Structure and Algorithm for Path of Solutions to a Class of Fused Lasso Problems
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We study a class of fused lasso problems where the estimated parameters in a sequence are regressed toward their respective observed values (fidelity loss), with ℓ_1 norm penalty (regularization loss) on the differences between successive parameters, which promotes local constancy. In many applications, there is a coefficient, often denoted as λ, on the regularization term, which adjusts the relative importance between the two losses. In this paper, we characterize how the optimal solution evolves with the increment of λ. We show that, if all fidelity loss functions are convex piecewise linear, the optimal value for each variable changes at most O(nq) times for a problem of n variables and total q breakpoints. On the other hand, we present an algorithm that solves the path of solutions of all variables in Õ(nq) time for all λ≥ 0. Interestingly, we find that the path of solutions for each variable can be divided into up to n locally convex-like segments. For problems of arbitrary convex loss functions, for a given solution accuracy, one can transform the loss functions into convex piecewise linear functions and apply the above results, giving pseudo-polynomial bounds as q becomes a pseudo-polynomial quantity. To our knowledge, this is the first work to solve the path of solutions for fused lasso of non-quadratic fidelity loss functions.
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