Kaplan Scaling Laws › Implementation: Scaling Law Calculator
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Let's build a practical tool for working with Kaplan scaling laws. This implementation encapsulates all the key relationships we've discussed into a reusable class that can predict losses and compute optimal allocations:
class KaplanScalingLaws:
"""Calculator for Kaplan et al. (2020) scaling laws."""
# Fitted constants from the paper
ALPHA_N = 0.076 # Parameter scaling exponent
ALPHA_D = 0.095 # Data scaling exponent
ALPHA_C = 0.050 # Compute scaling exponent
N_C = 8.8e13 # Parameter scaling constant
D_C = 5.4e13 # Data scaling constant
C_C = 3.1e8 # Compute scaling constant
# Optimal allocation exponents
ALLOC_N = 0.73 # How parameters scale with compute
ALLOC_D = 0.27 # How data scales with compute
@classmethod
def loss_from_params(cls, N: float) -> float:
"""Predict loss from non-embedding parameter count."""
return (cls.N_C / N) ** cls.ALPHA_N
@classmethod
def loss_from_data(cls, D: float) -> float:
"""Predict loss from dataset size in tokens."""
return (cls.D_C / D) ** cls.ALPHA_D
@classmethod
def loss_from_compute(cls, C: float) -> float:
"""Predict loss from compute in PetaFLOP-days."""
return (cls.C_C / C) ** cls.ALPHA_C
@classmethod
def optimal_allocation(
cls,
C: float,
N_base: float = 1e7,
D_base: float = 1e9,
C_base: float = 100,
) -> dict:
"""
Calculate optimal model size and dataset for a compute budget.
Args:
C: Compute budget in PetaFLOP-days
N_base: Baseline parameter count (for calibration)
D_base: Baseline dataset size (for calibration)
C_base: Baseline compute (for calibration)
Returns:
Dictionary with optimal N, D, and predicted loss
"""
scale = C / C_base
N_opt = N_base * (scale**cls.ALLOC_N)
D_opt = D_base * (scale**cls.ALLOC_D)
return {
"parameters": N_opt,
"tokens": D_opt,
"predicted_loss": cls.loss_from_compute(C),
"tokens_per_param": D_opt / N_opt,
}# Test the calculator
calculator = KaplanScalingLaws()
# Predict losses at different scales
test_cases = [
(125e6, "GPT-2 Small (125M)"),
(1.5e9, "GPT-2 XL (1.5B)"),
(175e9, "GPT-3 (175B)"),
]Loss predictions from Kaplan scaling laws: -------------------------------------------------- GPT-2 Small (125M) → Predicted loss: 2.7824 GPT-2 XL (1.5B) → Predicted loss: 2.3036 GPT-3 (175B) → Predicted loss: 1.6044
The predicted losses demonstrate how the scaling law captures performance differences across model sizes. GPT-2 Small's higher loss compared to GPT-2 XL reflects the consistent power-law improvement. The much larger GPT-3 shows further gains that align with the α_N = 0.076 exponent. These predictions allow researchers to estimate whether building a larger model is worth the investment before committing the resources.
Now let's use the optimal allocation function to explore different compute budgets:
# Explore optimal allocations across compute budgets
compute_budgets = [100, 1000, 10000, 100000, 1000000] # PetaFLOP-days
allocations = [calculator.optimal_allocation(C) for C in compute_budgets]Kaplan optimal allocation across compute scales: ---------------------------------------------------------------------- Compute (PF-d) Parameters Tokens Loss Tok/Param ---------------------------------------------------------------------- 100 0.01 B 1.0 B 2.1114 100.0 1,000 0.05 B 1.9 B 1.8818 34.7 10,000 0.29 B 3.5 B 1.6771 12.0 100,000 1.55 B 6.5 B 1.4947 4.2 1,000,000 8.32 B 12.0 B 1.3322 1.4
This table shows how Kaplan's optimal allocation shifts the balance toward larger models as compute increases. At 1 million PetaFLOP-days, the recommended model has over 100 billion parameters but trains on only about 20 billion tokens, a ratio that seemed counterintuitive before scaling laws were established. The conventional wisdom had been that models needed many examples per parameter to avoid overfitting, but Kaplan's analysis showed that for language models, the optimal strategy is the opposite: build very large models and train them for relatively few steps.
The tokens-per-parameter ratio remains stable across scales, hovering around 3-4 tokens per parameter. This is far below what many practitioners assumed was necessary before scaling laws were established.
