In this work, we study a new approach to optimizing the margin distribution realized by binary classifiers. The classical approach to this problem is simply maximization of the expected margin, while more recent proposals consider simultaneous variance control and proxy objectives based on robust location estimates, in the vein of keeping the margin distribution sharply concentrated in a desirable region. While conceptually appealing, these new approaches are often computationally unwieldy, and theoretical guarantees are limited. Given this context, we propose an algorithm which searches the hypothesis space in such a way that a pre-set "margin level" ends up being a distribution-robust estimator of the margin location. This procedure is easily implemented using gradient descent, and admits finite-sample bounds on the excess risk under unbounded inputs. Empirical tests on real-world benchmark data reinforce the basic principles highlighted by the theory, and are suggestive of a promising new technique for classification.

Learning effective visuomotor policies for robots purely from data is challenging, but also appealing since a learning-based system should not require manual tuning or calibration. In the case of a robot operating in a real environment the training process can be costly, time-consuming, and even dangerous since failures are common at the start of training. For this reason, it is desirable to be able to leverage \textit{simulation} and \textit{off-policy} data to the extent possible to train the robot. In this work, we introduce a robust framework that plans in simulation and transfers well to the real environment. Our model incorporates a gradient-descent based planning module, which, given the initial image and goal image, encodes the images to a lower dimensional latent state and plans a trajectory to reach the goal. The model, consisting of the encoder and planner modules, is trained through a meta-learning strategy in simulation first. We subsequently perform adversarial domain transfer on the encoder by using a bank of unlabelled but random images from the simulation and real environments to enable the encoder to map images from the real and simulated environments to a similarly distributed latent representation. By fine tuning the entire model (encoder + planner) with far fewer real world expert demonstrations, we show successful planning performances in different navigation tasks.

We wish to compute the gradient of an expectation over a finite or countably infinite sample space having $K \leq \infty$ categories. When $K$ is indeed infinite, or finite but very large, the relevant summation is intractable. Accordingly, various stochastic gradient estimators have been proposed. In this paper, we describe a technique that can be applied to reduce the variance of any such estimator, without changing its bias---in particular, unbiasedness is retained. We show that our technique is an instance of Rao-Blackwellization, and we demonstrate the improvement it yields in empirical studies on both synthetic and real-world data.

Significant advances have been made recently on training neural networks, where the main challenge is in solving an optimization problem with abundant critical points. However, existing approaches to address this issue crucially rely on a restrictive assumption: the training data is drawn from a Gaussian distribution. In this paper, we provide a novel unified framework to design loss functions with desirable landscape properties for a wide range of general input distributions. On these loss functions, remarkably, stochastic gradient descent theoretically recovers the true parameters with global initializations and empirically outperforms the existing approaches. Our loss function design bridges the notion of score functions with the topic of neural network optimization. Central to our approach is the task of estimating the score function from samples, which is of basic and independent interest to theoretical statistics. Traditional estimation methods (example: kernel based) fail right at the outset; we bring statistical methods of local likelihood to design a novel estimator of score functions, that provably adapts to the local geometry of the unknown density.

We formulate stochastic gradient descent (SGD) as a Bayesian filtering problem. Inference in the Bayesian setting naturally gives rise to BRMSprop and BAdam: Bayesian variants of RMSprop and Adam. Remarkably, the Bayesian approach recovers many features of state-of-the-art adaptive SGD methods, including amoungst others root-mean-square normalization, Nesterov acceleration and AdamW. As such, the Bayesian approach provides one explanation for the empirical effectiveness of state-of-the-art adaptive SGD algorithms. Empirically comparing BRMSprop and BAdam with naive RMSprop and Adam on MNIST, we find that Bayesian methods have the potential to considerably reduce test loss and classification error.

In empirical risk optimization, it has been observed that stochastic gradient implementations that rely on random reshuffling of the data achieve better performance than implementations that rely on sampling the data uniformly. Recent works have pursued justifications for this behavior by examining the convergence rate of the learning process under diminishing step-sizes. This work focuses on the constant step-size case and strongly convex loss function. In this case, convergence is guaranteed to a small neighborhood of the optimizer albeit at a linear rate. The analysis establishes analytically that random reshuffling outperforms uniform sampling by showing explicitly that iterates approach a smaller neighborhood of size $O(\mu^2)$ around the minimizer rather than $O(\mu)$. Furthermore, we derive an analytical expression for the steady-state mean-square-error performance of the algorithm, which helps clarify in greater detail the differences between sampling with and without replacement. We also explain the periodic behavior that is observed in random reshuffling implementations.

But a rather high variance introduced by the stochastic gradient in each step may slow down the convergence. In this paper, we propose the antithetic sampling strategy to reduce the variance by taking advantage of the internal structure in dataset. Under this new strategy, stochastic gradients in a mini-batch are no longer independent but negatively correlated as much as possible, while the mini-batch stochastic gradient is still an unbiased estimator of full gradient. For the binary classification problems, we just need to calculate the antithetic samples in advance, and reuse the result in each iteration, which is practical. Experiments are provided to confirm the effectiveness of the proposed method.

Most successful deep learning models use a number of parameters that is larger than the number of parameters that are needed to get zero-training error. This is typically referred to as overparameterization. Indeed, it can be argued that over-parameterization is one of the key techniques that has led to the remarkable success of neural networks. However, there is still no theoretical account for its effectiveness. One very intriguing observation in this context is that over-parameterized networks with ReLU activations often exhibit better generalization error than smaller networks (Neyshabur et al., 2014, 2018; Novak et al., 2018). This somewhat counterintuitive observation suggests that over-parameterized networks have an inductive bias towards solutions with better generalization performance. Understanding this inductive bias is a major theoretical challenge and is a necessary step towards a full understanding of neural networks in practice. To better understand this phenomenon, it is crucial to be able to reason about optimization and generalization properties of over-parameterized networks with ReLU activations, which are trained with gradient based methods. However, in the current state of affairs, these are far from understood. In fact, there do not exist optimization or generalization guarantees for these networks even in very simple learning tasks such as the classic XOR problem.

In this paper, we focus on approaches to parallelizing stochastic gradient descent (SGD) wherein data is farmed out to a set of workers, the results of which, after a number of updates, are then combined at a central master node. Although such synchronized SGD approaches parallelize well in idealized computing environments, they often fail to realize their promised computational acceleration in practical settings. One cause is slow workers, termed stragglers, who can cause the fusion step at the master node to stall, which greatly slowing convergence. In many straggler mitigation approaches work completed by these nodes, while only partial, is discarded completely. In this paper, we propose an approach to parallelizing synchronous SGD that exploits the work completed by all workers. The central idea is to fix the computation time of each worker and then to combine distinct contributions of all workers. We provide a convergence analysis and optimize the combination function. Our numerical results demonstrate an improvement of several factors of magnitude in comparison to existing methods.

Recent analyses of certain gradient descent optimization methods have shown that performance can degrade in some settings - such as with stochasticity or implicit momentum. In deep reinforcement learning (Deep RL), such optimization methods are often used for training neural networks via the temporal difference error or policy gradient. As an agent improves over time, the optimization target changes and thus the loss landscape (and local optima) change. Due to the failure modes of those methods, the ideal choice of optimizer for Deep RL remains unclear. As such, we provide an empirical analysis of the effects that a wide range of gradient descent optimizers and their hyperparameters have on policy gradient methods, a subset of Deep RL algorithms, for benchmark continuous control tasks. We find that adaptive optimizers have a narrow window of effective learning rates, diverging in other cases, and that the effectiveness of momentum varies depending on the properties of the environment. Our analysis suggests that there is significant interplay between the dynamics of the environment and Deep RL algorithm properties which aren't necessarily accounted for by traditional adaptive gradient methods. We provide suggestions for optimal settings of current methods and further lines of research based on our findings.