Numerical Methods

Numerical methods are algorithms for approximate computation when exact math is too slow or impossible. ML relies on numerical methods constantly: optimization, solving linear systems, stability of training, and efficient computation at scale.

Approximation & Error

• Floating point: limited precision can cause rounding and overflow/underflow.
• Stability: small numerical errors should not explode during computation.
• Conditioning: some problems amplify tiny input errors (ill-conditioned).

Solving Linear Systems

Many ML methods reduce to solving Ax = b (least squares, ridge regression). For large problems, iterative solvers can be faster than direct matrix inversion.

Numerical Integration

When modeling continuous-time dynamics (ODEs), we approximate integrals with methods like Euler or Runge–Kutta. Step size trades off speed vs accuracy.

Practical Tips For ML

• Normalize inputs; use stable loss implementations (log-sum-exp).
• Watch for NaNs/infs; use gradient clipping for exploding gradients.
• Prefer numerically stable decompositions (e.g. QR/SVD) over naive inversion.