We provide several algorithms for constrained optimization of a large class of convex problems, including softmax, $ell_p$ regression, and logistic regression. Central to our approach is the notion of width reduction, a technique which has proven immensely useful in the context of maximum flow [Christiano et al., STOC11] and, more recently, $ell_p$ regression [Adil et al., SODA19], in terms of improving the iteration complexity from $O(m^{1/2})$ to $tilde{O}(m^{1/3})$, where $m$ is the number of rows of the design matrix, and where each iteration amounts to a linear system solve. However, a considerable drawback is that these methods require both problem-specific potentials and individually tailored analyses. As our main contribution, we initiate a new direction of study by presenting the first unified approach to achieving $m^{1/3}$-type rates. Notably, our method goes beyond these previously considered problems to more broadly capture quasi-self-concordant losses, a class which has recently generated much interest and includes the well-studied problem of logistic regression, among others. In order to do so, we develop a unified width reduction method for carefully handling these losses based on a more general set of potentials. Additionally, we directly achieve $m^{1/3}$-type rates in the constrained setting without the need for any explicit acceleration schemes, thus naturally complementing recent work based on a ball-oracle approach [Carmon et al., NeurIPS20].
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