Let's break down estimating the FLOPs for GPT-3 training. Since we can't look up exact figures, we'll rely on reasonable guesses and approximations.

1. Model Size: GPT-3 has 175 billion parameters. This is our anchor point.

2. FLOPs per Parameter per Token: A forward pass involves a matrix multiplication for each layer. Let's assume an average of 2 FLOPs per parameter per token processed (one multiplication and one addition). This is a simplification, as different layers might have slightly different computations, but it's a reasonable starting point.

3. Dataset Size: GPT-3 was trained on a massive dataset. Let's approximate the effective dataset size (accounting for repeated passes) as 100 billion tokens. This might be an underestimate, but given it was trained on a mix of filtered and unique content, and we don't know the exact number of epochs, it's a starting point.

4. Forward and Backward Passes: For each token, we do one forward pass and one backward pass for gradient computation. So, we have 2 passes.

5. Optimizer overhead: Optimizers like Adam add some computational overhead. Let's add a 20% overhead factor to account for this.

Now, let's put it all together:

• FLOPs per token = 175B parameters * 2 FLOPs/parameter * 2 passes = 700B FLOPs/token

• Total FLOPs = 700B FLOPs/token * 100B tokens = 7 x 10^22 FLOPs

• Optimizer overhead: 7 x 10^22 * 1.2 = 8.4 x 10^22 FLOPs

Therefore, a very rough estimate is around 8.4 x 10^22 FLOPs to train GPT-3.

It's crucial to remember that this is a back-of-the-envelope calculation. The actual number could be higher or lower due to factors such as:

• Variations in layer computations: Different layers might have different computational complexity.
• Optimization techniques: Specific optimizations during training could reduce the actual FLOPs.
• Data preprocessing and tokenization overhead: We haven't accounted for these factors.
• Dataset repetitions/epochs: The assumed effective dataset size is a guess.
• Architecture specifics: The actual architecture and training process might involve more complexities not captured here.

So, while our estimate provides a general order of magnitude, it shouldn't be taken as a precise figure.

Let's walk through an estimation of how many FLOPs were required to train GPT-3, using only reasoning and not searching for the exact numbers.

Step 1: Understand Model Size

GPT-3 (the original) had 175 billion parameters.

Step 2: Estimate FLOPs per Forward/Backward Pass

• For transformer models, a rough rule of thumb is that a forward-and-backward pass for a parameter takes ~2 FLOPs per parameter per token (1 for forward, 1 for backward).
• Each token processed in a batch involves the whole model.

So, FLOPs per token = 2 × number of parameters.

Step 3: Estimate Total Number of Tokens Used in Training

• GPT-3 was reportedly trained on about 300 billion tokens.
• Let's use 300B tokens as our assumption.

Step 4: Calculate Total Number of FLOPs

Total FLOPs = FLOPs per token × number of tokens processed

• FLOPs per token = 2 × 175B = 350 billion FLOPs per token
• Number of tokens = 300 billion

Now, multiply:

[ \text{Total FLOPs} = 350,\text{B} \times 300,\text{B} = (3.5 \times 10^{11}) \times (3 \times 10^{11}) = 1.05 \times 10^{23} ]

So,

[ \boxed{ \approx 1 \times 10^{23} \text{ FLOPs} } ]

Step 5: Check and discussion

This estimate is rough, but matches the ballpark for very large models. (Other nuances, like multiple epochs or data repetition, may factor in, but using "number of tokens processed" as above is standard.)

Final Answer

It takes about ( 1 \times 10^{23} ) FLOPs to train GPT-3.

This comes from:

• ( 175 ) billion parameters
• Each token: ( 2 \times 175 ) billion = ( 350 ) billion FLOPs
• ( 300 ) billion training tokens
• ( 350 ) billion × ( 300 ) billion = ( 1 \times 10^{23} ) FLOPs

Assumptions:

• Each forward+backward pass needs ≈ 2×params FLOPs/token.
• 300B tokens in total.

Notes:

• Reality varies with model and optimization, but this is a widely accepted estimating method.

Sally has one sister. Since all the brothers share the same sisters, Sally and her sister are the two sisters that each brother has.

Let's break down the problem:

• Sally is a girl.
• Sally has 3 brothers.
• Each brother has 2 sisters.

Let's analyze:

Sally's siblings:

• 3 brothers (let's call them B1, B2, B3)
• Sally herself

The brothers' sisters:

• Each brother has 2 sisters.

Who are the possible sisters for each brother?

• Sally is a sister to each brother.
• Are there any other sisters? That's the question!

If Sally is the only girl, then each brother would only have 1 sister (Sally). But the problem says each brother has 2 sisters.

Therefore, there must be another sister besides Sally.

Let's check: Suppose Sally has 1 sister (besides herself). So the siblings are:

• Sally (girl)
• A second sister (let's call her Anna)
• 3 brothers

Children in the family:

• 2 girls (Sally and Anna)
• 3 boys (the brothers)

Now, for each brother:

• Their sisters are Sally and Anna: 2 sisters (matches the problem statement).

So, Sally has 1 sister.

Final answer:

[ oxed{1} ]

Sally has 1 sister.

No, 9.11 is less than 9.9. Think of it like this: 9.9 is the same as 9.90. 9.90 is larger than 9.11.

No, 9.11 is not greater than 9.9.

9.11 is less than 9.9.

This might seem counterintuitive, but when comparing decimal numbers:

• 9.11 and 9.9 both have 9 as the integer part.
• Now compare the decimal part:
  • 9.11 is 9 and 11 hundredths.
  • 9.9 is 9 and 9 tenths (which is 9.90).

So, compare 9.11 and 9.90:

• 9.11 < 9.90 (because 11 hundredths < 90 hundredths).

Therefore, 9.11 is less than 9.9.