Estimate Complexity

Let's estimate the FLOPs required to train GPT-3 without looking it up. We need to break this down into the key components: the model size and the amount of data used.

1. Model Size Estimation:

• Parameters: GPT-3 is famously large. Its largest version has 175 billion parameters. This is a key number to remember when thinking about its scale.
• Parameter Type: These parameters are typically stored as 32-bit floating-point numbers (FP32). So, each parameter is 4 bytes.
• Total Memory for Parameters: 175 billion parameters * 4 bytes/parameter = 700 billion bytes = 700 GB. This gives a sense of the model's memory footprint, which is related to the computational cost.

2. Data Size Estimation:

• Training Data: Large language models are trained on massive datasets of text. Commoncrawl, a large dataset of web pages, is often a significant component. Other sources like books and articles are also crucial. The exact size used for GPT-3 training is hard to pinpoint without knowing the data mix and filtering, but we're talking about hundreds of billions or even trillions of tokens. Let's estimate a plausible scale for a state-of-the-art model of that era. A common figure for data used for models around GPT-3's time was in the range of 300 billion to 500 billion tokens. Let's take a rough estimate of 500 billion tokens.
• Token Representation: Each token is processed in the model, and the computational cost per token depends on the model's complexity.

3. FLOPs per Operation:

• Matrix Multiplications: The core of transformer models like GPT-3 is matrix multiplication. In a forward pass through a layer, for a batch of size B, a sequence length L, and a hidden dimension H, matrix multiplications involve roughly ( 2 * H * H * L * B ) FLOPs. The self-attention mechanism adds further operations.
• Parameters and Operations: A rough rule of thumb for transformer models is that the forward pass through a single layer for a single token requires approximately ( 2 * ext{parameters_in_layer} ) FLOPs. Since the entire model has 175 billion parameters, and each parameter participates in computations, a simplified way to think about the computation per token is related to the total number of parameters. A common simplification is that the computational cost of a forward pass for one token is roughly ( 2 * ext{total_parameters} ). This comes from the dominant contribution of matrix multiplications where each parameter is involved in roughly two operations (a multiply and an add).

4. FLOPs per Token during Training:

• Training involves Forward and Backward Passes: Training involves a forward pass to calculate the output and loss, and a backward pass to calculate gradients. The backward pass is generally estimated to be about twice as computationally expensive as the forward pass. So, a forward pass + backward pass is roughly ( 3 imes ) the cost of a forward pass.
• FLOPs per Token: Using the simplified rule, the FLOPs per token during training would be approximately ( 3 imes (2 imes ext{total_parameters}) = 6 imes ext{total_parameters} ).
• Applying the Numbers: With 175 billion parameters, the FLOPs per training token is roughly ( 6 imes 175 imes 10^9 = 1050 imes 10^9 = 1.05 imes 10^{12} ) FLOPs per token.

5. Total Training FLOPs:

• Multiply by Data Size: To get the total training FLOPs, we multiply the FLOPs per token by the total number of tokens in the training data.
• Calculation: Total FLOPs ( approx ) (FLOPs per token) ( imes ) (Number of training tokens)
• Total FLOPs ( approx (1.05 imes 10^{12} ext{ FLOPs/token}) imes (500 imes 10^9 ext{ tokens}) )
• Total FLOPs ( approx (1.05 imes 500) imes 10^{12} imes 10^9 ) FLOPs
• Total FLOPs ( approx 525 imes 10^{21} ) FLOPs

6. Converting to ExaFLOPs:

• An exaFLOP (EFLOP) is ( 10^{18} ) FLOPs.
• Total EFLOPs ( approx 525 imes 10^{21} / 10^{18} ) EFLOPs
• Total EFLOPs ( approx 525 imes 10^3 ) EFLOPs = 525,000 EFLOPs

Refinement and Considerations:

• Simplifications: The "6 * parameters" rule is a simplification. The exact number of operations per token is more complex and depends on the specific architecture details (number of layers, attention heads, etc.), but this provides a reasonable order-of-magnitude estimate.
• Data Duplication/Epochs: Models are often trained for multiple "epochs," meaning they see the data multiple times. The 500 billion tokens is likely the size of the unique dataset. If it were trained for, say, an average of 2 epochs, the total effective tokens processed would be ( 2 imes 500 ) billion = 1 trillion tokens. This would double our FLOP estimate to around 1000-1050 EFLOPs. However, often the 500 billion figure already represents the total tokens processed over all epochs, or it's used as the scale number. Let's stick with the initial 500 billion tokens as the total processed amount for the estimate.
• Training Efficiency: Factors like computational efficiency of the hardware, batch size, and data parallelism can affect the time to train, but the total amount of computation (FLOPs) remains roughly the same for a given model and dataset size, assuming ideal scaling.

Conclusion and Final Estimate:

Based on the assumptions of 175 billion parameters trained on roughly 500 billion tokens, with the rule of thumb of 6 FLOPs per parameter per token during training, the estimated total FLOPs required to train GPT-3 is in the range of 500 to 600 ExaFLOPs.

Our calculation landed at around 525 ExaFLOPs. This aligns with general estimates floating around for models of this scale and era. The exact published number might differ slightly due to precise details of the architecture, data size, and training methodology, but this step-by-step reasoning provides a good order of magnitude estimate.

Estimated Total FLOPs: ~500-600 ExaFLOPs