Gradient descent and its variants are de facto standard algorithms for training machine learning models. As gradient descent is sensitive to its hyperparame-ters, we need to tune the hyperparameters carefully using a grid search.

The recently proposed stochastic Polyak stepsize (SPS) and stochastic line-search (SLS) for SGD have shown remarkable effectiveness when training over-parameterized models. However, two issues remain unsolved in this line of work.

The recently proposed stochastic Polyak stepsize (SPS) and stochastic line-search (SLS) for SGD have shown remarkable effectiveness when training over-parameterized models. However, two issues remain unsolved in this line of work. First, in non-interpolation settings, both algorithms only guarantee convergence to a neighborhood of a solution which may result in a worse output than the initial guess. While artificially decreasing the adaptive stepsize has been proposed to address this issue (Orvieto et al.), this approach results in slower convergence rates under interpolation. Second, intuitive line-search methods equipped with variance-reduction (VR) fail to converge (Dubois-Taine et al.).

Gradient descent and its variants are de facto standard algorithms for training machine learning models. As gradient descent is sensitive to its hyperparame-ters, we need to tune the hyperparameters carefully using a grid search.

We give exact expressions for the risk and learning rate curves in terms of a deterministic solution to a system of ODEs.

Composite optimization problems involve minimizing the composition of a smooth map with a convex function. Such objectives arise in numerous data science and signal processing applications, including phase retrieval, blind deconvolution, and collaborative filtering. The subgradient method achieves local linear convergence when the composite loss is well-conditioned. However, if the smooth map is, in a certain sense, ill-conditioned or overparameterized, the subgradient method exhibits much slower sublinear convergence even when the convex function is well-conditioned. To overcome this limitation, we introduce a Levenberg-Morrison-Marquardt subgradient method that converges linearly under mild regularity conditions at a rate determined solely by the convex function. Further, we demonstrate that these regularity conditions hold for several problems of practical interest, including square-variable formulations, matrix sensing, and tensor factorization. Numerical experiments illustrate the benefits of our method.

The recently proposed stochastic Polyak stepsize (SPS) and stochastic line-search (SLS) for SGD have shown remarkable effectiveness when training over-parameterized models. However, two issues remain unsolved in this line of work. First, in non-interpolation settings, both algorithms only guarantee convergence to a neighborhood of a solution which may result in a worse output than the initial guess. While artificially decreasing the adaptive stepsize has been proposed to address this issue (Orvieto et al.), this approach results in slower convergence rates under interpolation. Second, intuitive line-search methods equipped with variance-reduction (VR) fail to converge (Dubois-Taine et al.).