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 $O(1/\epsilon^{3/2})$ episodes to find an $\epsilon$-approximate stationary point of the nonconcave performance function $J(\boldsymbol{\theta})$ (i.e., $\boldsymbol{\theta}$ such that $\|\nabla J(\boldsymbol{\theta})\|_2^2\leq\epsilon$). This sample complexity improves the best known result $O(1/\epsilon^{5/3})$ for policy gradient algorithms by a factor of $O(1/\epsilon^{1/6})$. 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.

A Knowledge Transfer Framework for Differentially Private Sparse Learning

We study the problem of estimating high dimensional models with underlying sparse structures while preserving the privacy of each training example. We develop a differentially private high-dimensional sparse learning framework using the idea of knowledge transfer. More specifically, we propose to distill the knowledge from a "teacher" estimator trained on a private dataset, by creating a new dataset from auxiliary features, and then train a differentially private "student" estimator using this new dataset. In addition, we establish the linear convergence rate as well as the utility guarantee for our proposed method. For sparse linear regression and sparse logistic regression, our method achieves improved utility guarantees compared with the best known results (Kifer et al., 2012; Wang and Gu, 2019). We further demonstrate the superiority of our framework through both synthetic and real-world data experiments.

DP-LSSGD: A Stochastic Optimization Method to Lift the Utility in Privacy-Preserving ERM

Machine learning (ML) models trained by differentially private stochastic gradient descent (DP-SGD) has much lower utility than the non-private ones. To mitigate this degradation, we propose a DP Laplacian smoothing SGD (DP-LSSGD) for privacy-preserving ML. At the core of DP-LSSGD is the Laplace smoothing operator, which smooths out the Gaussian noise vector used in the Gaussian mechanism. Under the same amount of noise used in the Gaussian mechanism, DP-LSSGD attains the same differential privacy guarantee, but a strictly better utility guarantee, excluding an intrinsic term which is usually dominated by the other terms, for convex optimization than DP-SGD by a factor which is much less than one. In practice, DP-LSSGD makes training both convex and nonconvex ML models more efficient and enables the trained models to generalize better. For ResNet20, under the same strong differential privacy guarantee, DP-LSSGD can lift the testing accuracy of the trained private model by more than $8$\% compared with DP-SGD. The proposed algorithm is simple to implement and the extra computational complexity and memory overhead compared with DP-SGD are negligible. DP-LSSGD is applicable to train a large variety of ML models, including deep neural nets. The code is available at \url{https://github.com/BaoWangMath/DP-LSSGD}.

An Improved Analysis of Training Over-parameterized Deep Neural Networks

A recent line of research has shown that gradient-based algorithms with random initialization can converge to the global minima of the training loss for over-parameterized (i.e., sufficiently wide) deep neural networks. However, the condition on the width of the neural network to ensure the global convergence is very stringent, which is often a high-degree polynomial in the training sample size $n$ (e.g., $O(n^{24})$). In this paper, we provide an improved analysis of the global convergence of (stochastic) gradient descent for training deep neural networks, which only requires a milder over-parameterization condition than previous work in terms of the training sample size and other problem-dependent parameters. The main technical contributions of our analysis include (a) a tighter gradient lower bound that leads to a faster convergence of the algorithm, and (b) a sharper characterization of the trajectory length of the algorithm. By specializing our result to two-layer (i.e., one-hidden-layer) neural networks, it also provides a milder over-parameterization condition than the best-known result in prior work.

Generalization Bounds of Stochastic Gradient Descent for Wide and Deep Neural Networks

We study the training and generalization of deep neural networks (DNNs) in the over-parameterized regime, where the network width (i.e., number of hidden nodes per layer) is much larger than the number of training data points. We show that, the expected $0$-$1$ loss of a wide enough ReLU network trained with stochastic gradient descent (SGD) and random initialization can be bounded by the training loss of a random feature model induced by the network gradient at initialization, which we call a neural tangent random feature (NTRF) model. For data distributions that can be classified by NTRF model with sufficiently small error, our result yields a generalization error bound in the order of $\tilde{\mathcal{O}}(n^{-1/2})$ that is independent of the network width. Our result is more general and sharper than many existing generalization error bounds for over-parameterized neural networks. In addition, we establish a strong connection between our generalization error bound and the neural tangent kernel (NTK) proposed in recent work.

An Improved Convergence Analysis of Stochastic Variance-Reduced Policy Gradient

We revisit the stochastic variance-reduced policy gradient (SVRPG) method proposed by Papini et al. (2018) for reinforcement learning. We provide an improved convergence analysis of SVRPG and show that it can find an $\epsilon$-approximate stationary point of the performance function within $O(1/\epsilon^{5/3})$ trajectories. This sample complexity improves upon the best known result $O(1/\epsilon^2)$ by a factor of $O(1/\epsilon^{1/3})$. At the core of our analysis is (i) a tighter upper bound for the variance of importance sampling weights, where we prove that the variance can be controlled by the parameter distance between different policies; and (ii) a fine-grained analysis of the epoch length and batch size parameters such that we can significantly reduce the number of trajectories required in each iteration of SVRPG. We also empirically demonstrate the effectiveness of our theoretical claims of batch sizes on reinforcement learning benchmark tasks.

Generalization Error Bounds of Gradient Descent for Learning Over-parameterized Deep ReLU Networks

Empirical studies show that gradient-based methods can learn deep neural networks (DNNs) with very good generalization performance in the over-parameterization regime, where DNNs can easily fit a random labeling of the training data. While a line of recent work explains in theory that with over-parameterization and proper random initialization, gradient-based methods can find the global minima of the training loss for DNNs, it does not explain the good generalization performance of the gradient-based methods for learning over-parameterized DNNs. In this work, we take a step further, and prove that under certain assumption on the data distribution that is milder than linear separability, gradient descent (GD) with proper random initialization is able to train a sufficiently over-parameterized DNN to achieve arbitrarily small expected error (i.e., population error). This leads to an algorithmic-dependent generalization error bound for deep learning. To the best of our knowledge, this is the first result of its kind that can explain the good generalization performance of over-parameterized deep neural networks learned by gradient descent.

A Generalization Theory of Gradient Descent for Learning Over-parameterized Deep ReLU Networks

Empirical studies show that gradient based methods can learn deep neural networks (DNNs) with very good generalization performance in the over-parameterization regime, where DNNs can easily fit a random labeling of the training data. While a line of recent work explains in theory that gradient-based methods with proper random initialization can find the global minima of the training loss in over-parameterized DNNs, it does not explain the good generalization performance of the gradient-based methods for learning over-parameterized DNNs. In this work, we take a step further, and prove that under certain assumption on the data distribution that is milder than linear separability, gradient descent (GD) with proper random initialization is able to train a sufficiently over-parameterized DNN to achieve arbitrarily small expected error (i.e., population error). This leads to an algorithmic-dependent generalization error bound for deep learning. To the best of our knowledge, this is the first result of its kind that can explain the good generalization performance of over-parameterized deep neural networks learned by gradient descent.