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The Stackelberg prediction game (SPG) is popular in characterizing strategicinteractions between a learner and an attacker. As an important special case,the SPG with least squares loss (SPG-LS) has recently received much researchattention. Although initially formulated as a difficult bi-level optimizationproblem, SPG-LS admits tractable reformulations which can be polynomiallyglobally solved by semidefinite programming or second order cone programming.However, all the available approaches are not well-suited for handlinglarge-scale datasets, especially those with huge numbers of features. In thispaper, we explore an alternative reformulation of the SPG-LS. By a novelnonlinear change of variables, we rewrite the SPG-LS as a sphericallyconstrained least squares (SCLS) problem. Theoretically, we show that an$\epsilon$ optimal solution to the SCLS (and the SPG-LS) can be achieved in$\tilde{O}(N/\sqrt{\epsilon})$ floating-point operations, where $N$ is thenumber of nonzero entries in the data matrix. Practically, we apply twowell-known methods for solving this new reformulation, i.e., the Krylovsubspace method and the Riemannian trust region method. Both algorithms arefactorization free so that they are suitable for solving large scale problems.Numerical results on both synthetic and real-world datasets indicate that theSPG-LS, equipped with the SCLS reformulation, can be solved orders of magnitudefaster than the state of the art.

Solving Stackelberg Prediction Game with Least Squares Loss via Spherically Constrained Least Squares Reformulation