Equation 12 · Dense, Sparse, and Distilled: Comparing Approaches to Frontier Model Capacity
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Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
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Symbol N_a
is part of the quantity the equation computes from the expression on the right.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
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What the article says around this equation
Mixture-of-experts breaks the identity = 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 ] .
Sources cited in the surrounding passage
- [4] Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer ↗
- [5] Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity ↗
These citations give research context. Read each source to check which claims it supports.
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