**Summary:** This paper presents a theoretical analysis of Stochastic Dual Coordinate Ascent (SDCA), a stochastic optimization method for regularized loss minimization. The authors show that SDCA has strong theoretical guarantees comparable or better than Stochastic Gradient Descent (SGD), which is widely used for large-scale machine learning problems. SDCA works by optimizing dual variables in a stochastic manner, which allows it to achieve a duality gap that is an upper bound on the primal sub-optimality. The paper analyzes SDCA for two types of loss functions: L-Lipschitz and (1/γ)-smooth. For L-Lipschitz loss functions, SDCA achieves a convergence rate of ˜O(n + L²/(λε)), while for (1/γ)-smooth loss functions, the rate is ˜O((n + 1/(λγ)) log(1/ε)). These results are improved for almost smooth losses, where SDCA converges faster than SGD. The authors also compare SDCA with other methods, including SGD and the Stochastic Average Gradient (SAG) algorithm. They show that SDCA can outperform SGD in certain regimes, especially when high solution accuracy is required. Additionally, they propose a modified version of SGD that can be combined with SDCA to achieve faster convergence. The paper includes detailed proofs for the convergence of SDCA under different assumptions on the loss functions. It also provides empirical results showing that SDCA performs well in practice, especially for non-smooth losses like the hinge loss. The analysis highlights the importance of randomization in SDCA and shows that it can outperform cyclic dual coordinate ascent in terms of convergence speed. Overall, the paper provides a comprehensive theoretical analysis of SDCA, demonstrating its effectiveness for regularized loss minimization and showing that it can achieve better convergence rates than SGD in certain scenarios.**Summary:** This paper presents a theoretical analysis of Stochastic Dual Coordinate Ascent (SDCA), a stochastic optimization method for regularized loss minimization. The authors show that SDCA has strong theoretical guarantees comparable or better than Stochastic Gradient Descent (SGD), which is widely used for large-scale machine learning problems. SDCA works by optimizing dual variables in a stochastic manner, which allows it to achieve a duality gap that is an upper bound on the primal sub-optimality. The paper analyzes SDCA for two types of loss functions: L-Lipschitz and (1/γ)-smooth. For L-Lipschitz loss functions, SDCA achieves a convergence rate of ˜O(n + L²/(λε)), while for (1/γ)-smooth loss functions, the rate is ˜O((n + 1/(λγ)) log(1/ε)). These results are improved for almost smooth losses, where SDCA converges faster than SGD. The authors also compare SDCA with other methods, including SGD and the Stochastic Average Gradient (SAG) algorithm. They show that SDCA can outperform SGD in certain regimes, especially when high solution accuracy is required. Additionally, they propose a modified version of SGD that can be combined with SDCA to achieve faster convergence. The paper includes detailed proofs for the convergence of SDCA under different assumptions on the loss functions. It also provides empirical results showing that SDCA performs well in practice, especially for non-smooth losses like the hinge loss. The analysis highlights the importance of randomization in SDCA and shows that it can outperform cyclic dual coordinate ascent in terms of convergence speed. Overall, the paper provides a comprehensive theoretical analysis of SDCA, demonstrating its effectiveness for regularized loss minimization and showing that it can achieve better convergence rates than SGD in certain scenarios.