**Summary:** This paper investigates the statistical properties of classification algorithms that minimize convex surrogate losses of the 0-1 loss function. These algorithms, including support vector machines, boosting, and logistic regression, are computationally efficient due to convexity but introduce statistical trade-offs. The authors derive a general relationship between the risk under 0-1 loss and the risk under a convex surrogate loss function. This relationship provides nontrivial upper bounds on excess risk under the weakest possible condition: classification-calibration, a pointwise form of Fisher consistency. The paper introduces a variational transformation of the loss function that simplifies the computation of this relationship. It also presents a refined version of this result in the context of low noise, showing that minimizing the empirical surrogate risk leads to fast convergence rates under this assumption. The authors demonstrate that classification-calibrated loss functions, such as exponential, hinge, and sigmoid losses, satisfy the necessary conditions for this relationship to hold. The paper also explores the properties of the ψ-transform, a function derived from the surrogate loss, which quantifies the relationship between excess risk and excess surrogate risk. The ψ-transform is shown to be nonnegative and continuous, and its properties are used to establish bounds on the statistical error of classification algorithms. The results are applied to the estimation of convergence rates for function classes that are scaled convex hulls of a finite-dimensional base class, with a variety of commonly used loss functions. The paper concludes that minimizing the surrogate risk can provide a reasonable approximation of minimizing the 0-1 risk, especially under conditions of low noise and classification-calibration. This provides a theoretical foundation for the use of convex surrogate losses in statistical learning and classification.**Summary:** This paper investigates the statistical properties of classification algorithms that minimize convex surrogate losses of the 0-1 loss function. These algorithms, including support vector machines, boosting, and logistic regression, are computationally efficient due to convexity but introduce statistical trade-offs. The authors derive a general relationship between the risk under 0-1 loss and the risk under a convex surrogate loss function. This relationship provides nontrivial upper bounds on excess risk under the weakest possible condition: classification-calibration, a pointwise form of Fisher consistency. The paper introduces a variational transformation of the loss function that simplifies the computation of this relationship. It also presents a refined version of this result in the context of low noise, showing that minimizing the empirical surrogate risk leads to fast convergence rates under this assumption. The authors demonstrate that classification-calibrated loss functions, such as exponential, hinge, and sigmoid losses, satisfy the necessary conditions for this relationship to hold. The paper also explores the properties of the ψ-transform, a function derived from the surrogate loss, which quantifies the relationship between excess risk and excess surrogate risk. The ψ-transform is shown to be nonnegative and continuous, and its properties are used to establish bounds on the statistical error of classification algorithms. The results are applied to the estimation of convergence rates for function classes that are scaled convex hulls of a finite-dimensional base class, with a variety of commonly used loss functions. The paper concludes that minimizing the surrogate risk can provide a reasonable approximation of minimizing the 0-1 risk, especially under conditions of low noise and classification-calibration. This provides a theoretical foundation for the use of convex surrogate losses in statistical learning and classification.