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Register a new userAssessing Generalization of SGD via Disagreement
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Added by
Vaishnavh Nagarajan
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We empirically show that the test error of deep networks can be estimated by simply training the same architecture on the same training set but with a different run of Stochastic Gradient Descent (SGD), and measuring the disagreement rate between the two networks on unlabeled test data. This builds on -- and is a stronger version of -- the observation in Nakkiran & Bansal 20, which requires the second run to be on an altogether fresh training set. We further theoretically show that this peculiar phenomenon arises from the emph{well-calibrated} nature of emph{ensembles} of SGD-trained models. This finding not only provides a simple empirical measure to directly predict the test error using unlabeled test data, but also establishes a new conceptual connection between generalization and calibration.
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Efficient exploration is a long-standing problem in sensorimotor learning. Major advances have been demonstrated in noise-free, non-stochastic domains such as video games and simulation. However, most of these formulations either get stuck in environments with stochastic dynamics or are too inefficient to be scalable to real robotics setups. In this paper, we propose a formulation for exploration inspired by the work in active learning literature. Specifically, we train an ensemble of dynamics models and incentivize the agent to explore such that the disagreement of those ensembles is maximized. This allows the agent to learn skills by exploring in a self-supervised manner without any external reward. Notably, we further leverage the disagreement objective to optimize the agents policy in a differentiable manner, without using reinforcement learning, which results in a sample-efficient exploration. We demonstrate the efficacy of this formulation across a variety of benchmark environments including stochastic-Atari, Mujoco and Unity. Finally, we implement our differentiable exploration on a real robot which learns to interact with objects completely from scratch. Project videos and code are at https://pathak22.github.io/exploration-by-disagreement/
Recent empirical work on SGD applied to over-parameterized deep learning has shown that most gradient components over epochs are quite small. Inspired by such observations, we rigorously study properties of noisy truncated SGD (NT-SGD), a noisy gradient descent algorithm that truncates (hard thresholds) the majority of small gradient components to zeros and then adds Gaussian noise to all components. Considering non-convex smooth problems, we first establish the rate of convergence of NT-SGD in terms of empirical gradient norms, and show the rate to be of the same order as the vanilla SGD. Further, we prove that NT-SGD can provably escape from saddle points and requires less noise compared to previous related work. We also establish a generalization bound for NT-SGD using uniform stability based on discretized generalized Langevin dynamics. Our experiments on MNIST (VGG-5) and CIFAR-10 (ResNet-18) demonstrate that NT-SGD matches the speed and accuracy of vanilla SGD, and can successfully escape sharp minima while having better theoretical properties.
Distributional shift is one of the major obstacles when transferring machine learning prediction systems from the lab to the real world. To tackle this problem, we assume that variation across training domains is representative of the variation we might encounter at test time, but also that shifts at test time may be more extreme in magnitude. In particular, we show that reducing differences in risk across training domains can reduce a models sensitivity to a wide range of extreme distributional shifts, including the challenging setting where the input contains both causal and anti-causal elements. We motivate this approach, Risk Extrapolation (REx), as a form of robust optimization over a perturbation set of extrapolated domains (MM-REx), and propose a penalty on the variance of training risks (V-REx) as a simpler variant. We prove that variants of REx can recover the causal mechanisms of the targets, while also providing some robustness to changes in the input distribution (covariate shift). By appropriately trading-off robustness to causally induced distributional shifts and covariate shift, REx is able to outperform alternative methods such as Invariant Risk Minimization in situations where these types of shift co-occur.
We study local SGD (also known as parallel SGD and federated averaging), a natural and frequently used stochastic distributed optimization method. Its theoretical foundations are currently lacking and we highlight how all existing error guarantees in the convex setting are dominated by a simple baseline, minibatch SGD. (1) For quadratic objectives we prove that local SGD strictly dominates minibatch SGD and that accelerated local SGD is minimax optimal for quadratics; (2) For general convex objectives we provide the first guarantee that at least sometimes improves over minibatch SGD; (3) We show that indeed local SGD does not dominate minibatch SGD by presenting a lower bound on the performance of local SGD that is worse than the minibatch SGD guarantee.
Recurrent Neural Networks (RNNs) are among the most popular models in sequential data analysis. Yet, in the foundational PAC learning language, what concept class can it learn? Moreover, how can the same recurrent unit simultaneously learn functions from different input tokens to different output tokens, without affecting each other? Existing generalization bounds for RNN scale exponentially with the input length, significantly limiting their practical implications. In this paper, we show using the vanilla stochastic gradient descent (SGD), RNN can actually learn some notable concept class efficiently, meaning that both time and sample complexity scale polynomially in the input length (or almost polynomially, depending on the concept). This concept class at least includes functions where each output token is generated from inputs of earlier tokens using a smooth two-layer neural network.
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