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Equation 12 · Part 3 · Dense, Sparse, and Distilled: Comparing Approaches to Frontier Model Capacity

=

Na=NN_a = N
=

What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

Mixture-of-experts breaks the identity NaN_a = N . Shazeer and colleagues introduced the sparsely-gated MoE layer, in which a learned gate selects a small subset of expert sub-networks per example, allowing parameter counts far beyond what could be densely activated at the same compute [ 4 ] . Fedus, Zoph, and Shazeer simplified it decisively: the Switch layer routes each token to exactly one expert rather than the top- k , which reduced routing computation and communication cost while preserving quality, and they scaled the approach to trillion-parameter models [ 5 ] .

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.