\lceil x \rceil

Ceiling, floor, round

⌈x⌉ reads "ceiling of x" and rounds up. ⌊x⌋ reads "floor of x" and rounds down. The mixed ⌊x⌉ rounds to the nearest whole number.

Brackets with the hooks at the top on both sides round up; hooks at the bottom on both sides round down; one of each rounds to nearest.

Expert capacity in mixture-of-experts uses a ceiling because an expert cannot have a fraction of a token slot. Because capacity is a hard whole-number limit, tokens routed to an expert beyond it are dropped.

Example

\lceil 2.3 \rceil = 3, \quad \lfloor 2.7 \rfloor = 2, \quad \lfloor 2.7 \rceil = 3

Round up, round down, round to nearest.

Where the course uses it

17.5 Expert Capacity and Dropped Tokens · Module 17, Mixture of Expertsfirst use
24.2 Uniform Integer Quantization · Module 24, Quantization and Number Formats

Related terms

Integer quantizationConverts a real number x into a small integer q: divide by a scale s, round to the nearest integer, add a zero-point z, and clip to the allowed range.
Mixture-of-experts routingA small gate network scores every expert for a token, only the K highest-scoring experts run, and their outputs are combined using the gate's weights.