LLM Inference-Time Scaling Law

A LLM Inference-Time Scaling Law is an LLM scaling law that describes how language model performance relates to inference compute, model size, and inference strategy choices during model inference.

• AKA: Generation Scaling Law.
• Context:
  • It can (typically) predict LLM Performance based on inference compute budget allocation.
  • It can (typically) guide Model Size Selection through compute efficiency analysis.
  • It can (typically) optimize Inference Strategy for different deployment scenarios.
  • It can (typically) reveal Performance Saturation points for different model sizes.
  • ...
  • It can (often) demonstrate Power Law Decay in inference loss with increased sampling trials.
  • It can (often) show Smaller Model advantages under equivalent compute budgets.
  • It can (often) inform Resource Allocation for production deployments.
  • It can (often) guide Token Generation optimization through compute distribution.
  • ...
  • It can range from being a Simple Inference Law to being a Complex Inference Law, depending on its strategy complexity.
  • It can range from being a Single Token Law to being a Multi Token Law, depending on its generation scope.
  • It can range from being a Resource Usage Law to being a Quality Impact Law, depending on its measurement focus.
  • It can range from being a Model Size Specific Law to being a Size Independent Law, depending on its generalization level.
  • ...
  • It can have Performance Limits determined by correct answer likelihoods.
  • It can have Saturation Points specific to each model size.
  • It can have Strategy Effects on compute efficiency.
  • ...
• Examples:
  • LLM Inference Patterns, such as:
    • Token Generation Laws, such as:
      • Single Token Scaling showing compute impact on generation quality.
      • Multi Token Scaling relating sequence length to resource usage.
    • Strategy Impact Laws, such as:
      • Greedy Search Law demonstrating basic inference patterns.
      • Majority Voting Law showing ensemble method effects.
      • Best-of-N Law revealing sampling strategy impacts.
    • Resource Usage Laws, such as:
      • Memory Scaling Law relating model size to memory requirements.
      • Throughput Scaling Law showing batch size effects.
  • Performance Patterns, such as:
    • Llemma Model (2024) where 7B variants outperform 34B models at equivalent compute budgets.
    • Tree Search (2024) showing advanced algorithm benefits over basic sampling.
  • ...
• Counter-Examples:
  • Training Scaling Laws, which focus on model training rather than inference behavior.
  • Architecture Scaling Laws, which describe model structure rather than inference processs.
  • Data Scaling Laws, which relate to dataset size rather than inference compute.
• See: LLM Performance Optimization, Inference Strategy, Compute Budget Allocation, Model Deployment Efficiency, Generation Quality Metric.

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References

2024-01-04

• Perplexity.ai
  • Question: Please explain inference scaling laws for language models and their implications.
  • Answer: Inference scaling laws represent a crucial frontier in understanding how language model performance improves with computational resources during inference time, distinct from traditional training scaling laws.
    • Key Findings:
      • Model Size Trade-offs:
        • Smaller models can outperform larger models when given equivalent computation budgets.
        • The Llemma-7B model with advanced inference algorithms consistently outperforms Llemma-34B using standard majority voting across all FLOPS budgets.
      • Optimal Performance Patterns:
        • Performance typically improves with increased inference compute until reaching saturation.
        • While smaller models initially perform better, larger models show advantages after smaller models saturate.
    • Inference Strategies:
      • Advanced Methods:
        • Several inference approaches have been studied for optimizing performance:
          • Greedy search
          • Majority voting
          • Best-of-n sampling
          • Weighted voting
          • Tree search algorithms
      • Compute Optimization:
        • The optimal allocation of compute during inference depends on:
          • Model size selection
          • Number of generated tokens
          • Choice of inference strategy
    • Theoretical Framework:
      • Performance Limits:
        • Mathematical analysis shows that performance gains from additional sampling will eventually saturate for any fixed model.
        • The limit is determined by the likelihood of generating correct answers through all possible reasoning paths.
      • Power Law Behavior:
        • Inference loss exhibits power law decay as the number of trials increases, connecting with prompting costs.
        • This behavior has been demonstrated both theoretically and empirically in controlled settings.
    • Practical Implications:
      • Resource Allocation:
        • Real-world deployments typically operate below the saturation point of smaller models, making them more compute-optimal for many applications.
        • This suggests that investing in sophisticated inference algorithms for smaller models may be more effective than using larger models with simple inference strategies.
      • Future Directions:
        • The field of inference-time optimization remains relatively unexplored compared to training optimization.
        • Significant improvements may come from developing methods to spend orders of magnitude more resources on critical tokens during generation.
  • Citations:

[1] https://proceedings.mlr.press/v202/dettmers23a.html
[2] https://arxiv.org/html/2408.00724v2
[3] https://bohrium.dp.tech/paper/arxiv/2410.16377
[4] https://openreview.net/forum?id=j7DZWSc8qu
[5] https://openreview.net/forum?id=VNckp7JEHn
[6] https://arxiv.org/abs/2410.16377
[7] https://www.interconnects.ai/p/openai-strawberry-and-inference-scaling-laws
[8] https://openai.com/index/scaling-laws-for-neural-language-models/
[9] https://akashbajwa.substack.com/p/inference-time-scaling-laws
[10] https://epoch.ai/blog/scaling-laws-literature-review

2021

• (Jones, 2021) ⇒ Andy L. Jones. (2021). “Scaling Scaling Laws with Board Games.” doi:10.48550/arXiv.2104.03113
  • NOTES:
    • Introduction to Scaling Laws in Reinforcement Learning: Offers a lucid explanation of how scaling laws (originally prominent in NLP and vision) can apply to RL settings. This approach demonstrates the potential for scaling laws to unify multiple aspects of training, including the size of the model, the data, and even the complexity of the environment. The authors highlight how smaller-scale experiments can reliably inform predictions about more expensive or larger-scale tasks.
    • Hex Environment and Problem Scaling: Uses the strategic board game Hex (with adjustable board size) as a test bed to show that the performance laws identified on small boards generalize well to larger boards. This scaling of the problem itself—rather than only the model—reveals a deeper insight into how difficulty exponentially increases. It also showcases why simpler rule sets (compared to Go or Chess) are beneficial for reproducible and efficient experimentation.
    • Smooth Scaling Behavior Across Board Sizes: Finds that performance changes smoothly and predictably as board size increases, supporting the idea of consistent power-law or sigmoid-like curves in RL. This absence of abrupt performance “jumps” or “spikes” indicates that no hidden threshold or discrete insight is required to master larger boards. In turn, it suggests that scaling laws can bridge small and large tasks without discontinuous leaps in complexity or skill.
    • Implications for Future Research on Scaling in Reinforcement Learning: Discusses the broader significance of these results, hinting that problem scaling plus model scaling may generalize well beyond Hex to more complex multi-agent domains. If the phenomenon proves to be as universal as in language or vision tasks, it could enable small-scale RL work to profoundly impact large-scale game or simulation challenges. Moreover, the consistent success in forecasting performance opens up a wealth of possibilities for resource-limited labs to contribute to fundamental scaling research.

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