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The recent years have witnessed advances in parallel algorithms for largescale optimization problems. Notwithstanding demonstrated success, existingalgorithms that parallelize over features are usually limited by divergenceissues under high parallelism or require data preprocessing to alleviate theseproblems. In this work, we propose a Parallel Coordinate Descent Newtonalgorithm using multidimensional approximate Newton steps (PCDN), where theoff-diagonal elements of the Hessian are set to zero to enable parallelization.It randomly partitions the feature set into $b$ bundles/subsets with size of$P$, and sequentially processes each bundle by first computing the descentdirections for each feature in parallel and then conducting $P$-dimensionalline search to obtain the step size. We show that: (1) PCDN is guaranteed toconverge globally despite increasing parallelism; (2) PCDN converges to thespecified accuracy $\epsilon$ within the limited iteration number of$T_\epsilon$, and $T_\epsilon$ decreases with increasing parallelism (bundlesize $P$). Using the implementation technique of maintaining intermediatequantities, we minimize the data transfer and synchronization cost of the$P$-dimensional line search. For concreteness, the proposed PCDN algorithm isapplied to $\ell_1$-regularized logistic regression and $\ell_2$-loss SVM.Experimental evaluations on six benchmark datasets show that the proposed PCDNalgorithm exploits parallelism well and outperforms the state-of-the-artmethods in speed without losing accuracy.

Parallel Coordinate Descent Newton Method for Efficient $\ell_1$-Regularized Minimization