arXiv subjects

Quanquan Gu

Publications and source records attributed to Quanquan Gu.

At least 181 records · Page 10Linked to original sources

arXiv2021

Do Wider Neural Networks Really Help Adversarial Robustness?

Adversarial training is a powerful type of defense against adversarial examples. Previous empirical results suggest that adversarial training requires wider networks for better performances. However, it remains elusive how neural network width affects model robustness. In this paper, we carefully examine the relationship between network width and model robustness. Specifically, we show that the model robustness is closely related to the tradeoff between natural accuracy and perturbation stability, which is controlled by the robust regularization parameter $λ$. With the same $λ$, wider networks can achieve better natural accuracy but worse perturbation stability, leading to a potentially worse overall model robustness. To understand the origin of this phenomenon, we further relate the perturbation stability with the network's local Lipschitzness. By leveraging recent results on neural tangent kernels, we theoretically show that wider networks tend to have worse perturbation stability. Our analyses suggest that: 1) the common strategy of first fine-tuning $λ$ on small networks and then directly use it for wide model training could lead to deteriorated model robustness; 2) one needs to properly enlarge $λ$ to unleash the robustness potential of wider models fully. Finally, we propose a new Width Adjusted Regularization (WAR) method that adaptively enlarges $λ$ on wide models and significantly saves the tuning time.

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arXiv2021

Sample Efficient Policy Gradient Methods with Recursive Variance Reduction

Improving the sample efficiency in reinforcement learning has been a long-standing research problem. In this work, we aim to reduce the sample complexity of existing policy gradient methods. We propose a novel policy gradient algorithm called SRVR-PG, which only requires $O(1/ε^{3/2})$ episodes to find an $ε$-approximate stationary point of the nonconcave performance function $J(\boldsymbolθ)$ (i.e., $\boldsymbolθ$ such that $\|\nabla J(\boldsymbolθ)\|_2^2\leqε$). This sample complexity improves the existing result $O(1/ε^{5/3})$ for stochastic variance reduced policy gradient algorithms by a factor of $O(1/ε^{1/6})$. In addition, we also propose a variant of SRVR-PG with parameter exploration, which explores the initial policy parameter from a prior probability distribution. We conduct numerical experiments on classic control problems in reinforcement learning to validate the performance of our proposed algorithms.

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arXiv2021

Provable Robustness of Adversarial Training for Learning Halfspaces with Noise

We analyze the properties of adversarial training for learning adversarially robust halfspaces in the presence of agnostic label noise. Denoting $\mathsf{OPT}_{p,r}$ as the best robust classification error achieved by a halfspace that is robust to perturbations of $\ell_{p}$ balls of radius $r$, we show that adversarial training on the standard binary cross-entropy loss yields adversarially robust halfspaces up to (robust) classification error $\tilde O(\sqrt{\mathsf{OPT}_{2,r}})$ for $p=2$, and $\tilde O(d^{1/4} \sqrt{\mathsf{OPT}_{\infty, r}} + d^{1/2} \mathsf{OPT}_{\infty,r})$ when $p=\infty$. Our results hold for distributions satisfying anti-concentration properties enjoyed by log-concave isotropic distributions among others. We additionally show that if one instead uses a nonconvex sigmoidal loss, adversarial training yields halfspaces with an improved robust classification error of $O(\mathsf{OPT}_{2,r})$ for $p=2$, and $O(d^{1/4}\mathsf{OPT}_{\infty, r})$ when $p=\infty$. To the best of our knowledge, this is the first work to show that adversarial training provably yields robust classifiers in the presence of noise.

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arXiv2021

Direction Matters: On the Implicit Bias of Stochastic Gradient Descent with Moderate Learning Rate

Understanding the algorithmic bias of \emph{stochastic gradient descent} (SGD) is one of the key challenges in modern machine learning and deep learning theory. Most of the existing works, however, focus on \emph{very small or even infinitesimal} learning rate regime, and fail to cover practical scenarios where the learning rate is \emph{moderate and annealing}. In this paper, we make an initial attempt to characterize the particular regularization effect of SGD in the moderate learning rate regime by studying its behavior for optimizing an overparameterized linear regression problem. In this case, SGD and GD are known to converge to the unique minimum-norm solution; however, with the moderate and annealing learning rate, we show that they exhibit different \emph{directional bias}: SGD converges along the large eigenvalue directions of the data matrix, while GD goes after the small eigenvalue directions. Furthermore, we show that such directional bias does matter when early stopping is adopted, where the SGD output is nearly optimal but the GD output is suboptimal. Finally, our theory explains several folk arts in practice used for SGD hyperparameter tuning, such as (1) linearly scaling the initial learning rate with batch size; and (2) overrunning SGD with high learning rate even when the loss stops decreasing.

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arXiv2021

Batched Neural Bandits

In many sequential decision-making problems, the individuals are split into several batches and the decision-maker is only allowed to change her policy at the end of batches. These batch problems have a large number of applications, ranging from clinical trials to crowdsourcing. Motivated by this, we study the stochastic contextual bandit problem for general reward distributions under the batched setting. We propose the BatchNeuralUCB algorithm which combines neural networks with optimism to address the exploration-exploitation tradeoff while keeping the total number of batches limited. We study BatchNeuralUCB under both fixed and adaptive batch size settings and prove that it achieves the same regret as the fully sequential version while reducing the number of policy updates considerably. We confirm our theoretical results via simulations on both synthetic and real-world datasets.

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arXiv2021

Faster Convergence of Stochastic Gradient Langevin Dynamics for Non-Log-Concave Sampling

We provide a new convergence analysis of stochastic gradient Langevin dynamics (SGLD) for sampling from a class of distributions that can be non-log-concave. At the core of our approach is a novel conductance analysis of SGLD using an auxiliary time-reversible Markov Chain. Under certain conditions on the target distribution, we prove that $\tilde O(d^4ε^{-2})$ stochastic gradient evaluations suffice to guarantee $ε$-sampling error in terms of the total variation distance, where $d$ is the problem dimension. This improves existing results on the convergence rate of SGLD (Raginsky et al., 2017; Xu et al., 2018). We further show that provided an additional Hessian Lipschitz condition on the log-density function, SGLD is guaranteed to achieve $ε$-sampling error within $\tilde O(d^{15/4}ε^{-3/2})$ stochastic gradient evaluations. Our proof technique provides a new way to study the convergence of Langevin-based algorithms and sheds some light on the design of fast stochastic gradient-based sampling algorithms.

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arXiv2021

Provably Efficient Reinforcement Learning for Discounted MDPs with Feature Mapping

Modern tasks in reinforcement learning have large state and action spaces. To deal with them efficiently, one often uses predefined feature mapping to represent states and actions in a low-dimensional space. In this paper, we study reinforcement learning for discounted Markov Decision Processes (MDPs), where the transition kernel can be parameterized as a linear function of certain feature mapping. We propose a novel algorithm that makes use of the feature mapping and obtains a $\tilde O(d\sqrt{T}/(1-γ)^2)$ regret, where $d$ is the dimension of the feature space, $T$ is the time horizon and $γ$ is the discount factor of the MDP. To the best of our knowledge, this is the first polynomial regret bound without accessing the generative model or making strong assumptions such as ergodicity of the MDP. By constructing a special class of MDPs, we also show that for any algorithms, the regret is lower bounded by $Ω(d\sqrt{T}/(1-γ)^{1.5})$. Our upper and lower bound results together suggest that the proposed reinforcement learning algorithm is near-optimal up to a $(1-γ)^{-0.5}$ factor.

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arXiv2021

Logarithmic Regret for Reinforcement Learning with Linear Function Approximation

Reinforcement learning (RL) with linear function approximation has received increasing attention recently. However, existing work has focused on obtaining $\sqrt{T}$-type regret bound, where $T$ is the number of interactions with the MDP. In this paper, we show that logarithmic regret is attainable under two recently proposed linear MDP assumptions provided that there exists a positive sub-optimality gap for the optimal action-value function. More specifically, under the linear MDP assumption (Jin et al. 2019), the LSVI-UCB algorithm can achieve $\tilde{O}(d^{3}H^5/\text{gap}_{\text{min}}\cdot \log(T))$ regret; and under the linear mixture MDP assumption (Ayoub et al. 2020), the UCRL-VTR algorithm can achieve $\tilde{O}(d^{2}H^5/\text{gap}_{\text{min}}\cdot \log^3(T))$ regret, where $d$ is the dimension of feature mapping, $H$ is the length of episode, $\text{gap}_{\text{min}}$ is the minimal sub-optimality gap, and $\tilde O$ hides all logarithmic terms except $\log(T)$. To the best of our knowledge, these are the first logarithmic regret bounds for RL with linear function approximation. We also establish gap-dependent lower bounds for the two linear MDP models.

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arXiv2021

Provable Generalization of SGD-trained Neural Networks of Any Width in the Presence of Adversarial Label Noise

We consider a one-hidden-layer leaky ReLU network of arbitrary width trained by stochastic gradient descent (SGD) following an arbitrary initialization. We prove that SGD produces neural networks that have classification accuracy competitive with that of the best halfspace over the distribution for a broad class of distributions that includes log-concave isotropic and hard margin distributions. Equivalently, such networks can generalize when the data distribution is linearly separable but corrupted with adversarial label noise, despite the capacity to overfit. To the best of our knowledge, this is the first work to show that overparameterized neural networks trained by SGD can generalize when the data is corrupted with adversarial label noise.

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arXiv2021

Agnostic Learning of Halfspaces with Gradient Descent via Soft Margins

We analyze the properties of gradient descent on convex surrogates for the zero-one loss for the agnostic learning of linear halfspaces. If $\mathsf{OPT}$ is the best classification error achieved by a halfspace, by appealing to the notion of soft margins we are able to show that gradient descent finds halfspaces with classification error $\tilde O(\mathsf{OPT}^{1/2}) + \varepsilon$ in $\mathrm{poly}(d,1/\varepsilon)$ time and sample complexity for a broad class of distributions that includes log-concave isotropic distributions as a subclass. Along the way we answer a question recently posed by Ji et al. (2020) on how the tail behavior of a loss function can affect sample complexity and runtime guarantees for gradient descent.

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arXiv2021

Revisiting Membership Inference Under Realistic Assumptions

We study membership inference in settings where some of the assumptions typically used in previous research are relaxed. First, we consider skewed priors, to cover cases such as when only a small fraction of the candidate pool targeted by the adversary are actually members and develop a PPV-based metric suitable for this setting. This setting is more realistic than the balanced prior setting typically considered by researchers. Second, we consider adversaries that select inference thresholds according to their attack goals and develop a threshold selection procedure that improves inference attacks. Since previous inference attacks fail in imbalanced prior setting, we develop a new inference attack based on the intuition that inputs corresponding to training set members will be near a local minimum in the loss function, and show that an attack that combines this with thresholds on the per-instance loss can achieve high PPV even in settings where other attacks appear to be ineffective. Code for our experiments can be found here: https://github.com/bargavj/EvaluatingDPML.

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arXiv2021

Provable Multi-Objective Reinforcement Learning with Generative Models

Multi-objective reinforcement learning (MORL) is an extension of ordinary, single-objective reinforcement learning (RL) that is applicable to many real-world tasks where multiple objectives exist without known relative costs. We study the problem of single policy MORL, which learns an optimal policy given the preference of objectives. Existing methods require strong assumptions such as exact knowledge of the multi-objective Markov decision process, and are analyzed in the limit of infinite data and time. We propose a new algorithm called model-based envelop value iteration (EVI), which generalizes the enveloped multi-objective $Q$-learning algorithm in Yang et al., 2019. Our method can learn a near-optimal value function with polynomial sample complexity and linear convergence speed. To the best of our knowledge, this is the first finite-sample analysis of MORL algorithms.

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arXiv2021

Nearly Minimax Optimal Reinforcement Learning for Linear Mixture Markov Decision Processes

We study reinforcement learning (RL) with linear function approximation where the underlying transition probability kernel of the Markov decision process (MDP) is a linear mixture model (Jia et al., 2020; Ayoub et al., 2020; Zhou et al., 2020) and the learning agent has access to either an integration or a sampling oracle of the individual basis kernels. We propose a new Bernstein-type concentration inequality for self-normalized martingales for linear bandit problems with bounded noise. Based on the new inequality, we propose a new, computationally efficient algorithm with linear function approximation named $\text{UCRL-VTR}^{+}$ for the aforementioned linear mixture MDPs in the episodic undiscounted setting. We show that $\text{UCRL-VTR}^{+}$ attains an $\tilde O(dH\sqrt{T})$ regret where $d$ is the dimension of feature mapping, $H$ is the length of the episode and $T$ is the number of interactions with the MDP. We also prove a matching lower bound $Ω(dH\sqrt{T})$ for this setting, which shows that $\text{UCRL-VTR}^{+}$ is minimax optimal up to logarithmic factors. In addition, we propose the $\text{UCLK}^{+}$ algorithm for the same family of MDPs under discounting and show that it attains an $\tilde O(d\sqrt{T}/(1-γ)^{1.5})$ regret, where $γ\in [0,1)$ is the discount factor. Our upper bound matches the lower bound $Ω(d\sqrt{T}/(1-γ)^{1.5})$ proved by Zhou et al. (2020) up to logarithmic factors, suggesting that $\text{UCLK}^{+}$ is nearly minimax optimal. To the best of our knowledge, these are the first computationally efficient, nearly minimax optimal algorithms for RL with linear function approximation.

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arXiv2020

Neural Contextual Bandits with Deep Representation and Shallow Exploration

We study a general class of contextual bandits, where each context-action pair is associated with a raw feature vector, but the reward generating function is unknown. We propose a novel learning algorithm that transforms the raw feature vector using the last hidden layer of a deep ReLU neural network (deep representation learning), and uses an upper confidence bound (UCB) approach to explore in the last linear layer (shallow exploration). We prove that under standard assumptions, our proposed algorithm achieves $\tilde{O}(\sqrt{T})$ finite-time regret, where $T$ is the learning time horizon. Compared with existing neural contextual bandit algorithms, our approach is computationally much more efficient since it only needs to explore in the last layer of the deep neural network.

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arXiv2020

Double Explore-then-Commit: Asymptotic Optimality and Beyond

We study the multi-armed bandit problem with subgaussian rewards. The explore-then-commit (ETC) strategy, which consists of an exploration phase followed by an exploitation phase, is one of the most widely used algorithms in a variety of online decision applications. Nevertheless, it has been shown in Garivier et al. (2016) that ETC is suboptimal in the asymptotic sense as the horizon grows, and thus, is worse than fully sequential strategies such as Upper Confidence Bound (UCB). In this paper, we show that a variant of ETC algorithm can actually achieve the asymptotic optimality for multi-armed bandit problems as UCB-type algorithms do and extend it to the batched bandit setting. Specifically, we propose a double explore-then-commit (DETC) algorithm that has two exploration and exploitation phases and prove that DETC achieves the asymptotically optimal regret bound. To our knowledge, DETC is the first non-fully-sequential algorithm that achieves such asymptotic optimality. In addition, we extend DETC to batched bandit problems, where (i) the exploration process is split into a small number of batches and (ii) the round complexity is of central interest. We prove that a batched version of DETC can achieve the asymptotic optimality with only a constant round complexity. This is the first batched bandit algorithm that can attain the optimal asymptotic regret bound and optimal round complexity simultaneously.

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arXiv2020

Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization

We present a unified framework to analyze the global convergence of Langevin dynamics based algorithms for nonconvex finite-sum optimization with $n$ component functions. At the core of our analysis is a direct analysis of the ergodicity of the numerical approximations to Langevin dynamics, which leads to faster convergence rates. Specifically, we show that gradient Langevin dynamics (GLD) and stochastic gradient Langevin dynamics (SGLD) converge to the almost minimizer within $\tilde O\big(nd/(λε) \big)$ and $\tilde O\big(d^7/(λ^5ε^5) \big)$ stochastic gradient evaluations respectively, where $d$ is the problem dimension, and $λ$ is the spectral gap of the Markov chain generated by GLD. Both results improve upon the best known gradient complexity results (Raginsky et al., 2017). Furthermore, for the first time we prove the global convergence guarantee for variance reduced stochastic gradient Langevin dynamics (SVRG-LD) to the almost minimizer within $\tilde O\big(\sqrt{n}d^5/(λ^4ε^{5/2})\big)$ stochastic gradient evaluations, which outperforms the gradient complexities of GLD and SGLD in a wide regime. Our theoretical analyses shed some light on using Langevin dynamics based algorithms for nonconvex optimization with provable guarantees.

stat.ML↗

arXiv2020

Stochastic Variance-Reduced Hamilton Monte Carlo Methods

We propose a fast stochastic Hamilton Monte Carlo (HMC) method, for sampling from a smooth and strongly log-concave distribution. At the core of our proposed method is a variance reduction technique inspired by the recent advance in stochastic optimization. We show that, to achieve $ε$ accuracy in 2-Wasserstein distance, our algorithm achieves $\tilde O(n+κ^{2}d^{1/2}/ε+κ^{4/3}d^{1/3}n^{2/3}/ε^{2/3})$ gradient complexity (i.e., number of component gradient evaluations), which outperforms the state-of-the-art HMC and stochastic gradient HMC methods in a wide regime. We also extend our algorithm for sampling from smooth and general log-concave distributions, and prove the corresponding gradient complexity as well. Experiments on both synthetic and real data demonstrate the superior performance of our algorithm.

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arXiv2020

Stochastic Nested Variance Reduction for Nonconvex Optimization

We study finite-sum nonconvex optimization problems, where the objective function is an average of $n$ nonconvex functions. We propose a new stochastic gradient descent algorithm based on nested variance reduction. Compared with conventional stochastic variance reduced gradient (SVRG) algorithm that uses two reference points to construct a semi-stochastic gradient with diminishing variance in each iteration, our algorithm uses $K+1$ nested reference points to build a semi-stochastic gradient to further reduce its variance in each iteration. For smooth nonconvex functions, the proposed algorithm converges to an $ε$-approximate first-order stationary point (i.e., $\|\nabla F(\mathbf{x})\|_2\leq ε$) within $\tilde O(n\land ε^{-2}+ε^{-3}\land n^{1/2}ε^{-2})$ number of stochastic gradient evaluations. This improves the best known gradient complexity of SVRG $O(n+n^{2/3}ε^{-2})$ and that of SCSG $O(n\land ε^{-2}+ε^{-10/3}\land n^{2/3}ε^{-2})$. For gradient dominated functions, our algorithm also achieves better gradient complexity than the state-of-the-art algorithms. Thorough experimental results on different nonconvex optimization problems back up our theory.

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