{"ID":6620558,"CreatedAt":"2026-07-15T01:01:48.440468303Z","UpdatedAt":"2026-07-15T03:28:55.185153975Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.12501","arxiv_id":"2607.12501","title":"What Does Goodness Measure? A Likelihood-Ratio Account of Forward-Forward Learning","abstract":"The Forward-Forward (FF) algorithm trains each layer locally, so that a scalar goodness - the sum of squared activations - is high on real inputs and low on contrastive ones, with activations normalized between layers. Both choices are usually treated as heuristics. Under an explicit generative model they are not: the squared goodness is the sufficient statistic of a likelihood-ratio test between two zero-mean populations differing in scale, and the FF threshold is its boundary. It generalizes: anisotropic populations yield a Mahalanobis goodness, the plain square being its isotropic case; heavy-tailed populations yield a saturating statistic whose slope is a posterior precision - divisive normalization - with bounded evidence and an advantage only under aggregation. The same lens characterizes the inter-layer normalization: it must remove the length while preserving per-coordinate energy, explaining a depth collapse we observe under unit-norm normalization; and the pairwise objective admits a scale-inflation shortcut that a whitened goodness removes.","short_abstract":"The Forward-Forward (FF) algorithm trains each layer locally, so that a scalar goodness - the sum of squared activations - is high on real inputs and low on contrastive ones, with activations normalized between layers. Both choices are usually treated as heuristics. Under an explicit generative model they are not: the...","url_abs":"https://arxiv.org/abs/2607.12501","url_pdf":"https://arxiv.org/pdf/2607.12501v1","authors":"[\"Paolo Giannitrapani\"]","published":"2026-07-14T08:33:33Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"eess.IV\",\"stat.ML\"]","methods":"[]","has_code":false}