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We propose a new algorithm for minimizing regularized empirical loss:Stochastic Dual Newton Ascent (SDNA). Our method is dual in nature: in eachiteration we update a random subset of the dual variables. However, unlikeexisting methods such as stochastic dual coordinate ascent, SDNA is capable ofutilizing all curvature information contained in the examples, which leads tostriking improvements in both theory and practice - sometimes by orders ofmagnitude. In the special case when an L2-regularizer is used in the primal,the dual problem is a concave quadratic maximization problem plus a separableterm. In this regime, SDNA in each step solves a proximal subproblem involvinga random principal submatrix of the Hessian of the quadratic function; whencethe name of the method. If, in addition, the loss functions are quadratic, ourmethod can be interpreted as a novel variant of the recently introducedIterative Hessian Sketch.

SDNA: Stochastic Dual Newton Ascent for Empirical Risk Minimization