Finite-Time Analysis of Stochastic Gradient Descent under Markov Randomness

Motivated by broad applications in reinforcement learning and machine learning, this paper considers the popular stochastic gradient descent (SGD) when the gradients of the underlying objective function are sampled from Markov processes. This Markov sampling leads to the gradient samples being biased and not independent. The existing results for the convergence of SGD under Markov randomness are often established under the assumptions on the boundedness of either the iterates or the gradient samples. Our main focus is to study the finite-time convergence of SGD for different types of objective functions, without requiring these assumptions. We show that SGD converges nearly at the same rate with Markovian gradient samples as with independent gradient samples. The only difference is a logarithmic factor that accounts for the mixing time of the Markov chain.

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.

BUZz: BUffer Zones for defending adversarial examples in image classification

We propose a novel defense against all existing gradient based adversarial attacks on deep neural networks for image classification problems. Our defense is based on a combination of deep neural networks and simple image transformations. While straight forward in implementation, this defense yields a unique security property which we term buffer zones. In this paper, we formalize the concept of buffer zones. We argue that our defense based on buffer zones is secure against state-of-the-art black box attacks. We are able to achieve this security even when the adversary has access to the {\em entire} original training data set and unlimited query access to the defense. We verify our security claims through experimentation using FashionMNIST, CIFAR-10 and CIFAR-100. We demonstrate $<10\%$ attack success rate -- significantly lower than what other well-known defenses offer -- at only a price of a 15-20\% drop in clean accuracy. By using a new intuitive metric we explain why this trade-off offers a significant improvement over prior work.

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.

Hybrid Stochastic Gradient Descent Algorithms for Stochastic Nonconvex Optimization

We introduce a hybrid stochastic estimator to design stochastic gradient algorithms for solving stochastic optimization problems. Such a hybrid estimator is a convex combination of two existing biased and unbiased estimators and leads to some useful property on its variance. We limit our consideration to a hybrid SARAH-SGD for nonconvex expectation problems. However, our idea can be extended to handle a broader class of estimators in both convex and nonconvex settings. We propose a new single-loop stochastic gradient descent algorithm that can achieve $O(\max\{\sigma^3\varepsilon^{-1},\sigma\varepsilon^{-3}\})$-complexity bound to obtain an $\varepsilon$-stationary point under smoothness and $\sigma^2$-bounded variance assumptions. This complexity is better than $O(\sigma^2\varepsilon^{-4})$ often obtained in state-of-the-art SGDs when $\sigma < O(\varepsilon^{-3})$. We also consider different extensions of our method, including constant and adaptive step-size with single-loop, double-loop, and mini-batch variants. We compare our algorithms with existing methods on several datasets using two nonconvex models.

ProxSARAH: An Efficient Algorithmic Framework for Stochastic Composite Nonconvex Optimization

We propose a new stochastic first-order algorithmic framework to solve stochastic composite nonconvex optimization problems that covers both finite-sum and expectation settings. Our algorithms rely on the SARAH estimator introduced in (Nguyen et al, 2017) and consist of two steps: a proximal gradient and an averaging step making them different from existing nonconvex proximal-type algorithms. The algorithms only require an average smoothness assumption of the nonconvex objective term and additional bounded variance assumption if applied to expectation problems. They work with both constant and adaptive step-sizes, while allowing single sample and mini-batches. In all these cases, we prove that our algorithms can achieve the best-known complexity bounds. One key step of our methods is new constant and adaptive step-sizes that help to achieve desired complexity bounds while improving practical performance. Our constant step-size is much larger than existing methods including proximal SVRG schemes in the single sample case. We also specify the algorithm to the non-composite case that covers existing state-of-the-arts in terms of complexity bounds. Our update also allows one to trade-off between step-sizes and mini-batch sizes to improve performance. We test the proposed algorithms on two composite nonconvex problems and neural networks using several well-known datasets.

DTN: A Learning Rate Scheme with Convergence Rate of $\mathcal{O}(1/t)$ for SGD

We propose a novel diminishing learning rate scheme, coined Decreasing-Trend-Nature (DTN), which allows us to prove fast convergence of the Stochastic Gradient Descent (SGD) algorithm to a first-order stationary point for smooth general convex and some class of nonconvex including neural network applications for classification problems. We are the first to prove that SGD with diminishing learning rate achieves a convergence rate of $\mathcal{O}(1/t)$ for these problems. Our theory applies to neural network applications for classification problems in a straightforward way.