Estimate Complexity

Let's estimate the total number of floating-point operations (FLOPs) required to train GPT-3, based on what is known about its size, training data, and typical training procedures. Here's a step-by-step reasoning process:

1. Understand the model size and parameters:

• GPT-3 has approximately 175 billion parameters.
• Parameters are stored in FP32 or FP16 precision; often training uses mixed precision to reduce FLOPs slightly.

2. Determine the training dataset size:

• GPT-3 was trained on hundreds of billions of tokens. Reports suggest around 300 billion tokens.
• For simplicity, assume 300 billion tokens.

3. Estimate the number of training steps:

• To process 300 billion tokens, depending on batch size:

  Suppose:

  • Batch size = 1 million tokens (a plausible large batch size for such training, approximating high-performance hardware)

  Number of steps = Total tokens / Batch size:

  ( ext{Steps} = rac{300 imes 10^9}{10^6} = 300,000 )

  But in practice, the batch size per GPU (or per node) is smaller—say 1,000 tokens per batch, with multiple GPUs. The total number of steps is roughly on the order of 300,000 to 500,000.

  To be conservative, take approximately 300,000 steps for total training.

4. FLOPs per forward and backward pass:

• For each token, the transformer computes several matrix multiplications.
• Typically, the dominant cost per token in a transformer model scales with the model size.

A common approximation for FLOPs per token for training a transformer is:

( ext{FLOPs per token} approx 2 imes ext{number of parameters} imes ext{sequence length} )

But this depends on the specifics of the architecture. Historical estimates suggest:

• For large transformer models, about 6 FLOPs per parameter per token are required for training (this accounts for both forward and backward passes). This is a rough estimate from various literature.

Thus,

( ext{FLOPs per token} approx 6 imes ext{parameters} )

Plugging in GPT-3's parameters:

( 6 imes 175 imes 10^9 = 1.05 imes 10^{12} ext{ FLOPs per token} )

Note: This seems high; more refined estimates consider the overall cost per token is roughly on the order of a few trillion FLOPs, depending on the implementation.

Alternatively, a typical approximation used in literature (e.g., OpenAI's blog) is about 3 to 5 FLOPs per parameter per token.

Let's take 3 FLOPs per parameter per token for a more conservative estimate:

( 3 imes 175 imes 10^9 = 525 imes 10^9 = 5.25 imes 10^{11} ext{ FLOPs per token} )

Similarly, for total FLOPs:

( ext{Total FLOPs} approx ext{number of tokens} imes ext{FLOPs per token} )

Using 300 billion tokens and 5.25 × 10^{11} FLOPs per token:

( ext{Total FLOPs} approx 300 imes 10^9 imes 5.25 imes 10^{11} ) ( = (300 imes 5.25) imes 10^{20} ) ( = 1575 imes 10^{20} = 1.575 imes 10^{23} )

5. Adjust for training efficiency and hardware:

• Not every FLOP is used efficiently.
• Typically, the total number of FLOPs gives an upper bound.
• FLOPs estimates often consider the entire training process, including multiple epochs, but most models are trained with many passes over the data; in this case, since the tokens are large and the data volume is huge, training is roughly a single pass or a few passes.

6. Final estimate:

Given the approximations, a rough but reasonable total FLOPs estimate for training GPT-3:

( oxed{ ext{Total FLOPs} sim 10^{23} ext{ FLOPs} } )

Summary:

• Parameters: 175 billion
• Tokens processed: ~300 billion
• FLOPs per token: about 5 × 10^{11}
• Total FLOPs: approximately 10^{23} FLOPs

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Answer: The total number of FLOPs required to train GPT-3 is roughly on the order of 10^23 FLOPs. This estimate considers GPT-3's 175 billion parameters, training on approximately 300 billion tokens, and accounts for the computational cost per token during training.