "freqs_cis". Looking at the code, I can see this is related to rotary positional embeddings (RoPE). The "cis" stands for "cosine + i*sine", which is a mathematical notation for complex numbers in polar form. Let me look at the precompute_freqs_cis function to explain this: ```python def precompute_freqs_cis(dim: int, end: int, theta: float = 10000.0, rope_scaling_factor:int = 1) -> torch.Tensor: """Precomputes the frequency cis.""" freqs = 1.0 / (theta**(torch.arange(0, dim, 2)[:(dim // 2)].float() / dim)) freqs = freqs/rope_scaling_factor t = torch.arange(end, device=freqs.device) freqs = torch.outer(t, freqs).float() freqs_cis = torch.polar(torch.ones_like(freqs), freqs) # complex64 return freqs_cis ``` The key line is `torch.polar(torch.ones_like(freqs), freqs)` which creates complex numbers in polar form where the magnitude is 1 and the angle is the frequency. This creates e^(i*freq) = cos(freq) + i*sin(freq), which is why it's called "cis" (cosine + i*sine).
The "cis" in freqs_cis
stands for "cosine + i·sine", which is a mathematical notation for complex numbers in polar form. This is used for Rotary Position Embeddings (RoPE) in the model.
Looking at the precompute_freqs_cis
function:
def precompute_freqs_cis(dim: int, end: int, theta: float = 10000.0) -> torch.Tensor:
"""Precomputes the frequency cis."""
freqs = 1.0 / (theta**(torch.arange(0, dim, 2)[:(dim // 2)].float() / dim))
t = torch.arange(end, device=freqs.device)
freqs = torch.outer(t, freqs).float()
freqs_cis = torch.polar(torch.ones_like(freqs), freqs) # complex64
return freqs_cis
The key operation is torch.polar(torch.ones_like(freqs), freqs)
, which creates complex numbers:
This is often abbreviated as "cis(θ)" in mathematics, hence freqs_cis
.
How it's used in the model:
In apply_rotary_emb
, these complex numbers are used to rotate the query and key vectors:
def apply_rotary_emb(x: torch.Tensor, freqs_cis: torch.Tensor) -> torch.Tensor:
"""Applies the rotary embedding to the query and key tensors."""
x_ = torch.view_as_complex(
torch.stack(torch.chunk(x.transpose(1, 2).float(), 2, dim=-1), dim=-1))
x_out = torch.view_as_real(x_ * freqs_cis).type_as(x)
# ... reshaping logic ...
return x_out
The multiplication x_ * freqs_cis
performs a rotation in the complex plane, which encodes positional information into the attention mechanism. This is more efficient than traditional positional embeddings and allows the model to extrapolate to longer sequences than it was trained on.
For Gemma 2b-v2:
- Uses head dimension of 256
- Precomputes rotations for up to 2× max sequence length (16,384 positions)
- Default theta of 10,000 (standard RoPE base frequency)
Thinking Summary
The user is asking about the dimensions of Q, K, V in the Gemma 2b-v2 model. Let me trace through the dimensions carefully. From the config for 2b-v2: - hidden_size = 2304 - num_attention_heads = 8 - num_key_value_heads = 4 - head_dim = 256 In the GemmaAttention class: - self.q_size = self.num_heads * self.head_dim = 8 * 256 = 2048 - self.kv_size = self.num_kv_heads * self.head_dim = 4 * 256 = 1024 The QKV projection: - Input: [batch_size, input_len, hidden_size] = [batch_size, input_len, 2304] - QKV proj output: [batch_size, input_len, q_size + kv_size + kv_size] = [batch_size, input_len, 2048 + 1024 + 1024] = [batch_size, input_len, 4096] After split: - xq: [batch_size, input_len, 2048] - xk: [batch_size, input_len, 1024] - xv: [batch_size, input_len, 1024] After reshape: - xq: [batch_size, input_len, 8, 256] - xk: [batch_size, input_len, 4, 256] - xv: [batch_size, input_len, 4, 256] After transpose for attention computation: - q: [batch_size, 8, input_len, 256] - k: [batch_size, 4, max_seq_len, 256] (from cache) - v: [batch_size, 4, max_seq_len, 256] (from cache) I notice a key transformation occurs to align the head dimensions. The model uses repeat_interleave to expand the key and value tensors from 4 to 8 heads. This ensures K and V match Q's head count during attention computation. The process involves: - Repeating key tensor from [batch_size, max_seq_len, 4, 256] to [batch_size, max_seq_len, 8, 256] - Repeating value tensor similarly from 4 to 8 heads - Transposing to get final dimensions of [batch_size, 8, max_seq_len, 256] for both k and v
For the Gemma 2b-v2 model, here are the QKV dimensions at each stage:
Initial Configuration (2b-v2)
hidden_size
: 2304num_attention_heads
: 8num_key_value_heads
: 4head_dim
: 256
1. QKV Projection Input/Output
# Input to qkv_proj
hidden_states: [batch_size, input_len, 2304]
# After projection
qkv = self.qkv_proj(hidden_states)
# qkv: [batch_size, input_len, 4096]
# Where 4096 = q_size + kv_size + kv_size = 2048 + 1024 + 1024
2. After Splitting
xq, xk, xv = qkv.split([self.q_size, self.kv_size, self.kv_size], dim=-1)
# xq: [batch_size, input_len, 2048] # 8 heads × 256 dim
# xk: [batch_size, input_len, 1024] # 4 heads × 256 dim
# xv: [batch_size, input_len, 1024] # 4 heads × 256 dim
3. After Reshaping for Multi-Head
xq = xq.view(batch_size, -1, 8, 256) # [batch_size, input_len, 8, 256]
xk = xk.view(batch_size, -1, 4, 256) # [batch_size, input_len, 4, 256]
xv = xv.view(batch_size, -1, 4, 256) # [batch_size, input_len, 4, 256]
4. KV Cache Storage
k_cache: [batch_size, max_seq_len, 4, 256]
v_cache: [batch_size, max_seq_len, 4, 256]
5. For Attention Computation
# After transposing
q: [batch_size, 8, input_len, 256]
# K and V are expanded to match Q's head count
key = torch.repeat_interleave(key, 2, dim=2) # 4→8 heads
value = torch.repeat_interleave(value, 2, dim=2) # 4→8 heads
# After expansion and transpose
k: [batch_size, 8, max_seq_len, 256]
v: [batch_size, 8, max_seq_len, 256]
6. Attention Scores
scores = torch.matmul(q, k.transpose(2, 3))
# scores: [batch_size, 8, input_len, max_seq_len]
Key Points:
- Q uses 8 attention heads (full)
- K and V use only 4 heads (half) for memory efficiency
- Each KV head is shared by 2 Q heads (Multi-Query Attention variant)
- Total QKV parameters: 2304 × 4096 = 9,437,184 parameters
- Memory saved: Using 4 KV heads instead of 8 saves ~25% of KV cache memory
This is a form of Grouped Query Attention (GQA) where multiple query heads share the same key-value pairs, reducing memory requirements while maintaining model quality.
how does the kv cache work in general and in this implementation?
Thinking Summary
The user is asking about how KV cache works in general and specifically in the Gemma implementation. I should explain: 1. General concept of KV cache - why it's needed, what problem it solves 2. How it works conceptually 3. How it's implemented in this specific Gemma codebase 4. Show the specific code snippets and explain the flow
KV Cache: General Concept
The KV cache is an optimization technique for autoregressive generation in transformers that avoids redundant computation.
The Problem
During generation, each new token needs to attend to all previous tokens. Without caching:
- Token 1: Compute K,V for position 0
- Token 2: Recompute K,V for positions 0,1
- Token 3: Recompute K,V for positions 0,1,2
- ...
- Token N: Recompute K,V for positions 0,1,2,...,N-1
This leads to O(n²) complexity for generating n tokens.
The Solution
Cache the K and V projections after computing them once:
- Token 1: Compute and cache K,V for position 0
- Token 2: Use cached K,V for position 0, compute and cache for position 1
- Token 3: Use cached K,V for positions 0,1, compute and cache for position 2
This reduces complexity to O(n).
Gemma Implementation
1. KV Cache Initialization
# In generate() method
kv_caches = []
for _ in range(self.config.num_hidden_layers): # 26 layers for 2b-v2
size = (batch_size, max_seq_len, self.config.num_key_value_heads, self.config.head_dim)
# For 2b-v2: (batch_size, max_seq_len, 4, 256)
k_cache = torch.zeros(size=size, dtype=dtype, device=device)
v_cache = torch.zeros(size=size, dtype=dtype, device=device)
kv_caches.append((k_cache, v_cache))
Each layer gets its own K and V cache tensors, pre-allocated for the maximum sequence length.
2. Writing to Cache
In GemmaAttention.forward()
:
# Compute new K,V for current positions
xk = xk.view(batch_size, -1, self.num_kv_heads, self.head_dim)
xv = xv.view(batch_size, -1, self.num_kv_heads, self.head_dim)
# Apply rotary embeddings (positional encoding)
xk = apply_rotary_emb(xk, freqs_cis=freqs_cis)
# Write to cache at specific positions
k_cache, v_cache = kv_cache
k_cache.index_copy_(1, kv_write_indices, xk) # kv_write_indices = input_positions
v_cache.index_copy_(1, kv_write_indices, xv)
3. Reading from Cache
# Use the entire cache for attention
key = k_cache # [batch_size, max_seq_len, num_kv_heads, head_dim]
value = v_cache # [batch_size, max_seq_len, num_kv_heads, head_dim]
# Expand KV heads to match Q heads (4 → 8 for 2b-v2)
if self.num_kv_heads != self.num_heads:
key = torch.repeat_interleave(key, self.num_queries_per_kv, dim=2)
value = torch.repeat_interleave(value, self.num_queries_per_kv, dim=2)
4. Generation Flow
Prefill Phase (First Forward Pass):
# Process all prompt tokens at once
input_token_ids: [batch_size, prompt_len]
input_positions: [0, 1, 2, ..., prompt_len-1]
# Compute and cache K,V for all prompt positions
# k_cache[:, :prompt_len, :, :] now contains K values
# v_cache[:, :prompt_len, :, :] now contains V values
Generation Phase (Subsequent Passes):
# Process one new token at a time
input_token_ids: [batch_size, 1] # Just the last generated token
input_positions: [current_position] # e.g., [prompt_len], [prompt_len+1], ...
# Only compute K,V for the new position
# Cache is updated incrementally
5. Attention Computation with Cache
# Q is only computed for current tokens
q: [batch_size, num_heads, current_len, head_dim]
# K,V contain all positions up to max_seq_len
k: [batch_size, num_heads, max_seq_len, head_dim]
v: [batch_size, num_heads, max_seq_len, head_dim]
# Attention scores use full K cache
scores = torch.matmul(q, k.transpose(2, 3))
# scores: [batch_size, num_heads, current_len, max_seq_len]
# Mask ensures we only attend to valid positions
scores = scores + mask # Causal mask prevents attending to future
Key Implementation Details
- Pre-allocation: Caches are allocated for
max_seq_len
to avoid dynamic memory allocation - Sparse Updates: Only positions being processed get updated via
index_copy_
- Full Attention: During attention, the full cache is used but masked appropriately
- Memory Efficiency: Using 4 KV heads instead of 8 reduces cache size by 50%
Example: Generating 3 Tokens
Starting with prompt "The cat":
-
Prefill:
- Input: ["The", "cat"]
- Cache positions 0,1 with K,V for "The", "cat"
-
Generate "sat":
- Input: ["sat"] (from previous generation)
- Cache position 2 with K,V for "sat"
- Attention uses cached K,V from positions 0,1,2
-
Generate "on":
- Input: ["on"]
- Cache position 3 with K,V for "on"
- Attention uses cached K,V from positions 0,1,2,3
Without caching, step 3 would recompute K,V for all previous tokens. With caching, we only compute K,V for the new token.
where do the names Q K V come from? can you motivate the intuition?
Thinking Summary
The user is asking about the origins and intuition behind the Query, Key, Value terminology in attention mechanisms. This is a great question about the conceptual foundations of transformers.
Origin: Information Retrieval Systems
The Q, K, V terminology comes from database/information retrieval systems, where:
- Query: What you're searching for
- Key: Index/identifier for stored items
- Value: The actual content associated with each key
Database Analogy
# Traditional key-value store
database = {
"cat": "a small domesticated carnivorous mammal", # key: value
"dog": "a domesticated carnivorous mammal", # key: value
"bird": "a warm-blooded vertebrate with feathers" # key: value
}
query = "cat" # What we're looking for
result = database[query] # Retrieve value using exact key match
Attention as Soft Lookup
Attention transforms this into a soft, differentiable lookup:
# Hard lookup (database)
if query == key:
return value
# Soft lookup (attention)
similarity = dot_product(query, key)
weight = softmax(similarity)
return weight * value # Weighted combination
The Intuition
1. Query (Q): "What am I looking for?"
The query represents the current position asking: "What information do I need from other positions?"
Example: In "The cat sat on the ___", the blank position creates a query looking for relevant context.
2. Key (K): "What do I have to offer?"
Each position advertises what kind of information it contains via its key representation.
Example: The word "cat" might have a key that signals "I'm an animal/subject noun"
3. Value (V): "Here's my actual content"
The value is the actual information content that gets passed along if there's a match.
Example: The "cat" position's value might encode rich semantic information about cats, plurality, etc.
Visual Intuition
Think of a library:
QUERY: "I need books about cooking"
KEYS (spine labels):
- "French Cuisine"
- "Quantum Physics"
- "Italian Recipes"
- "Garden Design"
VALUES (book contents):
- [Full French cookbook]
- [Physics textbook]
- [Italian cookbook]
- [Gardening guide]
ATTENTION PROCESS:
1. Compare query with all keys
2. "French Cuisine" (high similarity: 0.4)
3. "Quantum Physics" (low similarity: 0.0)
4. "Italian Recipes" (high similarity: 0.5)
5. "Garden Design" (low similarity: 0.1)
RESULT: Weighted mix of cookbooks' contents
In Transformer Context
# For each token position i:
Q[i] = "What information do I (token i) need?"
K[j] = "What information does token j contain?"
V[j] = "Here's the actual information from token j"
# Attention computes:
attention[i,j] = softmax(Q[i] · K[j] / sqrt(d))
output[i] = Σ(attention[i,j] * V[j])
Real Example: "The cat sat"
For the word "sat":
- Q("sat"): "I'm a verb, I need a subject"
- K("the"): "I'm a determiner"
- K("cat"): "I'm a noun, likely a subject"
- V("cat"): [Semantic features of 'cat']
The high Q-K similarity between "sat" and "cat" means "sat" will strongly attend to "cat"'s value.
Why Separate K and V?
This separation is crucial because:
- Matching (via K) and Content (via V) are different tasks
- What makes a good search key isn't always what you want to retrieve
Example in language:
- Key: "I'm a pronoun referring to something"
- Value: "The actual entity being referred to"
Mathematical Foundation
The attention formula:
Can be read as:
- : Compare all queries with all keys
- : Convert similarities to probabilities
- Multiply by : Retrieve weighted combination of values
In Gemma's Implementation
# Each position creates its own Q,K,V
qkv = self.qkv_proj(hidden_states) # Linear projection
xq, xk, xv = qkv.split([...]) # Split into three parts
# Q asks, K advertises, V delivers
scores = torch.matmul(q, k.transpose(2, 3)) # Q-K matching
attention = F.softmax(scores) # Convert to weights
output = torch.matmul(attention, v) # Weighted V retrieval
The beauty is that these projections are learned - the model discovers what kinds of queries, keys, and values are useful for the task at hand.
where do the names Q K V come from? can you motivate the intuition? i understand databases so no superficial analogies, i want to understand the ML part
Thinking Summary
The user is asking about the origins and intuition behind Q, K, V naming in attention mechanisms. They specifically mention understanding databases, so they're likely familiar with key-value stores and query operations. They want the ML intuition, not superficial analogies. Let me explain: 1. The historical origin from the "Attention is All You Need" paper 2. The actual mathematical operation and why these names make sense 3. The retrieval/memory interpretation that motivated these names 4. How it differs from database operations despite similar naming I should focus on the mathematical operations and the retrieval metaphor that inspired the naming.
Origins and Deep Intuition of Q, K, V
The names come from the "Attention is All You Need" paper (2017) and represent a soft, differentiable memory retrieval mechanism.
The Core Operation
The attention mechanism computes:
Attention(Q, K, V) = softmax(QK^T / √d)V
This is fundamentally a weighted average of values, where weights are determined by similarity between queries and keys.
The Retrieval Metaphor
The authors conceptualized attention as a continuous addressing scheme for memory:
-
Query (Q): "What information am I looking for?"
- Each position in the sequence asks: "Given my current representation, what should I pay attention to?"
- In Gemma:
Q = X·W_q
where X is the current token's representation
-
Key (K): "What information do I contain/advertise?"
- Each position broadcasts: "This is what I represent"
- Keys are content descriptors - they encode what information is stored at each position
- In Gemma:
K = X·W_k
-
Value (V): "What information do I actually provide?"
- The actual content to be retrieved and aggregated
- Often different from the key - you search by one representation but retrieve another
- In Gemma:
V = X·W_v
Why This Separation Matters
Consider processing "The cat sat on the mat":
When processing "sat", it needs to understand its subject:
- Q (from "sat"): "I'm a verb, looking for my subject"
- K (from "cat"): "I'm a noun, could be a subject"
- K (from "the"): "I'm a determiner"
- V (from "cat"): [actual semantic content about 'cat']
The dot product Q·K
measures compatibility. High scores mean "this key matches what I'm querying for."
Mathematical Intuition
# For each query position i and key position j:
score[i,j] = dot(Q[i], K[j]) / sqrt(d)
# This measures: "How relevant is position j to position i's query?"
# Softmax converts scores to probabilities
weights[i,:] = softmax(scores[i,:])
# This means: "What fraction of attention should position i pay to each position?"
# Final output aggregates values
output[i] = sum(weights[i,j] * V[j] for j in positions)
# This means: "Weighted combination of all values, weighted by relevance"
Key Insights
-
Learned Projections: W_q, W_k, W_v are learned to create useful query/key/value spaces
- The model learns what aspects to query for
- What aspects to advertise as keys
- What information to actually pass as values
-
Asymmetry is Crucial:
- K ≠ V allows searching by one criterion but retrieving different information
- Like having an index on one column but retrieving entire rows
-
Soft Selection: Unlike hard database lookups, attention computes a weighted average
- Every position contributes, weighted by relevance
- Differentiable, allowing gradient-based learning
In Gemma's Implementation
# Project to different spaces
qkv = self.qkv_proj(hidden_states) # Single matrix multiply
xq, xk, xv = qkv.split([q_size, kv_size, kv_size], dim=-1)
# Q asks "what should I attend to?"
# K says "this is what I offer for matching"
# V says "this is what you get if you attend to me"
The beauty is that these projections are learned. The model discovers:
- What questions to ask (Q projection)
- How to advertise content for matching (K projection)
- What information to actually provide (V projection)
This creates a flexible, learnable attention pattern that can capture linguistic relationships, dependencies, and contextual information - far more powerful than fixed attention patterns or hard lookups.
Thinking Summary
The user is asking about the `apply_rotary_emb` function, which is used for rotary positional embeddings (RoPE) in the Gemma model. Let me look at the implementation and explain what it does.
The apply_rotary_emb
function implements Rotary Position Embeddings (RoPE), which encode positional information into the query and key vectors. Here's how it works:
Function Implementation
def apply_rotary_emb(x: torch.Tensor, freqs_cis: torch.Tensor) -> torch.Tensor:
"""Applies the rotary embedding to the query and key tensors."""
x_ = torch.view_as_complex(
torch.stack(torch.chunk(x.transpose(1, 2).float(), 2, dim=-1),
dim=-1))
x_out = torch.view_as_real(x_ * freqs_cis).type_as(x)
x_out = torch.cat(torch.chunk(x_out, 2, dim=-1), dim=-2)
x_out = x_out.reshape(x_out.shape[0], x_out.shape[1], x_out.shape[2],
-1).transpose(1, 2)
return x_out
Step-by-Step Breakdown
1. Input Reshaping
x.transpose(1, 2).float()
- Input
x
shape:[batch_size, seq_len, num_heads, head_dim]
- After transpose:
[batch_size, num_heads, seq_len, head_dim]
2. Split into Pairs
torch.chunk(x.transpose(1, 2).float(), 2, dim=-1)
- Splits the head_dim (256 for 2b-v2) into two halves
- Creates two tensors of shape
[batch_size, num_heads, seq_len, 128]
3. Create Complex Numbers
torch.view_as_complex(torch.stack(..., dim=-1))
- Stacks the two halves as real and imaginary parts
- Creates complex tensor:
[batch_size, num_heads, seq_len, 128]
(complex64) - Each pair
(x[2i], x[2i+1])
becomes a complex numberx[2i] + i*x[2i+1]
4. Apply Rotation
x_ * freqs_cis
freqs_cis
contains precomputed complex exponentials:- Multiplication in complex space performs rotation
- This rotates each 2D subspace by angle θ based on position
5. Convert Back to Real
torch.view_as_real(x_ * freqs_cis)
- Converts complex numbers back to real pairs
- Shape:
[batch_size, num_heads, seq_len, 128, 2]
6. Reshape to Original Format
torch.cat(torch.chunk(x_out, 2, dim=-1), dim=-2)
- Rearranges the real/imaginary pairs back to original dimension order
- Final transpose returns to original shape structure
The Mathematics
RoPE applies a rotation matrix to pairs of elements:
Where
Precomputed Frequencies
The freqs_cis
are precomputed in precompute_freqs_cis
:
def precompute_freqs_cis(dim: int, end: int, theta: float = 10000.0):
freqs = 1.0 / (theta**(torch.arange(0, dim, 2)[:(dim // 2)].float() / dim))
t = torch.arange(end, device=freqs.device)
freqs = torch.outer(t, freqs).float()
freqs_cis = torch.polar(torch.ones_like(freqs), freqs) # e^(i*freqs)
return freqs_cis
Why RoPE?
RoPE provides several advantages:
- Relative position encoding: The dot product between rotated queries and keys depends on their relative positions
- Long-range decay: More distant positions have less similarity
- Efficient computation: No additional parameters needed
- Extrapolation: Can handle sequences longer than training length
For Gemma 2b-v2, RoPE is applied to both queries and keys before attention computation, encoding positional information directly into the representations.
how does the attention computation in 3b flow through to the hidden states in 3a/c
Thinking Summary
The user is asking about how the attention computation output flows from step 3b back into the hidden states in steps 3a and 3c. Let me trace this carefully through the code. Looking at the Gemma2DecoderLayer.forward() method: ```python def forward( self, hidden_states: torch.Tensor, freqs_cis: torch.Tensor, kv_write_indices: torch.Tensor, kv_cache: Tuple[torch.Tensor, torch.Tensor], mask: torch.Tensor, local_mask: torch.Tensor, ) -> torch.Tensor: # Self Attention residual = hidden_states hidden_states = self.input_layernorm(hidden_states) hidden_states = self.self_attn( hidden_states=hidden_states, freqs_cis=freqs_cis, kv_write_indices=kv_write_indices, kv_cache=kv_cache, mask=mask, local_mask=local_mask, ) hidden_states = self.post_attention_layernorm(hidden_states) hidden_states = residual + hidden_states # MLP residual = hidden_states if self.pre_feedforward_layernorm is not None: hidden_states = self.pre_feedforward_layernorm(hidden_states) hidden_states = self.mlp(hidden_states) if self.post_feedforward_layernorm is not None: hidden_states = self.post_feedforward_layernorm(hidden_states) hidden_states = residual + hidden_states return hidden_states ``` So the flow is: 1. Save original hidden_states as residual 2. Apply input_layernorm to hidden_states 3. Pass normalized hidden_states through self_attn 4. Apply post_attention_layernorm to attention output 5. Add residual connection I track the attention transformation, where the original input is preserved through a clever residual connection mechanism. The self-attention output gets normalized and then added back to the initial input, allowing information flow while maintaining the original signal's integrity.
Let me trace how the attention output flows back into the hidden states through the residual connections:
Attention Block Flow in Gemma2DecoderLayer
Step 3a: Pre-Attention
# Save the original input for residual connection
residual = hidden_states # Shape: [batch_size, seq_len, 2304]
# Normalize for attention computation
hidden_states = self.input_layernorm(hidden_states) # RMSNorm
Step 3b: Attention Computation
# The attention mechanism processes the normalized hidden states
hidden_states = self.self_attn(
hidden_states=hidden_states, # Normalized input
freqs_cis=freqs_cis,
kv_write_indices=kv_write_indices,
kv_cache=kv_cache,
mask=mask,
local_mask=local_mask,
)
# Output shape: [batch_size, seq_len, 2304]
Step 3c: Post-Attention Integration
# Apply post-attention normalization
hidden_states = self.post_attention_layernorm(hidden_states)
# CRITICAL: Add residual connection
hidden_states = residual + hidden_states
The Residual Connection Pattern
The key insight is the residual connection that adds the original input back to the attention output:
Original Input (residual) ──────────────────┐
│ │
↓ │
Input LayerNorm │
│ │
↓ │
Self-Attention │
│ │
↓ │
Post-Attention LayerNorm │
│ │
↓ ↓
└──────────────────> Addition <────────┘
│
↓
Output (new hidden_states)
Why This Matters
-
Gradient Flow: The residual connection allows gradients to flow directly through the addition operation, bypassing the attention mechanism if needed.
-
Information Preservation: The original input information is preserved and combined with the attention-transformed information.
-
Mathematical Form:
hidden_states_out = hidden_states_in + LayerNorm(Attention(LayerNorm(hidden_states_in)))
The Same Pattern Repeats for MLP
# After attention block completes, save new residual
residual = hidden_states # This now includes attention output
# Pre-MLP normalization
hidden_states = self.pre_feedforward_layernorm(hidden_states)
# MLP computation
hidden_states = self.mlp(hidden_states)
# Post-MLP normalization
hidden_states = self.post_feedforward_layernorm(hidden_states)
# Another residual connection
hidden_states = residual + hidden_states
Data Flow Summary
For each Gemma2 layer:
- Input:
hidden_states
from previous layer - Attention Block:
- Save input as
residual
- Transform through normalized attention
- Add back
residual
- Save input as
- MLP Block:
- Save attention output as new
residual
- Transform through normalized MLP
- Add back
residual
- Save attention output as new
- Output: Enhanced
hidden_states
to next layer
This double-residual pattern with pre/post normalization is a key architectural feature of Gemma 2 that helps with training stability and allows the model to learn both local refinements (through attention/MLP) and preserve global information flow (through residuals).
If there are no initial tokens what is the initial state of attention?
Thinking Summary
The user is asking about the initial state of attention when there are no initial tokens. This is an interesting question about how the model starts generation from scratch. Let me think through this... In the context of Gemma models, typically there would always be at least one token - usually a BOS (beginning of sequence) token. But let me trace through what happens in the initial state. Looking at the code: 1. The tokenizer has a `bos_id` defined 2. In the `encode` method, there's a `bos` parameter that defaults to True 3. The model expects some input tokens But the question is about the attention mechanism's initial state. Let me look at the KV cache initialization and how attention works when starting fresh.
Looking at the code, Gemma models always start with at least one token - typically the BOS (Beginning of Sequence) token. But let me explain the initial attention state:
Initial KV Cache State
When the model starts, the KV caches are initialized to zeros:
# From generate() method
kv_caches = []
for _ in range(self.config.num_hidden_layers):
size = (batch_size, max_seq_len, self.config.num_key_value_heads,
self.config.head_dim)
dtype = self.config.get_dtype()
k_cache = torch.zeros(size=size, dtype=dtype, device=device)
v_cache = torch.zeros(size=size, dtype=dtype, device=device)
kv_caches.append((k_cache, v_cache))
First Token Processing
When processing the first token (usually BOS):
1. Tokenization
# From Tokenizer.encode()
def encode(self, s: str, bos: bool = True, eos: bool = False) -> List[int]:
t = self.sp_model.EncodeAsIds(s)
if bos:
t = [self.bos_id] + t # Prepends BOS token
return t
2. First Attention Computation
For the very first token at position 0:
# Input position is [0]
input_positions_tensor = torch.arange(0, 1) # Just position 0
# The attention mask allows the first token to only attend to itself
mask_tensor = torch.full((1, 1, max_seq_len, max_seq_len), -2.3819763e38)
mask_tensor = torch.triu(mask_tensor, diagonal=1)
# This creates a causal mask where position 0 can only see position 0
3. Attention Mechanics
When the first token goes through attention:
# Query from first token
q = xq[:, 0:1, :, :] # Shape: [batch, 1, heads, head_dim]
# Key and Value caches
# Initially all zeros except position 0 which gets updated:
k_cache[:, 0:1, :, :] = xk # First token's key
v_cache[:, 0:1, :, :] = xv # First token's value
# Attention scores
scores = torch.matmul(q, k.transpose(2, 3))
# Only position 0 has non-zero keys, so attention focuses entirely on position 0
The "Empty" Attention State
If hypothetically there were truly no tokens:
- The model couldn't function because embedding lookup would fail
- There would be no query vectors to compute attention
The closest conceptual "empty" state is:
- KV caches: All zeros (no previous context stored)
- Attention mask: Prevents looking at future positions
- First attention: Self-attention only (token attends to itself)
Practical Initialization
In practice, generation always starts with at least a BOS token:
# Example from the codebase
prompt_tokens = [self.tokenizer.encode(prompt) for prompt in prompts]
# This automatically includes BOS token due to default bos=True
So the initial attention state is:
- Position 0: Contains BOS token embeddings
- KV Cache: Zeros everywhere except position 0 after first forward pass
- Attention Pattern: First token can only attend to itself
- Subsequent tokens: Can attend to all previous positions including BOS
This is why language models can generate coherent text from just a BOS token - it provides the initial hidden state that seeds all subsequent generation.
can you explain the attention mask mechanism? why do we need to prevent the token form attending to future tokens? wouldn’t that be the case by default since they haven’t arrived yet? or is it that they’re all being preprocessed in this stage?
Thinking Summary
Great question! The user is asking about the attention mask mechanism and why we need to prevent tokens from attending to future tokens. They're wondering if this would happen naturally since future tokens haven't arrived yet, or if all tokens are being processed together. This is a fundamental concept in transformer architectures, especially for causal language models. Let me explain: 1. During training and when processing multiple tokens at once (like during prompt processing), all tokens ARE present in the sequence at the same time 2. The mask is needed to maintain causality - each token should only see previous tokens 3. During inference, especially token-by-token generation, the mask might seem redundant but it's still used for consistency Let me trace through the code to show this clearly.
Excellent question! You've hit on a key insight about how transformers process sequences. Let me explain why the attention mask is necessary.
The Key Insight: Parallel Processing
During both training and inference (especially prefill), all tokens in a sequence are processed simultaneously, not one by one. This is what makes transformers efficient.
Example: Processing "The cat sat"
When the model processes this sequence, all three tokens are in memory at once:
# All tokens are embedded simultaneously
input_token_ids = [234, 1876, 3457] # "The", "cat", "sat"
hidden_states = self.embedder(input_token_ids)
# Shape: [1, 3, 2304] - all 3 tokens have embeddings
Why We Need the Mask
Without a mask, when computing attention for position 0 ("The"), the attention mechanism would see:
- Position 0: "The" ✓
- Position 1: "cat" ✗ (shouldn't see this yet!)
- Position 2: "sat" ✗ (shouldn't see this yet!)
Here's what happens in the attention computation:
# Query from position 0 ("The")
q_0 = query[0] # Shape: [heads, head_dim]
# WITHOUT mask, attention scores would be:
scores = torch.matmul(q_0, all_keys.transpose())
# This computes attention to ALL positions including future ones!
# WITH mask:
mask_tensor = torch.triu(torch.full(size, -inf), diagonal=1)
# Creates:
# [[ 0, -inf, -inf],
# [ 0, 0, -inf],
# [ 0, 0, 0]]
scores = scores + mask[0] # Adds -inf to positions 1 and 2
# After softmax, -inf becomes 0 probability
The Mask Creation
# From the code:
mask_tensor = torch.full((1, 1, max_seq_len, max_seq_len), -2.3819763e38)
mask_tensor = torch.triu(mask_tensor, diagonal=1)
This creates an upper triangular matrix of -infinity values:
Position: 0 1 2 3 ...
0 [ 0 -inf -inf -inf ...] # Token 0 sees only position 0
1 [ 0 0 -inf -inf ...] # Token 1 sees positions 0,1
2 [ 0 0 0 -inf ...] # Token 2 sees positions 0,1,2
3 [ 0 0 0 0 ...] # Token 3 sees positions 0,1,2,3
Two Scenarios Where This Matters
1. Training/Prompt Processing (Parallel)
# Processing entire prompt at once
prompt = "The cat sat on the"
tokens = [234, 1876, 3457, 892, 234] # All 5 tokens
# Without mask, token 0 would attend to all 5 tokens!
# With mask, token 0 only attends to itself
2. Generation with KV Cache
# Even during generation, the KV cache contains ALL positions
k_cache = torch.zeros(batch, max_seq_len, heads, dim)
v_cache = torch.zeros(batch, max_seq_len, heads, dim)
# When generating token at position 5:
# - Positions 0-4: Have actual key/values
# - Positions 5-max: Are zeros
# The mask ensures we don't attend to the zero-filled future positions
The Sliding Window Complication
For Gemma 2's local attention, there's an additional mask:
if self.attn_type == AttentionType.LOCAL_SLIDING:
# Only attend to last 4096 tokens
sliding_mask = torch.triu(all_ones, -sliding_window_size + 1) *
torch.tril(all_ones, sliding_window_size - 1)
mask = torch.where(sliding_mask == 1, mask, -inf)
Summary
The attention mask is necessary because:
- Parallel Processing: Multiple tokens are processed simultaneously during training and prefill
- Causal Consistency: Ensures each token only sees its past, maintaining autoregressive property
- KV Cache Safety: Prevents attending to uninitialized future cache positions
- Training-Inference Parity: Same masking pattern works for both training and inference
Without the mask, the model would "cheat" by looking at future tokens during training, breaking the fundamental assumption that language models predict the next token based only on previous context.
Thinking Summary
The user wants me to explain the sliding window attention mechanism in Gemma 2. Let me look at the code carefully and explain how it works. From the code: ```python if self.attn_type == AttentionType.LOCAL_SLIDING and self.sliding_window_size is not None and local_mask is not None: mask = local_mask ``` And in the forward method: ```python local_mask_tensor = mask_tensor + torch.tril( torch.full((1, 1, max_seq_len, max_seq_len), -2.3819763e38, device=device), diagonal=-self.config.sliding_window_size, ) if self.config.sliding_window_size else None ``` For Gemma 2b-v2, the sliding window size is 4096. Let me break this down step by step.
Sliding Window Attention in Gemma 2
The sliding window mechanism limits how far back each token can "see" in the sequence. For Gemma 2b-v2, this window is 4096 tokens.
The Basic Concept
Instead of attending to ALL previous tokens, each token only attends to the most recent 4096 tokens:
Regular Attention (Global):
Token 5000 can see: [0, 1, 2, ..., 4998, 4999] ← All previous tokens
Sliding Window Attention (Local):
Token 5000 can see: [904, 905, ..., 4998, 4999] ← Only last 4096 tokens
↑
5000 - 4096 = 904
How the Mask is Created
Let's break down the mask creation code:
# Start with the regular causal mask
mask_tensor = torch.full((1, 1, max_seq_len, max_seq_len), -2.3819763e38)
mask_tensor = torch.triu(mask_tensor, diagonal=1) # Upper triangular with -inf
# Create the sliding window mask
local_mask_tensor = mask_tensor + torch.tril(
torch.full((1, 1, max_seq_len, max_seq_len), -2.3819763e38),
diagonal=-self.config.sliding_window_size, # -4096 for 2b-v2
)
Step-by-Step Mask Construction
- Start with causal mask:
Position: 0 1 2 3 4 5 ...
0 [ 0.0 -inf -inf -inf -inf -inf ...]
1 [ 0.0 0.0 -inf -inf -inf -inf ...]
2 [ 0.0 0.0 0.0 -inf -inf -inf ...]
3 [ 0.0 0.0 0.0 0.0 -inf -inf ...]
...
- Add lower triangular mask with diagonal=-4096:
torch.tril(full_matrix_of_-inf, diagonal=-4096)
This creates -inf values for positions more than 4096 tokens ago.
- Result for position 5000:
Position: 0 1 ... 903 904 905 ... 4999 5000
5000 [-inf -inf ... -inf 0.0 0.0 ... 0.0 -inf]
↑________________↑ ↑_____________________↑
Can't see these Can see these 4096 Can't see
(too far back) recent tokens future
Visual Example with Small Window (size=3)
Sliding window size = 3
Position: 0 1 2 3 4 5
0 [ 0.0 -inf -inf -inf -inf -inf] # Sees: [0]
1 [ 0.0 0.0 -inf -inf -inf -inf] # Sees: [0,1]
2 [ 0.0 0.0 0.0 -inf -inf -inf] # Sees: [0,1,2]
3 [ 0.0 0.0 0.0 0.0 -inf -inf] # Sees: [0,1,2,3] (full window)
4 [-inf 0.0 0.0 0.0 0.0 -inf] # Sees: [1,2,3,4] (slides!)
5 [-inf -inf 0.0 0.0 0.0 0.0] # Sees: [2,3,4,5]
↑
Can't see position 0 anymore (outside window)
In the Attention Computation
def forward(self, hidden_states, ...):
# ... compute queries, keys, values ...
# Compute attention scores
scores = torch.matmul(q, k.transpose(2, 3))
# Apply the appropriate mask
if self.attn_type == AttentionType.LOCAL_SLIDING:
mask = local_mask # Use sliding window mask
else:
mask = global_mask # Use full causal mask
scores = scores + mask # Add -inf to blocked positions
scores = F.softmax(scores, dim=-1) # -inf → 0 probability
Why Use Sliding Windows?
- Efficiency: Reduces computation from O(n²) to O(n·w) where w=window size
- Memory: Limits memory usage for very long sequences
- Local Context: Forces model to focus on recent context
- Layer Alternation: Gemma 2 alternates between local and global attention:
# From config for 2b-v2
attn_types=[AttentionType.LOCAL_SLIDING, AttentionType.GLOBAL] * 13
# Pattern: [Local, Global, Local, Global, ...] for 26 layers
This alternation allows the model to:
- Capture fine-grained local patterns (sliding window layers)
- Maintain long-range dependencies (global attention layers)
The 4096 token window is large enough to capture most relevant context while still providing computational benefits for very long sequences.
does this mean that when processing the first (non-bos) token there’s always a fixed contribution from attention?
what if there’s a fixed prefix?
Thinking Summary
The user is asking about two things: 1. Whether the first non-BOS token always gets a fixed contribution from attention (because it can only attend to the BOS token) 2. What happens with a fixed prefix Let me think through this: 1. For the first non-BOS token (at position 1): - It can attend to positions 0 and 1 (BOS and itself) - The BOS token embeddings are always the same for a given model - So yes, there's a fixed contribution from attending to the BOS token - But it also attends to itself, which varies based on the token 2. For a fixed prefix: - If you have a fixed prefix like "You are a helpful assistant", those tokens would be processed and stored in the KV cache - Subsequent tokens would attend to this fixed prefix - This would provide a consistent context/contribution to all following tokens
Great observation! Yes, you're right about the fixed contribution. Let me break this down:
First Non-BOS Token Attention
When processing the first "real" token (position 1), it attends to:
- Position 0 (BOS): Fixed contribution
- Position 1 (itself): Variable based on the token
# For token at position 1
# Q: query from current token (varies)
# K, V at position 0: always from BOS token (fixed)
# K, V at position 1: from current token (varies)
scores = torch.matmul(q, k.transpose(2, 3))
# scores[:, :, 0, 0] - attention to BOS (fixed pattern)
# scores[:, :, 0, 1] - self-attention (varies)
The attention weights might vary, but the BOS token's key-value contribution is always the same, creating a consistent "initialization signal" for all sequences.
Fixed Prefix Scenario
With a fixed prefix, this effect is amplified:
# Example: "You are a helpful assistant" as prefix
prefix_tokens = tokenizer.encode("You are a helpful assistant")
# Tokens: [BOS, 'You', 'are', 'a', 'helpful', 'assistant']
# After processing the prefix:
# KV cache positions 0-5 contain fixed representations
Impact on Generation
- Consistent Context: Every subsequent token attends to the same prefix representations
- Prefix as "Soft Prompt": Acts like learned prompt embeddings but in key-value space
- Attention Patterns:
# For any new token at position N:
for i in range(prefix_length):
# Attention to positions 0..prefix_length-1 gives fixed contributions
fixed_contribution += attention_weights[i] * v_cache[i]
for i in range(prefix_length, N):
# Attention to positions prefix_length..N-1 varies with context
variable_contribution += attention_weights[i] * v_cache[i]
Interesting Implications
-
BOS as Universal Context: The BOS token acts as a universal initialization that all sequences share
-
Prefix Caching: With fixed prefixes, you can:
# Precompute prefix KV cache once prefix_kv_cache = process_prefix(prefix_tokens) # Reuse for multiple completions for prompt in prompts: kv_cache = copy(prefix_kv_cache) generate_from_cache(prompt, kv_cache)
-
System Prompts: This is why system prompts work well - they provide consistent context:
system = "You are a helpful, harmless AI assistant." # This creates fixed KV contributions that shape all responses
Attention Distribution
For early tokens, the attention distribution is quite concentrated:
# Token at position 1: can only attend to 2 positions (BOS + self)
# Token at position 10: can attend to 11 positions
# Token at position 100: can attend to 101 positions
# Early tokens have more concentrated attention,
# making fixed contributions more influential
So yes, the BOS token (and any fixed prefix) provides a consistent "anchoring" signal that influences all subsequent generation. This is part of why models can maintain consistent behavior/style across a generation - they're partially conditioned on these fixed attention contributions.
what is the precise conceptual or mathematical definition of “attend”
Thinking Summary
The user is asking for the precise conceptual and mathematical definition of "attend" in the context of attention mechanisms. This is a great question because "attend" is often used loosely in explanations. I should provide both the mathematical formulation and the conceptual understanding.
Excellent question! Let me define "attend" precisely in the context of transformer attention.
Mathematical Definition
"Attend" means computing a weighted average of value vectors, where weights are determined by the similarity between query and key vectors:
# Precise mathematical operation:
Attention(Q, K, V) = softmax(QK^T / √d_k) V
Breaking this down step by step:
1. Similarity Computation
scores = torch.matmul(q, k.transpose(2, 3)) # QK^T
# scores[i,j] = dot product of query_i with key_j
# Measures: "How similar is my current state to past state j?"
2. Normalization to Weights
scores = scores / sqrt(head_dim) # Scale for stability
attention_weights = softmax(scores, dim=-1)
# attention_weights[i,j] = probability that position i should "look at" position j
# Constraint: sum(attention_weights[i,:]) = 1.0
3. Weighted Information Retrieval
output = torch.matmul(attention_weights, v)
# output[i] = Σ_j (attention_weights[i,j] * v[j])
# "Mix information from all positions, weighted by relevance"
Conceptual Definition
"Attend" means:
- Select: Choose which past positions to gather information from
- Weight: Determine how much to rely on each position
- Combine: Mix the information according to these weights
Concrete Example
Let's trace through a simple case:
# Position 3 attending to positions 0, 1, 2, 3
q3 = [0.5, 0.8] # Query from position 3
# Keys from all positions
k0 = [0.1, 0.2] # BOS token
k1 = [0.6, 0.7] # "The"
k2 = [0.4, 0.9] # "cat"
k3 = [0.5, 0.8] # "is" (current)
# Compute similarities (dot products)
score_0 = q3 · k0 = 0.5*0.1 + 0.8*0.2 = 0.21
score_1 = q3 · k1 = 0.5*0.6 + 0.8*0.7 = 0.86
score_2 = q3 · k2 = 0.5*0.4 + 0.8*0.9 = 0.92
score_3 = q3 · k3 = 0.5*0.5 + 0.8*0.8 = 0.89
# After softmax (simplified)
weights = [0.1, 0.3, 0.35, 0.25] # Position 2 gets highest weight
# Values to combine
v0 = [semantic representation of BOS]
v1 = [semantic representation of "The"]
v2 = [semantic representation of "cat"]
v3 = [semantic representation of "is"]
# Final output
output = 0.1*v0 + 0.3*v1 + 0.35*v2 + 0.25*v3
# "is" now contains a mixture emphasizing "cat"
Key Properties of "Attention"
- Permutation Equivariant: Order of K,V pairs doesn't matter (only their position encoding)
- Differentiable Selection: Soft selection via continuous weights vs hard selection
- Content-Based: Selection based on semantic similarity, not fixed positions
In Gemma's Implementation
# From GemmaAttention.forward()
q.mul_(self.scaling) # Scale queries
scores = torch.matmul(q, k.transpose(2, 3)) # Similarity
# Gemma 2 adds softcapping
scores = scores / 50.0
scores = torch.tanh(scores) # Bound similarities to [-1, 1]
scores = scores * 50.0
scores = scores + mask # Apply causal mask (-inf for future)
scores = F.softmax(scores.float(), dim=-1) # Convert to probabilities
output = torch.matmul(scores, v) # Weighted average
The Intuition
"Attention" is fundamentally about:
- Query: "What information am I looking for?"
- Keys: "What information is available?"
- Values: "What should I retrieve?"
- Weights: "How relevant is each piece?"
The mathematical precision: Attention is a differentiable database lookup where:
- Query = search term
- Keys = index/tags
- Values = stored content
- Output = relevance-weighted mixture of retrieved content