Estimate Complexity

To estimate the total number of FLOPs required to train GPT-3, we need to consider the model architecture, training process, and dataset size. GPT-3 has 175 billion parameters, a context length of 2048 tokens, 96 layers, a hidden dimension of 12,288, and a feed-forward dimension of 49,152 (4 × hidden dimension). The vocabulary size is approximately 50,257. The training dataset consists of 300 billion tokens.

Step 1: FLOPs per Token in Forward Pass

The forward pass involves computations for each transformer layer and the output projection. For one layer:

• Self-Attention:
  • Projections for Q, K, V: (3 \times \text{context length} \times \text{hidden dimension}^2 = 3 \times 2048 \times 12,288^2) FLOPs.
  • Attention scores: (\text{context length}^2 \times \text{hidden dimension} = 2048^2 \times 12,288) FLOPs.
  • Output projection: (\text{context length} \times \text{hidden dimension}^2 = 2048 \times 12,288^2) FLOPs.
• Feed-Forward Network:
  • First linear layer: (\text{context length} \times \text{hidden dimension} \times \text{feed-forward dimension} = 2048 \times 12,288 \times 49,152) FLOPs.
  • Second linear layer: (\text{context length} \times \text{feed-forward dimension} \times \text{hidden dimension} = 2048 \times 49,152 \times 12,288) FLOPs.

Summing these for one layer:

• Attention: (3 \times 2048 \times 12,288^2 + 2048^2 \times 12,288 + 2048 \times 12,288^2 = 2048 \times 12,288 \times (3 \times 12,288 + 2048 + 12,288) = 2048 \times 12,288 \times 49,152) FLOPs (simplified).
• Feed-forward: (2 \times 2048 \times 12,288 \times 49,152) FLOPs.
• Total per layer: (2048 \times 12,288 \times 49,152 + 2 \times 2048 \times 12,288 \times 49,152 = 3 \times 2048 \times 12,288 \times 49,152) FLOPs.

Per-token cost (dividing by context length 2048):

• Per token per layer: (3 \times 12,288 \times 49,152) FLOPs.
• For 96 layers: (96 \times 3 \times 12,288 \times 49,152) FLOPs.

Additionally, the output projection per token: (\text{hidden dimension} \times \text{vocabulary size} = 12,288 \times 50,257) FLOPs.

Total forward FLOPs per token: [ \text{FLOPs}_{\text{forward}} = 96 \times 3 \times 12,288 \times 49,152 + 12,288 \times 50,257 ] Calculating:

• (12,288 \times 49,152 = 603,979,776)
• (96 \times 3 \times 603,979,776 = 96 \times 1,811,939,328 = 173,946,175,488)
• (12,288 \times 50,257 = 617,558,016)
• Total: (173,946,175,488 + 617,558,016 = 174,563,733,504 \approx 1.7456 \times 10^{11}) FLOPs per token.

Step 2: Training FLOPs per Token

Training involves a forward pass and a backward pass. The backward pass typically requires twice the FLOPs of the forward pass due to gradient computation: [ \text{FLOPs}{\text{train}} = 3 \times \text{FLOPs}{\text{forward}} = 3 \times 1.7456 \times 10^{11} = 5.2368 \times 10^{11} \text{ FLOPs per token}. ]

Step 3: Total Training FLOPs

Multiply by the total number of training tokens (300 billion): [ \text{Total FLOPs} = 300 \times 10^9 \times 5.2368 \times 10^{11} = 1.57104 \times 10^{23} \text{ FLOPs}. ]

Step 4: Refinement and Approximation

The initial calculation includes only major operations. Layer normalization, residual connections, and other minor operations add a small overhead (approximately 5-10%). Accounting for this, we adjust:

• Overhead factor: 1.07 (7% overhead)
• Refined total: (1.57104 \times 10^{23} \times 1.07 \approx 1.681 \times 10^{23}) FLOPs.

However, given the complexity and approximations, a reasonable estimate is (1.6 \times 10^{23}) FLOPs. This aligns with known estimates for GPT-3 training, which range from (10^{23}) to (10^{24}) FLOPs.

Final Answer

The total FLOPs required to train GPT-3 is approximately (1.6 \times 10^{23}).

[ \boxed{1.6 \times 10^{23}} ]