The stochastic compositional optimization (SCO) is popular in many real-world applications, including risk management, reinforcement learning, and meta-learning. However, most of the previous methods for SCO require the smoothness assumption on both the outer and inner functions, which limits their applications to a wider range of problems. In this paper, we study the SCO problem in that both the outer and inner functions are Lipschitz continuous but possibly nonconvex and nonsmooth. In particular, we propose gradient-free stochastic methods for finding the $(\delta, \epsilon)$-Goldstein stationary points of such problems with non-asymptotic convergence rates. Our results also lead to an improved convergence rate for the convex nonsmooth SCO problem. Furthermore, we conduct numerical experiments to demonstrate the effectiveness of the proposed methods.

Multi-epoch, small-batch, Stochastic Gradient Descent (SGD) has been the method of choice for learning with large over-parameterized models. A popular theory for explaining why SGD works well in practice is that the algorithm has an implicit regularization that biases its output towards a good solution. Perhaps the theoretically most well understood learning setting for SGD is that of Stochastic Convex Optimization (SCO), where it is well known that SGD learns at a rate of $O(1/\sqrt{n})$, where $n$ is the number of samples. In this paper, we consider the problem of SCO and explore the role of implicit regularization, batch size and multiple epochs for SGD. Our main contributions are threefold: * We show that for any regularizer, there is an SCO problem for which Regularized Empirical Risk Minimzation fails to learn.

Our main contributions are threefold: 1. We show that for any regularizer, there is an SCO problem for which Regularized Empirical Risk Minimzation fails to learn.

The stochastic compositional optimization (SCO) is popular in many real-world applications, including risk management, reinforcement learning, and meta-learning. However, most of the previous methods for SCO require the smoothness assumption on both the outer and inner functions, which limits their applications to a wider range of problems. In this paper, we study the SCO problem in that both the outer and inner functions are Lipschitz continuous but possibly nonconvex and nonsmooth. In particular, we propose gradient-free stochastic methods for finding the (\delta, \epsilon) -Goldstein stationary points of such problems with non-asymptotic convergence rates. Our results also lead to an improved convergence rate for the convex nonsmooth SCO problem. Furthermore, we conduct numerical experiments to demonstrate the effectiveness of the proposed methods.

Multi-epoch, small-batch, Stochastic Gradient Descent (SGD) has been the method of choice for learning with large over-parameterized models. A popular theory for explaining why SGD works well in practice is that the algorithm has an implicit regularization that biases its output towards a good solution. Perhaps the theoretically most well understood learning setting for SGD is that of Stochastic Convex Optimization (SCO), where it is well known that SGD learns at a rate of O(1/\sqrt{n}), where n is the number of samples. In this paper, we consider the problem of SCO and explore the role of implicit regularization, batch size and multiple epochs for SGD. Our main contributions are threefold: * We show that for any regularizer, there is an SCO problem for which Regularized Empirical Risk Minimzation fails to learn.

Simulating a single trajectory of a dynamical system under some state-dependent policy is a core bottleneck in policy optimization algorithms. The many inherently serial policy evaluations that must be performed in a single simulation constitute the bulk of this bottleneck. To wit, in applying policy optimization to supply chain optimization (SCO) problems, simulating a single month of a supply chain can take several hours. We present an iterative algorithm for policy simulation, which we dub Picard Iteration. This scheme carefully assigns policy evaluation tasks to independent processes. Within an iteration, a single process evaluates the policy only on its assigned tasks while assuming a certain 'cached' evaluation for other tasks; the cache is updated at the end of the iteration. Implemented on GPUs, this scheme admits batched evaluation of the policy on a single trajectory. We prove that the structure afforded by many SCO problems allows convergence in a small number of iterations, independent of the horizon. We demonstrate practical speedups of 400x on large-scale SCO problems even with a single GPU, and also demonstrate practical efficacy in other RL environments.

Applying iterative solvers on sparsity-constrained optimization (SCO) requires tedious mathematical deduction and careful programming/debugging that hinders these solvers' broad impact. In the paper, the library skscope is introduced to overcome such an obstacle. With skscope, users can solve the SCO by just programming the objective function. The convenience of skscope is demonstrated through two examples in the paper, where sparse linear regression and trend filtering are addressed with just four lines of code. More importantly, skscope's efficient implementation allows state-of-the-art solvers to quickly attain the sparse solution regardless of the high dimensionality of parameter space. Numerical experiments reveal the available solvers in skscope can achieve up to 80x speedup on the competing relaxation solutions obtained via the benchmarked convex solver.

In this work, we investigate the interplay between memorization and learning in the context of \emph{stochastic convex optimization} (SCO). We define memorization via the information a learning algorithm reveals about its training data points. We then quantify this information using the framework of conditional mutual information (CMI) proposed by Steinke and Zakynthinou (2020). Our main result is a precise characterization of the tradeoff between the accuracy of a learning algorithm and its CMI, answering an open question posed by Livni (2023). We show that, in the $L^2$ Lipschitz--bounded setting and under strong convexity, every learner with an excess error $\varepsilon$ has CMI bounded below by $\Omega(1/\varepsilon^2)$ and $\Omega(1/\varepsilon)$, respectively. We further demonstrate the essential role of memorization in learning problems in SCO by designing an adversary capable of accurately identifying a significant fraction of the training samples in specific SCO problems. Finally, we enumerate several implications of our results, such as a limitation of generalization bounds based on CMI and the incompressibility of samples in SCO problems.