arXiv subjects

Lam M. Nguyen

Publications and source records attributed to Lam M. Nguyen.

At least 55 records · Page 3Linked to original sources

arXiv2021

SMG: A Shuffling Gradient-Based Method with Momentum

We combine two advanced ideas widely used in optimization for machine learning: shuffling strategy and momentum technique to develop a novel shuffling gradient-based method with momentum, coined Shuffling Momentum Gradient (SMG), for non-convex finite-sum optimization problems. While our method is inspired by momentum techniques, its update is fundamentally different from existing momentum-based methods. We establish state-of-the-art convergence rates of SMG for any shuffling strategy using either constant or diminishing learning rate under standard assumptions (i.e.$L$-smoothness and bounded variance). When the shuffling strategy is fixed, we develop another new algorithm that is similar to existing momentum methods, and prove the same convergence rates for this algorithm under the $L$-smoothness and bounded gradient assumptions. We demonstrate our algorithms via numerical simulations on standard datasets and compare them with existing shuffling methods. Our tests have shown encouraging performance of the new algorithms.

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arXiv2021

Hogwild! over Distributed Local Data Sets with Linearly Increasing Mini-Batch Sizes

Hogwild! implements asynchronous Stochastic Gradient Descent (SGD) where multiple threads in parallel access a common repository containing training data, perform SGD iterations and update shared state that represents a jointly learned (global) model. We consider big data analysis where training data is distributed among local data sets in a heterogeneous way -- and we wish to move SGD computations to local compute nodes where local data resides. The results of these local SGD computations are aggregated by a central "aggregator" which mimics Hogwild!. We show how local compute nodes can start choosing small mini-batch sizes which increase to larger ones in order to reduce communication cost (round interaction with the aggregator). We improve state-of-the-art literature and show $O(\sqrt{K}$) communication rounds for heterogeneous data for strongly convex problems, where $K$ is the total number of gradient computations across all local compute nodes. For our scheme, we prove a \textit{tight} and novel non-trivial convergence analysis for strongly convex problems for {\em heterogeneous} data which does not use the bounded gradient assumption as seen in many existing publications. The tightness is a consequence of our proofs for lower and upper bounds of the convergence rate, which show a constant factor difference. We show experimental results for plain convex and non-convex problems for biased (i.e., heterogeneous) and unbiased local data sets.

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arXiv2020

A Scalable MIP-based Method for Learning Optimal Multivariate Decision Trees

Several recent publications report advances in training optimal decision trees (ODT) using mixed-integer programs (MIP), due to algorithmic advances in integer programming and a growing interest in addressing the inherent suboptimality of heuristic approaches such as CART. In this paper, we propose a novel MIP formulation, based on a 1-norm support vector machine model, to train a multivariate ODT for classification problems. We provide cutting plane techniques that tighten the linear relaxation of the MIP formulation, in order to improve run times to reach optimality. Using 36 data-sets from the University of California Irvine Machine Learning Repository, we demonstrate that our formulation outperforms its counterparts in the literature by an average of about 10% in terms of mean out-of-sample testing accuracy across the data-sets. We provide a scalable framework to train multivariate ODT on large data-sets by introducing a novel linear programming (LP) based data selection method to choose a subset of the data for training. Our method is able to routinely handle large data-sets with more than 7,000 sample points and outperform heuristics methods and other MIP based techniques. We present results on data-sets containing up to 245,000 samples. Existing MIP-based methods do not scale well on training data-sets beyond 5,500 samples.

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arXiv2020

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

We develop a novel and single-loop variance-reduced algorithm to solve a class of stochastic nonconvex-convex minimax problems involving a nonconvex-linear objective function, which has various applications in different fields such as machine learning and robust optimization. This problem class has several computational challenges due to its nonsmoothness, nonconvexity, nonlinearity, and non-separability of the objective functions. Our approach relies on a new combination of recent ideas, including smoothing and hybrid biased variance-reduced techniques. Our algorithm and its variants can achieve $\mathcal{O}(T^{-2/3})$-convergence rate and the best known oracle complexity under standard assumptions, where $T$ is the iteration counter. They have several computational advantages compared to existing methods such as simple to implement and less parameter tuning requirements. They can also work with both single sample or mini-batch on derivative estimators, and with constant or diminishing step-sizes. We demonstrate the benefits of our algorithms over existing methods through two numerical examples, including a nonsmooth and nonconvex-non-strongly concave minimax model.

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arXiv2020

Convergence Rates of Accelerated Markov Gradient Descent with Applications in Reinforcement Learning

Motivated by broad applications in machine learning, we study the popular accelerated stochastic gradient descent (ASGD) algorithm for solving (possibly nonconvex) optimization problems. We characterize the finite-time performance of this method when the gradients are sampled from Markov processes, and hence biased and dependent from time step to time step; in contrast, the analysis in existing work relies heavily on the stochastic gradients being independent and sometimes unbiased. Our main contributions show that under certain (standard) assumptions on the underlying Markov chain generating the gradients, ASGD converges at the nearly 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. One of the key motivations for this study are complicated control problems that can be modeled by a Markov decision process and solved using reinforcement learning. We apply the accelerated method to several challenging problems in the OpenAI Gym and Mujoco, and show that acceleration can significantly improve the performance of the classic temporal difference learning and REINFORCE algorithms.

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arXiv2020

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.

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arXiv2020

Inexact SARAH Algorithm for Stochastic Optimization

We develop and analyze a variant of the SARAH algorithm, which does not require computation of the exact gradient. Thus this new method can be applied to general expectation minimization problems rather than only finite sum problems. While the original SARAH algorithm, as well as its predecessor, SVRG, require an exact gradient computation on each outer iteration, the inexact variant of SARAH (iSARAH), which we develop here, requires only stochastic gradient computed on a mini-batch of sufficient size. The proposed method combines variance reduction via sample size selection and iterative stochastic gradient updates. We analyze the convergence rate of the algorithms for strongly convex and non-strongly convex cases, under smooth assumption with appropriate mini-batch size selected for each case. We show that with an additional, reasonable, assumption iSARAH achieves the best known complexity among stochastic methods in the case of non-strongly convex stochastic functions.

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arXiv2020

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.

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arXiv2020

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.

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arXiv2020

Stochastic Gauss-Newton Algorithms for Nonconvex Compositional Optimization

We develop two new stochastic Gauss-Newton algorithms for solving a class of non-convex stochastic compositional optimization problems frequently arising in practice. We consider both the expectation and finite-sum settings under standard assumptions, and 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 value and its Jacobian, where $\varepsilon$ is a desired accuracy. In the finite sum case, we also estimate $\mathcal{O}(\varepsilon^{-2})$ 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. Finally, we illustrate our theoretical results via two numerical examples on both synthetic and real datasets.

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arXiv2020

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 straightforward in implementation, this defense yields a unique security property which we term buffer zones. We argue that our defense based on buffer zones offers significant improvements over state-of-the-art defenses. We are able to achieve this improvement even when the adversary has access to the {\em entire} original training data set and unlimited query access to the defense. We verify our claim through experimentation using Fashion-MNIST and CIFAR-10: We demonstrate $<11\%$ attack success rate -- significantly lower than what other well-known state-of-the-art defenses offer -- at only a price of a $11-18\%$ drop in clean accuracy. By using a new intuitive metric, we explain why this trade-off offers a significant improvement over prior work.

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arXiv2020

A Hybrid Stochastic Optimization Framework for Stochastic Composite Nonconvex Optimization

We introduce a new approach to develop stochastic optimization algorithms for a class of stochastic composite and possibly nonconvex optimization problems. The main idea is to combine two stochastic estimators to create a new hybrid one. We first introduce our hybrid estimator and then investigate its fundamental properties to form a foundational 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 $\mathcal{O}(σ^3\varepsilon^{-1} + σ\varepsilon^{-3})$-oracle complexity bound, matching the best-known ones from state-of-the-art double-loop algorithms in the literature, where $σ> 0$ is the variance and $\varepsilon$ is a desired accuracy. Then, we consider two different variants of our method: adaptive step-size and restarting schemes that have similar theoretical guarantees as in our first algorithm. We also study two mini-batch variants of the proposed methods. 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.

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arXiv2020

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.

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arXiv2019

Tight Dimension Independent Lower Bound on the Expected Convergence Rate for Diminishing Step Sizes in SGD

We study the convergence of Stochastic Gradient Descent (SGD) for strongly convex objective functions. We prove for all $t$ a lower bound on the expected convergence rate after the $t$-th SGD iteration; the lower bound is over all possible sequences of diminishing step sizes. It implies that recently proposed sequences of step sizes at ICML 2018 and ICML 2019 are {\em universally} close to optimal in that the expected convergence rate after {\em each} iteration is within a factor $32$ of our lower bound. This factor is independent of dimension $d$. We offer a framework for comparing with lower bounds in state-of-the-art literature and when applied to SGD for strongly convex objective functions our lower bound is a significant factor $775\cdot d$ larger compared to existing work.

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arXiv2019

New Convergence Aspects of Stochastic Gradient Algorithms

The classical convergence analysis of SGD is carried out under the assumption that the norm of the stochastic gradient is uniformly bounded. While this might hold for some loss functions, it is violated for cases where the objective function is strongly convex. In Bottou et al. (2018), a new analysis of convergence of SGD is performed under the assumption that stochastic gradients are bounded with respect to the true gradient norm. We show that for stochastic problems arising in machine learning such bound always holds; and we also propose an alternative convergence analysis of SGD with diminishing learning rate regime. We then move on to the asynchronous parallel setting, and prove convergence of Hogwild! algorithm in the same regime in the case of diminished learning rate. It is well-known that SGD converges if a sequence of learning rates $\{η_t\}$ satisfies $\sum_{t=0}^\infty η_t \rightarrow \infty$ and $\sum_{t=0}^\infty η^2_t < \infty$. We show the convergence of SGD for strongly convex objective function without using bounded gradient assumption when $\{η_t\}$ is a diminishing sequence and $\sum_{t=0}^\infty η_t \rightarrow \infty$. In other words, we extend the current state-of-the-art class of learning rates satisfying the convergence of SGD.

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arXiv2019

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 $O(\max\{σ^3\varepsilon^{-1},σ\varepsilon^{-3}\})$-complexity bound to obtain an $\varepsilon$-stationary point under smoothness and $σ^2$-bounded variance assumptions. This complexity is better than $O(σ^2\varepsilon^{-4})$ often obtained in state-of-the-art SGDs when $σ< O(\varepsilon^{-3})$. 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.

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arXiv2019

Characterization of Convex Objective Functions and Optimal Expected Convergence Rates for SGD

We study Stochastic Gradient Descent (SGD) with diminishing step sizes for convex objective functions. We introduce a definitional framework and theory that defines and characterizes a core property, called curvature, of convex objective functions. In terms of curvature we can derive a new inequality that can be used to compute an optimal sequence of diminishing step sizes by solving a differential equation. Our exact solutions confirm known results in literature and allows us to fully characterize a new regularizer with its corresponding expected convergence rates.

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arXiv2019

Finite-Sum Smooth Optimization with SARAH

The total complexity (measured as the total number of gradient computations) of a stochastic first-order optimization algorithm that finds a first-order stationary point of a finite-sum smooth nonconvex objective function $F(w)=\frac{1}{n} \sum_{i=1}^n f_i(w)$ has been proven to be at least $Ω(\sqrt{n}/ε)$ for $n \leq \mathcal{O}(ε^{-2})$ where $ε$ denotes the attained accuracy $\mathbb{E}[ \|\nabla F(\tilde{w})\|^2] \leq ε$ for the outputted approximation $\tilde{w}$ (Fang et al., 2018). In this paper, we provide a convergence analysis for a slightly modified version of the SARAH algorithm (Nguyen et al., 2017a;b) and achieve total complexity that matches the lower-bound worst case complexity in (Fang et al., 2018) up to a constant factor when $n \leq \mathcal{O}(ε^{-2})$ for nonconvex problems. For convex optimization, we propose SARAH++ with sublinear convergence for general convex and linear convergence for strongly convex problems; and we provide a practical version for which numerical experiments on various datasets show an improved performance.

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