Dualnormer

Dualnormer is a term used in convex analysis and machine learning to describe an operator or component that computes or applies dual norms. It functions within optimization pipelines to enforce or exploit dual-norm constraints and regularizers, and can be implemented as a software module or a theoretical construct in proofs and algorithms.

In normed vector spaces, the dual norm of a norm ||·|| is defined by ||y||_ = sup{ y^T x : ||x|| ≤ 1 }. The dualnormer thus maps a vector to its dual-norm value with respect to a specified base norm, and can expose the dual ball or other dual-geometry features to aid dual updates and projections in optimization.

For many common norms there are closed-form duals: the L1 norm has dual L∞, the L2 norm is self-dual, and the nuclear norm has the operator norm as its dual. For other norms, the dual norm may be computed by solving a convex optimization problem or via established analytical formulas. A dualnormer may also provide subgradients and proximal operators related to dual norms, which are useful in iterative methods.

Applications span regularized learning and regression, where dual norms appear in dual formulations and subgradient methods; robust optimization; and neural network training that imposes norm-based constraints on weights or activations. In optimization algorithms, the dualnormer supports dual ascent, primal–dual methods, and computations of Fenchel conjugates and their proximals.

Variants include group norms and mixed norms, whose duals are likewise characterized. The term dualnormer is used in literature to denote either a theoretical construct or a practical implementation within numerical solvers and optimization libraries.