We present a method for the synthesis of polynomial lasso programs. These programs consist of a program stem, a set of transitions, and an exit condition, all in the form of algebraic assertions (conjunctions of polynomial equalities). Central to this approach is the discovery of non-linear (algebraic) loop invariants. We extend Sankaranarayanan, Sipma, and Manna's template-based approach and prove a completeness criterion. We perform program synthesis by generating a constraint whose solution is a synthesized program together with a loop invariant that proves the program's correctness. This constraint is non-linear and is passed to an SMT solver. Moreover, we can enforce the termination of the synthesized program with the support of test cases.
Polynomial lasso programs have also received some attention regarding the analysis of their termination properties. Bradley, Manna and Sipma use finite difference arithmetic to compute lexicographic polynomial ranking functions for polynomial lasso programs @cite_16 .
@cite_16
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The Lasso is a very well-known penalized regression model, which adds an L1 penalty with parameter λ1 on the coefficients to the squared error loss function. The Fused Lasso extends this model by also putting an L1 penalty with parameter λ2 on the difference of neighboring coefficients, assuming there is a natural ordering. In this article, we develop a path algorithm for solving the Fused Lasso Signal Approximator that computes the solutions for all values of λ1 and λ2. We also present an approximate algorithm that has considerable speed advantages for a moderate trade-off in accuracy. In the Online Supplement for this article, we provide proofs and further details for the methods developed in the article.