An Optimal Hybrid Variance-Reduced Algorithm for Stochastic Composite Nonconvex Optimization

In this note we propose a new variant of the hybrid variance-reduced proximal gradient method in [7] to solve a common stochastic composite nonconvex optimization problem under standard assumptions. We simply replace the independent unbiased estimator in our hybrid- SARAH estimator introduced in [7] by the stochastic gradient evaluated at the same sample, leading to the identical momentum-SARAH estimator introduced in [2]. This allows us to save one stochastic gradient per iteration compared to [7], and only requires two samples per iteration. Our algorithm is very simple and achieves optimal stochastic oracle complexity bound in terms of stochastic gradient evaluations (up to a constant factor). Our analysis is essentially inspired by [7], but we do not use two different step-sizes.

Asynchronous Federated Learning with Reduced Number of Rounds and with Differential Privacy from Less Aggregated Gaussian Noise

The feasibility of federated learning is highly constrained by the server-clients infrastructure in terms of network communication. Most newly launched smartphones and IoT devices are equipped with GPUs or sufficient computing hardware to run powerful AI models. However, in case of the original synchronous federated learning, client devices suffer waiting times and regular communication between clients and server is required. This implies more sensitivity to local model training times and irregular or missed updates, hence, less or limited scalability to large numbers of clients and convergence rates measured in real time will suffer. We propose a new algorithm for asynchronous federated learning which eliminates waiting times and reduces overall network communication - we provide rigorous theoretical analysis for strongly convex objective functions and provide simulation results. By adding Gaussian noise we show how our algorithm can be made differentially private -- new theorems show how the aggregated added Gaussian noise is significantly reduced.

Hybrid Variance-Reduced SGD Algorithms For Nonconvex-Concave Minimax Problems

We develop a novel variance-reduced algorithm to solve a stochastic nonconvex-concave minimax problem which has various applications in different fields. This problem has several computational challenges due to its nonsmoothness, nonconvexity, nonlinearity, and non-separability of the objective functions. Our approach relies on a novel combination of recent ideas, including smoothing and hybrid stochastic variance-reduced techniques. Our algorithm and its variants can achieve $\mathcal{O}(T^{-2/3})$-convergence rate in $T$, and the best-known oracle complexity under standard assumptions. They have several computational advantages compared to existing methods. They can also work with both single sample or mini-batch on derivative estimators, with constant or diminishing step-sizes. We demonstrate the benefits of our algorithms over existing methods through two numerical examples.

Randomized Primal-Dual Algorithms for Composite Convex Minimization with Faster Convergence Rates

We develop two novel randomized primal-dual algorithms to solve nonsmooth composite convex optimization problems. The first algorithm is fully randomized, i.e., it has randomized updates on both primal and dual variables, while the second one is a semi-randomized scheme which only has one randomized update on the primal (or dual) variable while using the full update for the other. Both algorithms achieve the best-known $\mathcal{O}(1/k)$ or $\mathcal{O}(1/k^2)$ convergence rates in expectation under either only convexity or strong convexity, respectively, where $k$ is the iteration counter. Interestingly, with new parameter update rules, our algorithms can achieve $o(1/k)$ or $o(1/k^2)$ best-iterate convergence rate in expectation under either convexity or strong convexity, respectively. These rates can be obtained for both the primal and dual problems. To the best of our knowledge, this is the first time such faster convergence rates are shown for randomized primal-dual methods. Finally, we verify our theoretical results via two numerical examples and compare them with the state-of-the-art.

A Hybrid Stochastic Policy Gradient Algorithm for Reinforcement Learning

We propose a novel hybrid stochastic policy gradient estimator by combining an unbiased policy gradient estimator, the REINFORCE estimator, with another biased one, an adapted SARAH estimator for policy optimization. The hybrid policy gradient estimator is shown to be biased, but has variance reduced property. Using this estimator, we develop a new Proximal Hybrid Stochastic Policy Gradient Algorithm (ProxHSPGA) to solve a composite policy optimization problem that allows us to handle constraints or regularizers on the policy parameters. We first propose a single-looped algorithm then introduce a more practical restarting variant. We prove that both algorithms can achieve the best-known trajectory complexity $\mathcal{O}\left(\varepsilon^{-3}\right)$ to attain a first-order stationary point for the composite problem which is better than existing REINFORCE/GPOMDP $\mathcal{O}\left(\varepsilon^{-4}\right)$ and SVRPG $\mathcal{O}\left(\varepsilon^{-10/3}\right)$ in the non-composite setting. We evaluate the performance of our algorithm on several well-known examples in reinforcement learning. Numerical results show that our algorithm outperforms two existing methods on these examples. Moreover, the composite settings indeed have some advantages compared to the non-composite ones on certain problems.

A Unified Convergence Analysis for Shuffling-Type Gradient Methods

In this paper, we provide a unified convergence analysis for a class of shuffling-type gradient methods for solving a well-known finite-sum minimization problem commonly used in machine learning. This algorithm covers various variants such as randomized reshuffling, single shuffling, and cyclic/incremental gradient schemes. We consider two different settings: strongly convex and non-convex problems. Our main contribution consists of new non-asymptotic and asymptotic convergence rates for a general class of shuffling-type gradient methods to solve both non-convex and strongly convex problems. While our rate in the non-convex problem is new (i.e. not known yet under standard assumptions), the rate on the strongly convex case matches (up to a constant) the best-known results. However, unlike existing works in this direction, we only use standard assumptions such as smoothness and strong convexity. Finally, we empirically illustrate the effect of learning rates via a non-convex logistic regression and neural network examples.

Stochastic Gauss-Newton Algorithms for Nonconvex Compositional Optimization

We develop two new stochastic Gauss-Newton algorithms for solving a class of stochastic nonconvex compositional optimization problems frequently arising in practice. We consider both the expectation and finite-sum settings under standard assumptions. We use both classical stochastic and SARAH estimators for approximating function values and Jacobians. In the expectation case, we establish $\mathcal{O}(\varepsilon^{-2})$ iteration complexity to achieve a stationary point in expectation and estimate the total number of stochastic oracle calls for both function values and its Jacobian, where $\varepsilon$ is a desired accuracy. In the finite sum case, we also estimate the same iteration complexity and the total oracle calls with high probability. To our best knowledge, this is the first time such global stochastic oracle complexity is established for stochastic Gauss-Newton methods. We illustrate our theoretical results via numerical examples on both synthetic and real datasets.

A Newton Frank-Wolfe Method for Constrained Self-Concordant Minimization

We demonstrate how to scalably solve a class of constrained self-concordant minimization problems using linear minimization oracles (LMO) over the constraint set. We prove that the number of LMO calls of our method is nearly the same as that of the Frank-Wolfe method in the L-smooth case. Specifically, our Newton Frank-Wolfe method uses $\mathcal{O}(\epsilon^{-\nu})$ LMO's, where $\epsilon$ is the desired accuracy and $\nu:= 1 + o(1)$. In addition, we demonstrate how our algorithm can exploit the improved variants of the LMO-based schemes, including away-steps, to attain linear convergence rates. We also provide numerical evidence with portfolio design with the competitive ratio, D-optimal experimental design, and logistic regression with the elastic net where Newton Frank-Wolfe outperforms the state-of-the-art.

Using positive spanning sets to achieve stationarity with the Boosted DC Algorithm

The Difference of Convex function Algorithm (DCA) is widely used for minimizing the difference of two convex functions. A recently proposed accelerated version, termed BDCA for Boosted DC Algorithm, incorporates a line search step to achieve a larger decrease of the objective value at each iteration. Thanks to this step, BDCA usually converges much faster than DCA in practice. The solutions found by DCA are guaranteed to be critical points of the problem, but they are rarely local minima. Although BDCA tends to improve the objective value of the solutions it finds, these are frequently just critical points as well. In this paper we combine BDCA with a simple Derivative-Free Optimization (DFO) algorithm to force the stationarity (lack of descent direction) at the point obtained. The potential of this approach is illustrated through some computational experiments on a Minimum-Sum-of-Squares problem. Our numerical results demonstrate that the new method provides better solutions while still remains faster than DCA in the majority of test cases.

A Hybrid Stochastic Optimization Framework for Stochastic Composite Nonconvex Optimization

In this paper, we introduce a new approach to develop stochastic optimization algorithms for solving stochastic composite and possibly nonconvex optimization problems. The main idea is to combine two stochastic estimators to form a new hybrid one. We first introduce our hybrid estimator and then investigate its fundamental properties to form a foundation theory for algorithmic development. Next, we apply our theory to develop several variants of stochastic gradient methods to solve both expectation and finite-sum composite optimization problems. Our first algorithm can be viewed as a variant of proximal stochastic gradient methods with a single-loop, but can achieve $\mathcal{O}(\sigma^3\varepsilon^{-1} + \sigma\varepsilon^{-3})$ complexity bound that is significantly better than the $\mathcal{O}(\sigma^2\varepsilon^{-4})$-complexity in state-of-the-art stochastic gradient methods, where $\sigma$ is the variance and $\varepsilon$ is a desired accuracy. Then, we consider two different variants of our method: adaptive step-size and double-loop schemes that have the same theoretical guarantees as in our first algorithm. We also study two mini-batch variants and develop two hybrid SARAH-SVRG algorithms to solve the finite-sum problems. In all cases, we achieve the best-known complexity bounds under standard assumptions. We test our methods on several numerical examples with real datasets and compare them with state-of-the-arts. Our numerical experiments show that the new methods are comparable and, in many cases, outperform their competitors.