# Datafits

A datafit defines the loss on the complete linear predictor, including the intercept. Matrix operations belong to the design. Loss values and derivatives use the same averaging convention. 

| Datafit | Response | Loss | Compatible solvers |
| --- | --- | --- | --- |
| [Gaussian](<gaussian>) | One finite real response per observation | Averaged squared error | Coordinate descent for L1, proximal gradient, IRLS, hybrid for supported penalties |
| [Binomial](<binomial>) | Finite values in $[0,1]$ | Averaged Bernoulli negative log likelihood | Proximal gradient, IRLS, hybrid for supported penalties |
| [Multivariate Gaussian](<multivariate-gaussian>) | Multiple finite response columns | Squared error, summed over responses and averaged over observations | Proximal gradient with AxisPenalty |

All three fit unpenalized intercepts by default. Response dimensions must match nonempty training observations. Data and parameters are validated when losses are evaluated or models are fitted. 

[`shrinkage::Datafit`](<https://shrinkage.cc/packages/rust/reference/9d16367f3210bbbb9bf039290cf9c94f447e51d4ea33aea446a7f30e965d873a>) supplies loss evaluation. `DifferentiableDatafit` adds gradients and observation metadata. [`shrinkage::SmoothDatafit`](<https://shrinkage.cc/packages/rust/reference/f6f2ab3078b0ea56f174e2981b999f9be72be3c00ca00ad6485be1ed341db8c1>) asserts convexity, and [`shrinkage::DiagonalHessianDatafit`](<https://shrinkage.cc/packages/rust/reference/0c7a483f4e622180bfcc5efab524b3377e2104e91e0d263eb5a204d4a4ab722e>) supplies exact, potentially signed second derivatives. `QuadraticModelDatafit` supplies nonnegative working curvature to IRLS. These capabilities determine which [solvers](<../reference/solvers/>) can fit a composition. 

Gaussian and binomial models also support configurable links through the typed Rust IRLS API using `.with_link(...)`.