Thompson sampling is a well-known approach for balancing exploration and exploitation in reinforcement learning. It requires the posterior distribution of value-action functions to be maintained; this is generally intractable for tasks that have a high dimensional state-action space. We derive a variational Thompson sampling approximation for DQNs which uses a deep network whose parameters are perturbed by a learned variational noise distribution. We interpret the successful NoisyNets method \cite{fortunato2018noisy} as an approximation to the variational Thompson sampling method that we derive. Further, we propose State Aware Noisy Exploration (SANE) which seeks to improve on NoisyNets by allowing a non-uniform perturbation, where the amount of parameter perturbation is conditioned on the state of the agent. This is done with the help of an auxiliary perturbation module, whose output is state dependent and is learnt end to end with gradient descent. We hypothesize that such state-aware noisy exploration is particularly useful in problems where exploration in certain \textit{high risk} states may result in the agent failing badly. We demonstrate the effectiveness of the state-aware exploration method in the off-policy setting by augmenting DQNs with the auxiliary perturbation module.

Deep neural networks with batch normalization (BN-DNNs) are invariant to weight rescaling due to their normalization operations. However, using weight decay (WD) benefits these weight-scale-invariant networks, which is often attributed to an increase of the effective learning rate when the weight norms are decreased. In this paper, we demonstrate the insufficiency of the previous explanation and investigate the implicit biases of stochastic gradient descent (SGD) on BN-DNNs to provide a theoretical explanation for the efficacy of weight decay. We identity two implicit biases of SGD on BN-DNNs: 1) the weight norms in SGD training remain constant in the continuous-time domain and keep increasing in the discrete-time domain; 2) SGD optimizes weight vectors in fully-connected networks or convolution kernels in convolution neural networks by updating components lying in the input feature span, while leaving those components orthogonal to the input feature span unchanged. Thus, SGD without WD accumulates weight noise orthogonal to the input feature span, and cannot eliminate such noise. Our empirical studies corroborate the hypothesis that weight decay suppresses weight noise that is left untouched by SGD. Furthermore, we propose to use weight rescaling (WRS) instead of weight decay to achieve the same regularization effect, while avoiding performance degradation of WD on some momentum-based optimizers. Our empirical results on image recognition show that regardless of optimization methods and network architectures, training BN-DNNs using WRS achieves similar or better performance compared with using WD. We also show that training with WRS generalizes better compared to WD, on other computer vision tasks.

Motivated by the recent successes of neural networks that have the ability to fit the data perfectly and generalize well, we study the noiseless model in the fundamental least-squares setup. We assume that an optimum predictor fits perfectly inputs and outputs $\langle \theta_* , \phi(X) \rangle = Y$, where $\phi(X)$ stands for a possibly infinite dimensional non-linear feature map. To solve this problem, we consider the estimator given by the last iterate of stochastic gradient descent (SGD) with constant step-size. In this context, our contribution is two fold: (i) from a (stochastic) optimization perspective, we exhibit an archetypal problem where we can show explicitly the convergence of SGD final iterate for a non-strongly convex problem with constant step-size whereas usual results use some form of average and (ii) from a statistical perspective, we give explicit non-asymptotic convergence rates in the over-parameterized setting and leverage a fine-grained parameterization of the problem to exhibit polynomial rates that can be faster than $O(1/T)$. The link with reproducing kernel Hilbert spaces is established.

Factorization-based gradient descent is a scalable and efficient algorithm for solving low-rank matrix completion. Recent progress in structured non-convex optimization has offered global convergence guarantees for gradient descent under certain statistical assumptions on the low-rank matrix and the sampling set. However, while the theory suggests gradient descent enjoys fast linear convergence to a global solution of the problem, the universal nature of the bounding technique prevents it from obtaining an accurate estimate of the rate of convergence. In this paper, we perform a local analysis of the exact linear convergence rate of gradient descent for factorization-based matrix completion for symmetric matrices. Without any additional assumptions on the underlying model, we identify the deterministic condition for local convergence of gradient descent, which only depends on the solution matrix and the sampling set. More crucially, our analysis provides a closed-form expression of the asymptotic rate of convergence that matches exactly with the linear convergence observed in practice. To the best of our knowledge, our result is the first one that offers the exact rate of convergence of gradient descent for matrix factorization in Euclidean space for matrix completion.

Understanding the generalization capability of learning algorithms is at the heart of statistical learning theory. In this paper, we investigate the generalization gap of stochastic gradient Langevin dynamics (SGLD), a widely used optimizer for training deep neural networks (DNNs). We derive an algorithm-dependent generalization bound by analyzing SGLD through an information-theoretic lens. Our analysis reveals an intricate trade-off between learning and information dissipation: SGLD learns from data by updating parameters at each iteration while dissipating information from early training stages. Our bound also involves the variance of gradients which captures a particular kind of "sharpness" of the loss landscape. The main proof techniques in this paper rely on strong data processing inequalities--a fundamental concept in information theory--and Otto-Villani's HWI inequality. Finally, we demonstrate our bound through numerical experiments, showing that it can predict the behavior of the true generalization gap. Keywords: Algorithm-dependent generalization bound, non-convex learning, information theory, stochastic gradient Langevin dynamics.

It is well-known that simple short-sighted algorithms, such as gradient descent, generalize well in the over-parameterized learning tasks, due to their implicit regularization. However, it is unknown whether the implicit regularization of these algorithms can be extended to robust learning tasks, where a subset of samples may be grossly corrupted with noise. In this work, we provide a positive answer to this question in the context of robust matrix recovery problem. In particular, we consider the problem of recovering a low-rank matrix from a number of linear measurements, where a subset of measurements are corrupted with large noise. We show that a simple sub-gradient method converges to the true low-rank solution efficiently, when it is applied to the over-parameterized l1-loss function without any explicit regularization or rank constraint. Moreover, by building upon a new notion of restricted isometry property, called sign-RIP, we prove the robustness of the sub-gradient method against outliers in the over-parameterized regime. In particular, we show that, with Gaussian measurements, the sub-gradient method is guaranteed to converge to the true low-rank solution, even if an arbitrary fraction of the measurements are grossly corrupted with noise.

Algorithmic stability has emerged over the last two decades as a central tool in generalization analysis of learning algorithms. While the classical approach in generalization theory originating in the PAC learning framework appeal to uniform convergence arguments, more recent progress on stochastic convex optimization models, starting with the pioneering work of Bousquet and Elisseeff (2002) and Shalev-Shwartz et al. (2009), has relied on stability analysis for deriving tight generalization results for convex risk minimizing algorithms. Perhaps the most common form of algorithmic stability is the so called uniform stability (Bousquet and Elisseeff, 2002). Roughly, the uniform stability of a learning algorithm is the worst-case change in its output model, in terms of its loss on an arbitrary example, when replacing a single sample in the data set used for training. Bousquet and Elisseeff (2002) initially used uniform stability to argue about the generalization of empirical risk minimization with strongly convex losses.

Stochastic gradient Markov Chain Monte Carlo algorithms are popular samplers for approximate inference, but they are generally biased. We show that many recent versions of these methods (e.g. Chen et al. (2014)) cannot be corrected using Metropolis-Hastings rejection sampling, because their acceptance probability is always zero. We can fix this by employing a sampler with realizable backwards trajectories, such as Gradient-Guided Monte Carlo (Horowitz, 1991), which generalizes stochastic gradient Langevin dynamics (Welling and Teh, 2011) and Hamiltonian Monte Carlo. We show that this sampler can be used with stochastic gradients, yielding nonzero acceptance probabilities, which can be computed even across multiple steps.

We give a new separation result between the generalization performance of stochastic gradient descent (SGD) and of full-batch gradient descent (GD) in the fundamental stochastic convex optimization model. While for SGD it is well-known that $O(1/\epsilon^2)$ iterations suffice for obtaining a solution with $\epsilon$ excess expected risk, we show that with the same number of steps GD may overfit and emit a solution with $\Omega(1)$ generalization error. Moreover, we show that in fact $\Omega(1/\epsilon^4)$ iterations are necessary for GD to match the generalization performance of SGD, which is also tight due to recent work by Bassily et al. (2020). We further discuss how regularizing the empirical risk minimized by GD essentially does not change the above result, and revisit the concepts of stability, implicit bias and the role of the learning algorithm in generalization.

We study the generalization properties of the popular stochastic gradient descent method for optimizing general non-convex loss functions. Our main contribution is providing upper bounds on the generalization error that depend on local statistics of the stochastic gradients evaluated along the path of iterates calculated by SGD. The key factors our bounds depend on are the variance of the gradients (with respect to the data distribution) and the local smoothness of the objective function along the SGD path, and the sensitivity of the loss function to perturbations to the final output. Our key technical tool is combining the information-theoretic generalization bounds previously used for analyzing randomized variants of SGD with a perturbation analysis of the iterates.