Engineering / advanced
Numerical behavior of learning computations
Trace rounding and conditioning through reductions, losses, gradients, optimizer state, and update accumulation with explicit error checks.
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Inside this module
Module 01 advanced
Numerical behavior of learning computations
Trace rounding and conditioning through reductions, losses, gradients, optimizer state, and update accumulation with explicit error checks.
- 01
Unit roundoff is half the spacing convention, not the whole error story
Separate significand precision, local spacing, exponent range, and rounding mode before interpreting numerical tolerances in learning kernels.
28 min - 02
Cancellation can expose errors that happened before subtraction
Separate a sensitive small difference from its computation, then compare raw moments and centered statistics on an exactly specified batch.
29 min - 03
Reduction order and compensation solve related but different problems
Compare sequential and balanced sums, inspect a genuine Kahan counterexample, and explain what a Neumaier correction recovers in one controlled case.
30 min - 04
Shift exponential losses before they overflow
Derive shifted log-sum-exp and direct log-probability losses while preserving finite-input domains and recognizing already lost logits.
29 min - 05
Scale the backward path before tiny gradients disappear
Explain when loss scaling preserves gradient range, why unscaling order matters, and how overflow checks change whether an optimizer step occurs.
29 min - 06
A nonzero gradient can still produce no stored update
Compare update size with the parameter's local ulp, inspect optimizer moments, and separate master-state preservation from visible low-precision copies.
29 min - 07
Small positive contributions can disappear systematically in a coarse accumulator
Distinguish input quantization from reduction error and diagnose order-dependent stagnation in low-precision sums used by learning computations.
30 min - 08
Stochastic rounding is conditionally unbiased only under a precise rule
Derive neighboring-value probabilities and distinguish a locally unbiased rounding operation from independence, reproducibility, and final-model accuracy.
32 min - 09
Parallel reduction order can change a valid floating-point answer
Trace nonassociative sums through parallel and distributed execution while separating determinism, accuracy, conditioning, and actual implementation defects.
31 min - 10
Approve a precision change with separate quality and cost evidence
Build mechanism-level references and controlled end-task experiments before accepting a faster or smaller learning computation.
34 min