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Register a new userGeneralization Bounds of SGLD for Non-convex Learning: Two Theoretical Viewpoints
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Wenlong Mou
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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Algorithm-dependent generalization error bounds are central to statistical learning theory. A learning algorithm may use a large hypothesis space, but the limited number of iterations controls its model capacity and generalization error. The impacts of stochastic gradient methods on generalization error for non-convex learning problems not only have important theoretical consequences, but are also critical to generalization errors of deep learning. In this paper, we study the generalization errors of Stochastic Gradient Langevin Dynamics (SGLD) with non-convex objectives. Two theories are proposed with non-asymptotic discrete-time analysis, using Stability and PAC-Bayesian results respectively. The stability-based theory obtains a bound of $Oleft(frac{1}{n}Lsqrt{beta T_k}right)$, where $L$ is uniform Lipschitz parameter, $beta$ is inverse temperature, and $T_k$ is aggregated step sizes. For PAC-Bayesian theory, though the bound has a slower $O(1/sqrt{n})$ rate, the contribution of each step is shown with an exponentially decaying factor by imposing $ell^2$ regularization, and the uniform Lipschitz constant is also replaced by actual norms of gradients along trajectory. Our bounds have no implicit dependence on dimensions, norms or other capacity measures of parameter, which elegantly characterizes the phenomenon of Fast Training Guarantees Generalization in non-convex settings. This is the first algorithm-dependent result with reasonable dependence on aggregated step sizes for non-convex learning, and has important implications to statistical learning aspects of stochastic gradient methods in complicated models such as deep learning.
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Generalization performance of stochastic optimization stands a central place in learning theory. In this paper, we investigate the excess risk performance and towards improved learning rates for two popular approaches of stochastic optimization: empirical risk minimization (ERM) and stochastic gradient descent (SGD). Although there exists plentiful generalization analysis of ERM and SGD for supervised learning, current theoretical understandings of ERM and SGD either have stronger assumptions in convex learning, e.g., strong convexity, or show slow rates and less studied in nonconvex learning. Motivated by these problems, we aim to provide improved rates under milder assumptions in convex learning and derive faster rates in nonconvex learning. It is notable that our analysis span two popular theoretical viewpoints: emph{stability} and emph{uniform convergence}. Specifically, in stability regime, we present high probability learning rates of order $mathcal{O} (1/n)$ w.r.t. the sample size $n$ for ERM and SGD with milder assumptions in convex learning and similar high probability rates of order $mathcal{O} (1/n)$ in nonconvex learning, rather than in expectation. Furthermore, this type of learning rate is improved to faster order $mathcal{O} (1/n^2)$ in uniform convergence regime. To our best knowledge, for ERM and SGD, the learning rates presented in this paper are all state-of-the-art.
Generalization error (also known as the out-of-sample error) measures how well the hypothesis learned from training data generalizes to previously unseen data. Proving tight generalization error bounds is a central question in statistical learning theory. In this paper, we obtain generalization error bounds for learning general non-convex objectives, which has attracted significant attention in recent years. We develop a new framework, termed Bayes-Stability, for proving algorithm-dependent generalization error bounds. The new framework combines ideas from both the PAC-Bayesian theory and the notion of algorithmic stability. Applying the Bayes-Stability method, we obtain new data-dependent generalization bounds for stochastic gradient Langevin dynamics (SGLD) and several other noisy gradient methods (e.g., with momentum, mini-batch and acceleration, Entropy-SGD). Our result recovers (and is typically tighter than) a recent result in Mou et al. (2018) and improves upon the results in Pensia et al. (2018). Our experiments demonstrate that our data-dependent bounds can distinguish randomly labelled data from normal data, which provides an explanation to the intriguing phenomena observed in Zhang et al. (2017a). We also study the setting where the total loss is the sum of a bounded loss and an additional ell_2 regularization term. We obtain new generalization bounds for the continuous Langevin dynamic in this setting by developing a new Log-Sobolev inequality for the parameter distribution at any time. Our new bounds are more desirable when the noisy level of the process is not small, and do not become vacuous even when T tends to infinity.
We investigate 1) the rate at which refined properties of the empirical risk---in particular, gradients---converge to their population counterparts in standard non-convex learning tasks, and 2) the consequences of this convergence for optimization. Our analysis follows the tradition of norm-based capacity control. We propose vector-valued Rademacher complexities as a simple, composable, and user-friendly tool to derive dimension-free uniform convergence bounds for gradients in non-convex learning problems. As an application of our techniques, we give a new analysis of batch gradient descent methods for non-convex generalized linear models and non-convex robust regression, showing how to use any algorithm that finds approximate stationary points to obtain optimal sample complexity, even when dimension is high or possibly infinite and multiple passes over the dataset are allowed. Moving to non-smooth models we show----in contrast to the smooth case---that even for a single ReLU it is not possible to obtain dimension-independent convergence rates for gradients in the worst case. On the positive side, it is still possible to obtain dimension-independent rates under a new type of distributional assumption.
One popular trend in meta-learning is to learn from many training tasks a common initialization for a gradient-based method that can be used to solve a new task with few samples. The theory of meta-learning is still in its early stages, with several recent learning-theoretic analyses of methods such as Reptile [Nichol et al., 2018] being for convex models. This work shows that convex-case analysis might be insufficient to understand the success of meta-learning, and that even for non-convex models it is important to look inside the optimization black-box, specifically at properties of the optimization trajectory. We construct a simple meta-learning instance that captures the problem of one-dimensional subspace learning. For the convex formulation of linear regression on this instance, we show that the new task sample complexity of any initialization-based meta-learning algorithm is $Omega(d)$, where $d$ is the input dimension. In contrast, for the non-convex formulation of a two layer linear network on the same instance, we show that both Reptile and multi-task representation learning can have new task sample complexity of $mathcal{O}(1)$, demonstrating a separation from convex meta-learning. Crucially, analyses of the training dynamics of these methods reveal that they can meta-learn the correct subspace onto which the data should be projected.
Gradient clipping is commonly used in training deep neural networks partly due to its practicability in relieving the exploding gradient problem. Recently, citet{zhang2019gradient} show that clipped (stochastic) Gradient Descent (GD) converges faster than vanilla GD/SGD via introducing a new assumption called $(L_0, L_1)$-smoothness, which characterizes the violent fluctuation of gradients typically encountered in deep neural networks. However, their iteration complexities on the problem-dependent parameters are rather pessimistic, and theoretical justification of clipping combined with other crucial techniques, e.g. momentum acceleration, are still lacking. In this paper, we bridge the gap by presenting a general framework to study the clipping algorithms, which also takes momentum methods into consideration. We provide convergence analysis of the framework in both deterministic and stochastic setting, and demonstrate the tightness of our results by comparing them with existing lower bounds. Our results imply that the efficiency of clipping methods will not degenerate even in highly non-smooth regions of the landscape. Experiments confirm the superiority of clipping-based methods in deep learning tasks.
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