To estimate the total number of FLOPs (floating-point operations) required to train GPT-3 without searching online, we rely on known architectural details and standard computational models for transformer training. Here's the step-by-step reasoning:

Step 1: Key Parameters of GPT-3

• Number of parameters (P): GPT-3 has 175 billion parameters (175 × 10⁹).
• Training dataset size (D): The model was trained on approximately 300 billion tokens (300 × 10⁹ tokens). This is a standard figure associated with GPT-3.

Step 2: FLOPs per Token Calculation

Training a transformer model involves two main operations per token:

1. Forward pass: Computes the model's output.
2. Backward pass: Computes gradients for backpropagation.

For transformer architectures, the combined FLOPs for forward and backward passes per token is empirically estimated as 6 × P. This is derived from:

• Forward pass: Requires ~2 × P FLOPs per token (matrix multiplications dominate, with each parameter used in multiply-add operations).
• Backward pass: Typically requires ~2× the FLOPs of the forward pass due to gradient calculations (~4 × P FLOPs per token).
• Total per token: 2P (forward) + 4P (backward) = 6P FLOPs per token.

Step 3: Total FLOPs for the Entire Dataset

The total FLOPs scales linearly with the dataset size:

• Total FLOPs = FLOPs per token × Number of tokens.
• Substituting the values:
  Total FLOPs = (6 × P) × D = 6 × (175 × 10⁹) × (300 × 10⁹).

Step 4: Calculation

• Compute the product:
  6 × 175 × 300 × 10⁹ × 10⁹ = 6 × 52,500 × 10¹⁸.
• Simplify:
  6 × 52,500 = 315,000.
  Total FLOPs = 315,000 × 10¹⁸ = 3.15 × 10²³ FLOPs.

Step 5: Validation with Known Compute

• GPT-3 training reportedly used ~3,640 petaflop-days of compute.
• Convert petaflop-days to FLOPs:
  1 petaflop-day = 10¹⁵ FLOPs × 86,400 seconds/day = 8.64 × 10¹⁹ FLOPs.
  3,640 petaflop-days = 3,640 × 8.64 × 10¹⁹ ≈ 3.14 × 10²³ FLOPs.
• This closely matches our estimate of 3.15 × 10²³ FLOPs, confirming the calculation.

Step 6: Batch Size and Steps Consideration

• Batch size: GPT-3 used a batch size of 3.2 million tokens per step (80 batches × 4096 tokens/batch).
• Number of training steps:
  Steps = Total tokens / Batch size = 300 × 10⁹ / 3.2 × 10⁶ = 93,750 steps.
• FLOPs per step:
  Per step FLOPs = 6 × P × Batch size = 6 × 175 × 10⁹ × 3.2 × 10⁶ = 3.36 × 10¹⁸ FLOPs/step.
• Total FLOPs:
  93,750 steps × 3.36 × 10¹⁸ FLOPs/step ≈ 3.15 × 10²³ FLOPs.
  This consistency reinforces the estimate.

Step 7: Negligible Factors

• Parameter updates: Optimizer steps (e.g., Adam) add O(P) FLOPs per step. For 93,750 steps:
  Update FLOPs ≈ 93,750 × 175 × 10⁹ ≈ 1.64 × 10¹⁶ FLOPs (negligible vs. 3.15 × 10²³).
• Embedding/softmax layers: Included in the 6P per-token estimate.
• Attention mechanisms: Captured by the 6P heuristic (validated for transformers).

Final Estimate

The total FLOPs required to train GPT-3 is 3.15 × 10²³ FLOPs. This is derived from:

• Standard transformer FLOPs per token (6P).
• GPT-3 parameters (175B) and dataset size (300B tokens).
• Cross-validation with reported compute usage (3,640 petaflop-days).

To estimate the total number of FLOPs required to train GPT-3, let's break it down into key components.

1. Model Size and Parameters

GPT-3 has 175 billion parameters.

2. FLOPs per Forward and Backward Pass

In deep learning, the FLOPs required for a forward pass and backward pass are estimated as:

• Forward pass: ( 2 imes ) the number of parameters
• Backward pass: ( 2 imes ) the FLOPs of the forward pass

Thus, total FLOPs per training step:

[ ext{Total FLOPs per step} = 6 imes ext{number of parameters} imes ext{tokens per batch} ]

For GPT-3, let's assume 175B parameters and a batch size of 2048 tokens (common for large-scale training).

[ ext{FLOPs per training step} = 6 imes 175 imes 10^9 imes 2048 ]

[ = 2.15 imes 10^{15} ext{ FLOPs per step} ]

3. Total Training Steps

GPT-3 was trained with 300 billion tokens (as per OpenAI's paper).

If the batch size is 2048 tokens, the total number of steps is:

[ ext{Total steps} = rac{300 imes 10^9}{2048} approx 1.46 imes 10^8 ]

4. Total FLOPs Required

Multiplying the FLOPs per step by the total number of steps:

[ ext{Total FLOPs} = (2.15 imes 10^{15}) imes (1.46 imes 10^8) ]

[ approx 3.14 imes 10^{23} ext{ FLOPs} ]

Final Answer:

The estimated total number of FLOPs required to train GPT-3 is about ( 3.14 imes 10^{23} ) floating point operations.

Sally has 3 brothers, meaning there are 3 boys in the family. Each brother has 2 sisters, indicating that there are exactly 2 girls in the family. Since Sally is one of the girls, there must be one additional sister. Therefore, Sally has one sister.

This can be verified by considering the family composition:

• Sisters: Sally and one other girl.
• Brothers: 3 boys.

From Sally's perspective:

• She has 3 brothers.
• She has 1 sister (the other girl).

From each brother's perspective:

• They have 2 sisters (Sally and the other girl).

This satisfies both conditions: Sally has 3 brothers, and each brother has 2 sisters.

If there were only one sister (Sally), then each brother would have only one sister, which contradicts the information that each brother has 2 sisters. If there were three sisters, then each brother would have three sisters, which also contradicts the information that each brother has 2 sisters. Thus, there must be exactly two sisters, meaning Sally has one sister.

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Sally has only one sister.

Explanation:

• Sally has 3 brothers.
• Each brother has 2 sisters.
• The two sisters are Sally and one other girl.

Thus, Sally has 1 sister.