low-precision training

Low precision means representing weights, activations, and gradients in fewer bits than FP32 — FP16, BF16, FP8, FP4, or INT8 — to shrink memory footprint and run more math per second on specialized hardware. The choice is never just 'how many bits' but how those bits split between range and precision.\n\n**A float is a sign, an exponent, and a mantissa — and each format spends its bits differently.** The exponent sets dynamic range (how large and small a value can be); the mantissa sets precision (how finely values are resolved). FP32 has 8 exponent and 23 mantissa bits. BF16 keeps all 8 exponent bits but drops to 7 mantissa, so it matches FP32's range while sacrificing precision — which is why it trains stably as a near drop-in. FP16 keeps 10 mantissa but only 5 exponent bits, so it is precise but underflows without loss scaling.\n\n**Below 16 bits the split gets sharper.** FP8 comes in two flavors: E4M3 leans on precision for forward passes, E5M2 leans on range for gradients. FP4 has just 16 representable levels, so it only works with fine-grained block scales that restore local range. INT8 abandons the float split entirely — a uniform grid plus one scale factor — which is cheapest to multiply but least forgiving of outliers.\n\n| Format | Bits | Exp / Mant | Strength | Main risk |\n|---|---|---|---|---|\n| FP32 | 32 | 8 / 23 | baseline accuracy | 4x the memory & bandwidth |\n| FP16 | 16 | 5 / 10 | precise | narrow range, needs loss scaling |\n| BF16 | 16 | 8 / 7 | FP32 range, stable | coarser mantissa |\n| FP8 (E4M3/E5M2) | 8 | 4/3 or 5/2 | fast train + infer | tight range, per-tensor scales |\n| FP4 / INT4 | 4 | 2/1 or integer | max compression | needs block/group scaling |\n\n```svg\n\n```\n\n**Range and precision fail in different ways, so mixed precision is the norm.** Practical pipelines keep a high-precision master copy for the optimizer, compute the heavy matmuls in BF16 or FP8, and reserve FP32 for accumulation and sensitive reductions. The art is matching each format's bit-split to the tensor's statistics: wide-dynamic-range gradients want exponent bits, tightly clustered weights want mantissa bits or a uniform integer grid.\n\nRead low precision through a quant lens rather than an accuracy-loss lens: every bit removed multiplies effective bandwidth and compute throughput, so per the roofline it directly moves a memory-bound kernel toward its compute roof. The engineering question is not 'is FP8 lossy' but which field — range or precision — a given tensor can afford to shorten before error crosses tolerance, measured rather than assumed.