Estimate Complexity

Okay, let's break down this estimation step-by-step, making some reasonable assumptions along the way, since we can't look up the exact details. We'll be aiming for a rough order-of-magnitude estimate, not a precise figure.

1. Key Parameters & Assumptions (and why we need them):

To estimate FLOPs, we need to consider several factors influencing computational cost:

• Model Size (Parameters): This is the most crucial factor. GPT-3 is famously large. Let's assume it has 175 billion parameters (this is the generally cited number, and we're working without looking things up). Crucially, each parameter is not a single FLOP. Each parameter participates in many FLOPs during both forward and backward passes.
• Training Data Size (Tokens): GPT-3 was trained on a massive dataset. We need a rough estimate of the number of tokens in the training data. Let's assume it was trained on roughly 300 billion tokens. This is based on the common crawl dataset being cited, and estimates for the number of tokens are in this region.
• Training Epochs: How many times did the model see the entire training dataset? Large language models are typically trained for a relatively small number of epochs, sometimes even less than one full epoch (i.e., they don't see every token). Let's assume an effective number of epochs of 0.5, which takes into account how the model likely saw the data.
• Sequence Length: The context window the model processes at a time. A longer sequence length increases computational cost. Let's assume an average sequence length of 2048 tokens during training (This is a typical size for these models).
• Batch Size: How many sequences are processed in parallel. Larger batch sizes generally mean more efficient use of hardware, but there are practical limits. We don't have enough information on this to factor it directly, so we will use the rule of thumb in step 2.
• Forward Pass vs. Backward Pass: Backpropagation (used for calculating gradients and updating weights) is computationally more expensive than the forward pass. A common rule of thumb is that the backward pass takes roughly twice the computation of the forward pass.

2. The "6N" Rule of Thumb (and why it's useful):

A widely used (and relatively accurate) approximation for the FLOPs required to train a transformer model is:

FLOPs ≈ 6 * Number of Parameters * Number of Tokens

• Why 6? This factor accounts for several things:

  • Forward Pass (2): Consider the matrix multiplications involved in a transformer. For each token in a sequence, you're essentially doing a matrix multiplication (embedding size * number of parameters). This gives us a factor of roughly 2 (simplified).
  • Backward Pass (4): The backward pass involves calculating gradients, which roughly doubles the computation of the forward pass, so 2 * 2 = 4.
  • Total (6): 2 (forward) + 4 (backward) = 6

• Why this is useful: This rule of thumb avoids having to delve into the specifics of the transformer architecture (number of layers, attention heads, etc.). It captures the dominant computational cost, which is the interaction of every parameter with every token.

3. Calculation:

Let's plug in our estimates:

• Number of Parameters (N) = 175 billion = 1.75 x 10^11
• Number of Tokens = 300 billion = 3 x 10^11
• Number of Epochs = 0.5

Total number of tokens seen: 3 x 10^11 * 0.5 = 1.5 x 10^11

FLOPs ≈ 6 * (1.75 x 10^11) * (1.5 x 10^11) FLOPs ≈ 6 * 2.625 * 10^22 FLOPs ≈ 15.75 x 10^22 FLOPs ≈ 1.575 x 10^23

4. Final Answer and Considerations:

Our estimated total number of FLOPs required to train GPT-3 is approximately 1.6 x 10^23 FLOPs. This is 16 zettaFLOPs.

• Order of Magnitude: This is the most important takeaway. We're in the ballpark of 10^23. The precise number could be somewhat higher or lower, but this gives us a good sense of the scale.
• Simplifications: We've made many simplifying assumptions. We didn't explicitly account for:
  • Optimizer FLOPs (e.g., Adam optimizer updates).
  • Specific architecture details (number of layers, attention heads, feedforward network dimensions).
  • Data loading and preprocessing overhead.
  • The "effective" number of epochs is a very rough estimate.
  • Variations in sequence length during training.
• The 6N rule of thumb The rule is not perfect, but the main limiting factor is likely the number of tokens.

In conclusion, our reasoned estimate places the FLOPs required for training GPT-3 in the range of zettaFLOPs, highlighting the immense computational resources needed for such a large language model. This demonstrates that the number of parameters and the number of tokens are crucial.