We develop a variant of the stochastic prox-linear method for minimizing the Conditional Value-at-Risk (CVaR) objective. CVaR is a risk measure focused on minimizing worst-case performance, defined as the average of the top quantile of the losses. In machine learning, such a risk measure is useful to train more robust models. Although the stochastic subgradient method (SGM) is a natural choice for minimizing the CVaR objective, we show that our stochastic prox-linear (SPL+) algorithm can better exploit the structure of the objective, while still providing a convenient closed form update. Our SPL+ method also adapts to the scaling of the loss function, which allows for easier tuning. We then specialize a general convergence theorem for SPL+ to our setting, and show that it allows for a wider selection of step sizes compared to SGM. We support this theoretical finding experimentally.

Adaptive gradient methods are typically used for training over-parameterized models capable of exactly fitting the data; we thus study their convergence in this interpolation setting. Under an interpolation assumption, we prove that AMSGrad with a constant step-size and momentum can converge to the minimizer at the faster $O(1/T)$ rate for smooth, convex functions. Furthermore, in this setting, we show that AdaGrad can achieve an $O(1)$ regret in the online convex optimization framework. When interpolation is only approximately satisfied, we show that constant step-size AMSGrad converges to a neighbourhood of the solution. On the other hand, we prove that AdaGrad is robust to the violation of interpolation and converges to the minimizer at the optimal rate. However, we demonstrate that even for simple, convex problems satisfying interpolation, the empirical performance of these methods heavily depends on the step-size and requires tuning. We alleviate this problem by using stochastic line-search (SLS) and Polyak's step-sizes (SPS) to help these methods adapt to the function's local smoothness. By using these techniques, we prove that AdaGrad and AMSGrad do not require knowledge of problem-dependent constants and retain the convergence guarantees of their constant step-size counterparts. Experimentally, we show that these techniques help improve the convergence and generalization performance across tasks, from binary classification with kernel mappings to classification with deep neural networks.

We consider stochastic second order methods for minimizing strongly-convex functions under an interpolation condition satisfied by over-parameterized models. Under this condition, we show that the regularized sub-sampled Newton method (R-SSN) achieves global linear convergence with an adaptive step size and a constant batch size. By growing the batch size for both the sub-sampled gradient and Hessian, we show that R-SSN can converge at a quadratic rate in a local neighbourhood of the solution. We also show that R-SSN attains local linear convergence for the family of self-concordant functions. Furthermore, we analyse stochastic BFGS algorithms in the interpolation setting and prove their global linear convergence. We empirically evaluate stochastic L-BFGS and a "Hessian-free" implementation of R-SSN for binary classification on synthetic, linearly-separable datasets and consider real medium-size datasets under a kernel mapping. Our experimental results show the fast convergence of these methods both in terms of the number of iterations and wall-clock time.