Learning to learn is a powerful paradigm for enabling models to learn from data more effectively and efficiently. A popular approach to meta-learning is to train a recurrent model to read in a training dataset as input and output the parameters of a learned model, or output predictions for new test inputs. Alternatively, a more recent approach to meta-learning aims to acquire deep representations that can be effectively fine-tuned, via standard gradient descent, to new tasks. In this paper, we consider the meta-learning problem from the perspective of universality, formalizing the notion of learning algorithm approximation and comparing the expressive power of the aforementioned recurrent models to the more recent approaches that embed gradient descent into the meta-learner. In particular, we seek to answer the following question: does deep representation combined with standard gradient descent have sufficient capacity to approximate any learning algorithm? We find that this is indeed true, and further find, in our experiments, that gradient-based meta-learning consistently leads to learning strategies that generalize more widely compared to those represented by recurrent models.

We consider two questions at the heart of machine learning; how can we predict if a minimum will generalize to the test set, and why does stochastic gradient descent find minima that generalize well? Our work responds to Zhang et al. (2016), who showed deep neural networks can easily memorize randomly labeled training data, despite generalizing well on real labels of the same inputs. We show that the same phenomenon occurs in small linear models. These observations are explained by the Bayesian evidence, which penalizes sharp minima but is invariant to model parameterization. We also demonstrate that, when one holds the learning rate fixed, there is an optimum batch size which maximizes the test set accuracy. We propose that the noise introduced by small mini-batches drives the parameters towards minima whose evidence is large. Interpreting stochastic gradient descent as a stochastic differential equation, we identify the "noise scale" $g = \epsilon (\frac{N}{B} - 1) \approx \epsilon N/B$, where $\epsilon$ is the learning rate, $N$ the training set size and $B$ the batch size. Consequently the optimum batch size is proportional to both the learning rate and the size of the training set, $B_{opt} \propto \epsilon N$. We verify these predictions empirically.

Stochastic gradient descent (SGD) is an immensely popular approach for online learning in settings where data arrives in a stream or data sizes are very large. However, despite an ever-increasing volume of work on SGD, much less is known about the statistical inferential properties of SGD-based predictions. Taking a fully inferential viewpoint, this paper introduces a novel procedure termed HiGrad to conduct statistical inference for online learning, without incurring additional computational cost compared with SGD. The HiGrad procedure begins by performing SGD updates for a while and then splits the single thread into several threads, and this procedure hierarchically operates in this fashion along each thread. With predictions provided by multiple threads in place, a t-based confidence interval is constructed by decorrelating predictions using covariance structures given by the Ruppert--Polyak averaging scheme. Under certain regularity conditions, the HiGrad confidence interval is shown to attain asymptotically exact coverage probability. Finally, the performance of HiGrad is evaluated through extensive simulation studies and a real data example. An R package higrad has been developed to implement the method.

We analyze stochastic gradient algorithms for optimizing nonconvex, nonsmooth finite-sum problems. In particular, the objective function is given by the summation of a differentiable (possibly nonconvex) component, together with a possibly non-differentiable but convex component. We propose a proximal stochastic gradient algorithm based on variance reduction, called ProxSVRG+. The algorithm is a slight variant of the ProxSVRG algorithm [Reddi et al., 2016b]. Our main contribution lies in the analysis of ProxSVRG+. It recovers several existing convergence results (in terms of the number of stochastic gradient oracle calls and proximal operations), and improves/generalizes some others. In particular, ProxSVRG+ generalizes the best results given by the SCSG algorithm, recently proposed by [Lei et al., 2017] for the smooth nonconvex case. ProxSVRG+ is more straightforward than SCSG and yields simpler analysis. Moreover, ProxSVRG+ outperforms the deterministic proximal gradient descent (ProxGD) for a wide range of minibatch sizes, which partially solves an open problem proposed in [Reddi et al., 2016b]. Finally, for nonconvex functions satisfied Polyak-{\L}ojasiewicz condition, we show that ProxSVRG+ achieves global linear convergence rate without restart. ProxSVRG+ is always no worse than ProxGD and ProxSVRG/SAGA, and sometimes outperforms them (and generalizes the results of SCSG) in this case.

We propose a meta-learning approach for learning gradient-based reinforcement learning (RL) algorithms. The idea is to evolve a differentiable loss function, such that an agent, which optimizes its policy to minimize this loss, will achieve high rewards. The loss is parametrized via temporal convolutions over the agent's experience. Because this loss is highly flexible in its ability to take into account the agent's history, it enables fast task learning and eliminates the need for reward shaping at test time. Empirical results show that our evolved policy gradient algorithm achieves faster learning on several randomized environments compared to an off-the-shelf policy gradient method. Moreover, at test time, our learner optimizes only its learned loss function, and requires no explicit reward signal. In effect, the agent internalizes the reward structure, suggesting a direction toward agents that learn to solve new tasks simply from intrinsic motivation.

A major challenge in understanding the generalization of deep learning is to explain why (stochastic) gradient descent can exploit the network architecture to find solutions that have good generalization performance when using high capacity models. We find simple but realistic examples showing that this phenomenon exists even when learning linear classifiers --- between two linear networks with the same capacity, the one with a convolutional layer can generalize better than the other when the data distribution has some underlying spatial structure. We argue that this difference results from a combination of the convolution architecture, data distribution and gradient descent, all of which are necessary to be included in a meaningful analysis. We provide a general analysis of the generalization performance as a function of data distribution and convolutional filter size, given gradient descent as the optimization algorithm, then interpret the results using concrete examples. Experimental results show that our analysis is able to explain what happens in our introduced examples.

We provide a numerically robust and fast method capable of exploiting the local geometry when solving large-scale stochastic optimisation problems. Our key innovation is an auxiliary variable construction coupled with an inverse Hessian approximation computed using a receding history of iterates and gradients. It is the Markov chain nature of the classic stochastic gradient algorithm that enables this development. The construction offers a mechanism for stochastic line search adapting the step length. We numerically evaluate and compare against current state-of-the-art with encouraging performance on real-world benchmark problems where the number of observations and unknowns is in the order of millions.

The diverse world of non-convex machine learning tasks have given rise to numerous non-convex optimization problems. Some of them are perhaps as hard as minimizing general non-convex objectives (such as deep learning), but some others may be only slightly harder than convex optimization (such as matrix completion, principal component analysis, dictionary learning, etc). Therefore, it is both interesting and challenging to identify classes of optimization problems that interplay between non-convex and convex optimization, and (hopefully) optimally and practically solving them. At least tracing back to 2015, Shalev-Shwartz [27] identified a class of functions that are convex, but can be written as finite average of non-convex functions.

Stochastic gradient descent (SGD) is the optimization algorithm of choice in many machine learning applications such as regularized empirical risk minimization and training deep neural networks. The classical analysis of convergence 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 always violated for cases where the objective function is strongly convex. In (Bottou et al.,2016) a new analysis of convergence of SGD is performed under the assumption that stochastic gradients are bounded with respect to the true gradient norm. Here we show that for stochastic problems arising in machine learning such bound always holds. Moreover, we propose an alternative convergence analysis of SGD with diminishing learning rate regime, which is results in more relaxed conditions that those in (Bottou et al.,2016). We then move on the asynchronous parallel setting, and prove convergence of the Hogwild! algorithm in the same regime, obtaining the first convergence results for this method in the case of diminished learning rate.

It has been widely observed that many activation functions and pooling methods of neural network models have (positive-) rescaling-invariant property, including ReLU, PReLU, max-pooling, and average pooling, which makes fully-connected neural networks (FNNs) and convolutional neural networks (CNNs) invariant to (positive) rescaling operation across layers. This may cause unneglectable problems with their optimization: (1) different NN models could be equivalent, but their gradients can be very different from each other; (2) it can be proven that the loss functions may have many spurious critical points in the redundant weight space. To tackle these problems, in this paper, we first characterize the rescaling-invariant properties of NN models using equivalent classes and prove that the dimension of the equivalent class space is significantly smaller than the dimension of the original weight space. Then we represent the loss function in the compact equivalent class space and develop novel algorithms that conduct optimization of the NN models directly in the equivalent class space. We call these algorithms Equivalent Class Optimization (abbreviated as EC-Opt) algorithms. Moreover, we design efficient tricks to compute the gradients in the equivalent class, which almost have no extra computational complexity as compared to standard back-propagation (BP). We conducted experimental study to demonstrate the effectiveness of our proposed new optimization algorithms. In particular, we show that by using the idea of EC-Opt, we can significantly improve the accuracy of the learned model (for both FNN and CNN), as compared to using conventional stochastic gradient descent algorithms.