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How Numerical Errors Scale with Hardware

Niklas Baumgarten

arXiv:2610.35413v1cs.DC

TL;DR

Computational resources and energy constrain numerical results, while complexity analysis alone does not answer what accuracy a given system can achieve. The note combines complexity analysis with parallelization laws to relate parallel hardware to error reduction; it focuses on time constraints, though memory or data can instead be limiting.

  • Problem

    Complexity analysis alone leaves open what error is achievable within a time frame or how additional hardware changes results, despite energy and hardware constraints limiting computational outcomes.

    Source: §1 · p. 1, §2 · p. 2

  • Method

    The note combines error-versus-cost complexity relations with parallelization laws to connect hardware resources and energy budgets to achievable accuracy.

    Source: §1 · pp. 1–2

  • Results

    The proposed comparison uses relative error reduction on parallel systems rather than speedup, and the proposition is supported when the stated inverse-function condition holds and the right Figure 1 plot is observed.

    Source: §1 · p. 1, §2 · p. 3

  • Takeaways & Limitations

    The framework advocates evaluating computational problems under physical resource constraints, particularly how more computing units affect numerical error.

    Source: §4 · p. 5

    The analysis focuses on computing time, although memory capacity or data availability may instead be the limiting constraint for an application.

    Source: §1 · p. 2, §4 · p. 5

Abstract

from arXiv · show

In this short technical note, we outline how to bound numerical errors with respect to computational resources. We consider algorithms with known complexity estimates that can be executed in parallel and combine these estimates with parallelization laws. We suppose that the proposed approach generalizes beyond the presented examples and argue that it provides a straightforward and fair framework for comparing algorithms, hardware systems, and software implementations. This is achieved by imposing identical resource or energy budgets on each configuration, and comparing the resulting numerical error evaluations.

1 Introduction

The introduction frames computational resources and energy as limits on numerical results, and proposes connecting complexity analysis with parallelization laws to relate hardware resources to achievable accuracy. It motivates this framework as a way to compare resource needs while accounting for time, memory, and data constraints.

1 Introduction

  • Amdahl’s law treats fixed-size problems and Gustafson’s law scales problem size with processing units, while numerical applications can link problem size nonlinearly to error.Neural scaling laws relate model parameters and training data to training loss, which can be expressed in terms of total training compute under a compute-optimal strategy.Source: §1 · p. 1
  • The note connects parallelization laws with error-versus-cost complexity relations to relate hardware resources to achievable accuracy and invested energy.It proposes relative error reduction on parallel systems rather than speedup.Source: §1 · p. 1
  • Finding an algorithm that minimizes numerical error under computational constraints is framed as a knapsack problem over decisions such as resource allocation.Constraints can include computing time, memory or data availability, and the number of processing units.Source: §1 · pp. 1–2
  • Choosing an optimal sequence of resource-allocation decisions is NP-hard, while complexity analysis bounds the operational cost needed to reach a specified error.The complexity bound has an unknown constant that may depend on the problem’s condition number and the implementation.Source: §1 · p. 2
  • Memory and data constraints can determine an algorithm’s smallest achievable error, but the note does not make them its primary investigation goal.Limited training data can introduce bias or overfitting, and insufficient memory can prevent larger finite element bases.Source: §1 · p. 2

2 Proposition

The section combines complexity estimates with parallelization laws to connect hardware resources to achievable numerical error. Under stated workload conditions, it bounds error at a fixed time budget and supports comparisons under energy budgets.

2 Proposition

  • Because c is usually unknown, the note instead relies on algorithms solving Theorem 1.1 through greedy or dynamic programming, whose information exchange makes λs nonzero.The theorem-based approach avoids needing c but requires these algorithmic strategies.Source: §2 · pp. 2, 4
  • With f*(P)=f^-1(P) and a sufficiently large workload per processing unit, parallel error is bounded by R times the serial error achievable within T.The bound combines the parallel reduction factor R with the serial accuracy estimate f^-1(T).Source: §2 · p. 3
  • As P increases, the non-parallelizable component leaves an error floor εs; more computing time, not more hardware, reduces that summand.The bound approaches εs as P→∞.Source: §2 · pp. 3–4
  • Figure 1 shows diminishing error reductions as P grows, because the P-independent term becomes increasingly important.The right panel fits achieved error estimates at fixed final time T against P.Source: §2 · p. 3, Figure 1 · p. 4
  • Under a fixed Watt-hour budget, achieved errors can be compared to identify the best algorithm, hardware system, or software implementation.The bound also permits monetary or carbon-dioxide budgets based on electricity prices and energy mix.Source: §2 · p. 4

3 Examples

The examples examine how Theorem 2.1 applies to Monte Carlo estimation, finite element methods, neural-network training, and gradient descent, with applicability depending on algorithm structure.

Monte Carlo Estimation

  • f(ϵ)=ϵ^-2; Monte Carlo estimation can be implemented greedily and embarrassingly parallel, satisfying the assumptions of Theorem 2.1.Numerical verification is reported for multilevel Monte Carlo applied to acoustic wave equations [7].Source: §3 · p. 4

Gradient Descent

  • Gradient descent under strong convexity, smoothness, and sufficient step size achieves O(log(1/ϵ)) steps, but its sequential iteration does not directly satisfy Theorem 2.1’s assumptions.Theorem 2.1 can apply when other resource-consuming tasks, such as function evaluation or batch estimation, are aligned with the iteration index.Source: §3 · pp. 4–5

Finite Element Methods

  • FEM work scales as O(h^-d) while discretization error decays as O(h^α), yielding f(ϵ)=ϵ^-d/α for error below ϵ.Geometric mesh refinement makes constraints require adaptation; in three dimensions, halving h requires eightfold P for fixed T, a weak-scaling experiment.Source: §3 · p. 5

Training of Neural Networks

  • Neural scaling laws relate training loss to compute after balancing dataset size and trainable parameters, and splitting Cmin into P·T is argued to clarify hardware effects.The passage argues Theorem 2.1 applies to compute-related neural scaling laws.Source: §3 · p. 5

4 Conclusion and Discussion

The discussion frames numerical error and system constraints together, while emphasizing that the note focuses on computing time and needs broader validation.

4 Conclusion and Discussion

  • Theorem 1.1 pairs a numerical-error performance measure with system constraints such as processing units, time, memory, data, or energy budgets.The passage relates this framing to approaches in the Theory of Constraints.Source: §4 · p. 5
  • Theorem 2.1’s validity requires examination beyond the examples presented, so its generality remains an open scope boundary.Source: §4 · p. 5
  • The note combines complexity analysis with parallelization laws to study computing-time constraints and how increasing computing units affects numerical errors.Memory capacity or data availability may instead be the dominant constraint, depending on the application.Source: §4 · p. 5

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