Stochastic bilevel optimization generalizes the classic stochastic optimization from the minimization of a single objective to the minimization of an objective function that depends on the solution of another optimization problem. Recently, bilevel optimization is regaining popularity in emerging machine learning applications such as hyper-parameter optimization and model-agnostic meta learning. To solve this class of optimization problems, existing methods require either double-loop or two-timescale updates, which are sometimes less efficient. This paper develops a new optimization method for a class of stochastic bilevel problems that we term Single-Timescale stochAstic BiLevEl optimization (STABLE) method. STABLE runs in a single loop fashion, and uses a single-timescale update with a fixed batch size. To achieve an $\epsilon$-stationary point of the bilevel problem, STABLE requires ${\cal O}(\epsilon^{-2})$ samples in total; and to achieve an $\epsilon$-optimal solution in the strongly convex case, STABLE requires ${\cal O}(\epsilon^{-1})$ samples. To the best of our knowledge, when STABLE was proposed, it is the first bilevel optimization algorithm achieving the same order of sample complexity as SGD for single-level stochastic optimization.

Stochastic gradient descent (SGD) has taken the stage as the primary workhorse for largescale machine learning. It is often used with its adaptive variants such as AdaGrad, Adam, and AMSGrad. This paper proposes an adaptive stochastic gradient descent method for distributed machine learning, which can be viewed as the communicationadaptive counterpart of the celebrated Adam method — justifying its name CADA. The key components of CADA are a set of new rules tailored for adaptive stochastic gradients that can be implemented to save communication upload. The new algorithms adaptively reuse the stale Adam gradients, thus saving communication, and still have convergence rates comparable to original Adam. In numerical experiments, CADA achieves impressive empirical performance in terms of total communication round reduction.

Stochastic compositional optimization generalizes classic (non-compositional) stochastic optimization to the minimization of com-positions of functions. Each composition may introduce an additional expectation. The series of expectations may be nested. Stochastic compositional optimization is gaining popularity in applications such as meta learning. This paper presents a new Stochastically Corrected Stochastic Compositional gradient method (SCSC). SCSC runs in a single-time scale with a single loop, uses a fixed batch size, and guarantees to converge at the same rate as the stochastic gradient descent (SGD) method for non-compositional stochastic optimization. It is easy to apply SGD-improvement techniques to accelerate SCSC. This helps SCSC achieve state-of-the-art performance for stochastic compositional optimization. In particular, we apply Adam to SCSC, and the exhibited rate of convergence matches that of the original Adam on non-compositional optimization. We test SCSC using the model-agnostic meta-learning tasks.

Stochastic nested optimization, including stochastic compositional, min-max, and bilevel optimization, is gaining popularity in many machine learning applications. While the three problems share a nested structure, existing works often treat them separately, thus developing problem-specific algorithms and analyses. Among various exciting developments, simple SGD-type updates (potentially on multiple variables) are still prevalent in solving this class of nested problems, but they are believed to have a slower convergence rate than non-nested problems. This paper unifies several SGD-type updates for stochastic nested problems into a single SGD approach that we term ALternating Stochastic gradient dEscenT (ALSET) method. By leveraging the hidden smoothness of the problem, this paper presents a tighter analysis of ALSET for stochastic nested problems. Under the new analysis, to achieve an $\epsilon$-stationary point of the nested problem, it requires ${\cal O}(\epsilon^{-2})$ samples in total. Under certain regularity conditions, applying our results to stochastic compositional, min-max, and reinforcement learning problems either improves or matches the best-known sample complexity in the respective cases. Our results explain why simple SGD-type algorithms in stochastic nested problems all work very well in practice without the need for further modifications.

The incremental aggregated gradient algorithm is popular in network optimization and machine learning research. However, the current convergence results require the objective function to be strongly convex. And the existing convergence rates are also limited to linear convergence. Due to the mathematical techniques, the stepsize in the algorithm is restricted by the strongly convex constant, which may make the stepsize be very small (the strongly convex constant may be small). In this paper, we propose a general proximal incremental aggregated gradient algorithm, which contains various existing algorithms including the basic incremental aggregated gradient method. Better and new convergence results are proved even with the general scheme. The novel results presented in this paper, which have not appeared in previous literature, include: a general scheme, nonconvex analysis, the sublinear convergence rates of the function values, much larger stepsizes that guarantee the convergence, the convergence when noise exists, the line search strategy of the proximal incremental aggregated gradient algorithm and its convergence.

Stochastic gradient methods are the workhorse (algorithms) of large-scale optimization problems in machine learning, signal processing, and other computational sciences and engineering. This paper studies Markov chain gradient descent, a variant of stochastic gradient descent where the random samples are taken on the trajectory of a Markov chain. Existing results of this method assume convex objectives and a reversible Markov chain and thus have their limitations. We establish new non-ergodic convergence under wider step sizes, for nonconvex problems, and for non-reversible finite-state Markov chains. Nonconvexity makes our method applicable to broader problem classes. Non-reversible finite-state Markov chains, on the other hand, can mix substatially faster. To obtain these results, we introduce a new technique that varies the mixing levels of the Markov chains. The reported numerical results validate our contributions.