Exploring with Sticky Mittens: Reinforcement Learning with Expert Interventions via Option Templates

Environments with sparse rewards and long horizons pose a significant challenge for current reinforcement learning algorithms. A key feature enabling humans to learn challenging control tasks is that they often receive expert intervention that enables them to understand the high-level structure of the task before mastering low-level control actions. We propose a framework for leveraging expert intervention to solve long-horizon reinforcement learning tasks. We consider option templates, which are specifications encoding a potential option that can be trained using reinforcement learning. We formulate expert intervention as allowing the agent to execute option templates before learning an implementation. This enables them to use an option, before committing costly resources to learning it. We evaluate our approach on three challenging reinforcement learning problems, showing that it outperforms state of-the-art approaches by an order of magnitude. Project website at https://sites.google.com/view/stickymittens

Bayes-Optimal Classifiers under Group Fairness

Machine learning algorithms are becoming integrated into more and more high-stakes decision-making processes, such as in social welfare issues. Due to the need of mitigating the potentially disparate impacts from algorithmic predictions, many approaches have been proposed in the emerging area of fair machine learning. However, the fundamental problem of characterizing Bayes-optimal classifiers under various group fairness constraints is not well understood as a theoretical benchmark. Based on the classical Neyman-Pearson argument (Neyman and Pearson, 1933; Shao, 2003) for optimal hypothesis testing, this paper provides a general framework for deriving Bayes-optimal classifiers under group fairness. This enables us to propose a group-based thresholding method that can directly control disparity, and more importantly, achieve an optimal fairness-accuracy tradeoff. These advantages are supported by experiments.

iDECODe: In-distribution Equivariance for Conformal Out-of-distribution Detection

Machine learning methods such as deep neural networks (DNNs), despite their success across different domains, are known to often generate incorrect predictions with high confidence on inputs outside their training distribution. The deployment of DNNs in safety-critical domains requires detection of out-of-distribution (OOD) data so that DNNs can abstain from making predictions on those. A number of methods have been recently developed for OOD detection, but there is still room for improvement. We propose the new method iDECODe, leveraging in-distribution equivariance for conformal OOD detection. It relies on a novel base non-conformity measure and a new aggregation method, used in the inductive conformal anomaly detection framework, thereby guaranteeing a bounded false detection rate. We demonstrate the efficacy of iDECODe by experiments on image and audio datasets, obtaining state-of-the-art results. We also show that iDECODe can detect adversarial examples.

Learning Augmentation Distributions using Transformed Risk Minimization

Adapting to the structure of data distributions (such as symmetry and transformation invariances) is an important challenge in machine learning. Invariances can be built into the learning process by architecture design, or by augmenting the dataset. Both require a priori knowledge about the exact nature of the symmetries. Absent this knowledge, practitioners resort to expensive and time-consuming tuning. To address this problem, we propose a new approach to learn distributions of augmentation transforms, in a new \emph{Transformed Risk Minimization} (TRM) framework. In addition to predictive models, we also optimize over transformations chosen from a hypothesis space. As an algorithmic framework, our TRM method is (1) efficient (jointly learns augmentations and models in a \emph{single training loop}), (2) modular (works with \emph{any} training algorithm), and (3) general (handles \emph{both discrete and continuous} augmentations). We theoretically compare TRM with standard risk minimization, and give a PAC-Bayes upper bound on its generalization error. We propose to optimize this bound over a rich augmentation space via a new parametrization over compositions of blocks, leading to the new \emph{Stochastic Compositional Augmentation Learning} (SCALE) algorithm. We compare SCALE experimentally with prior methods (Fast AutoAugment and Augerino) on CIFAR10/100, SVHN . Additionally, we show that SCALE can correctly learn certain symmetries in the data distribution (recovering rotations on rotated MNIST) and can also improve calibration of the learned model.

Solon: Communication-efficient Byzantine-resilient Distributed Training via Redundant Gradients

There has been a growing need to provide Byzantine-resilience in distributed model training. Existing robust distributed learning algorithms focus on developing sophisticated robust aggregators at the parameter servers, but pay less attention to balancing the communication cost and robustness. In this paper, we propose Solon, an algorithmic framework that exploits gradient redundancy to provide communication efficiency and Byzantine robustness simultaneously. Our theoretical analysis shows a fundamental trade-off among computational load, communication cost, and Byzantine robustness. We also develop a concrete algorithm to achieve the optimal trade-off, borrowing ideas from coding theory and sparse recovery. Empirical experiments on various datasets demonstrate that Solon provides significant speedups over existing methods to achieve the same accuracy, over 10 times faster than Bulyan and 80% faster than Draco. We also show that carefully designed Byzantine attacks break Signum and Bulyan, but do not affect the successful convergence of Solon.

Comparing Classes of Estimators: When does Gradient Descent Beat Ridge Regression in Linear Models?

Modern methods for learning from data depend on many tuning parameters, such as the stepsize for optimization methods, and the regularization strength for regularized learning methods. Since performance can depend strongly on these parameters, it is important to develop comparisons between \emph{classes of methods}, not just for particularly tuned ones. Here, we take aim to compare classes of estimators via the relative performance of the \emph{best method in the class}. This allows us to rigorously quantify the tuning sensitivity of learning algorithms. As an illustration, we investigate the statistical estimation performance of ridge regression with a uniform grid of regularization parameters, and of gradient descent iterates with a fixed stepsize, in the standard linear model with a random isotropic ground truth parameter. (1) For orthogonal designs, we find the \emph{exact minimax optimal classes of estimators}, showing they are equal to gradient descent with a polynomially decaying learning rate. We find the exact suboptimalities of ridge regression and gradient descent with a fixed stepsize, showing that they decay as either $1/k$ or $1/k^2$ for specific ranges of $k$ estimators. (2) For general designs with a large number of non-zero eigenvalues, we find that gradient descent outperforms ridge regression when the eigenvalues decay slowly, as a power law with exponent less than unity. If instead the eigenvalues decay quickly, as a power law with exponent greater than unity or exponentially, we find that ridge regression outperforms gradient descent. Our results highlight the importance of tuning parameters. In particular, while optimally tuned ridge regression is the best estimator in our case, it can be outperformed by gradient descent when both are restricted to being tuned over a finite regularization grid.

PAC Prediction Sets Under Covariate Shift

An important challenge facing modern machine learning is how to rigorously quantify the uncertainty of model predictions. Conveying uncertainty is especially important when there are changes to the underlying data distribution that might invalidate the predictive model. Yet, most existing uncertainty quantification algorithms break down in the presence of such shifts. We propose a novel approach that addresses this challenge by constructing \emph{probably approximately correct (PAC)} prediction sets in the presence of covariate shift. Our approach focuses on the setting where there is a covariate shift from the source distribution (where we have labeled training examples) to the target distribution (for which we want to quantify uncertainty). Our algorithm assumes given importance weights that encode how the probabilities of the training examples change under the covariate shift. In practice, importance weights typically need to be estimated; thus, we extend our algorithm to the setting where we are given confidence intervals for the importance weights rather than their true value. We demonstrate the effectiveness of our approach on various covariate shifts designed based on the DomainNet and ImageNet datasets.

Understanding Generalization in Adversarial Training via the Bias-Variance Decomposition

Adversarially trained models exhibit a large generalization gap: they can interpolate the training set even for large perturbation radii, but at the cost of large test error on clean samples. To investigate this gap, we decompose the test risk into its bias and variance components. We find that the bias increases monotonically with perturbation size and is the dominant term in the risk. Meanwhile, the variance is unimodal, peaking near the interpolation threshold for the training set. In contrast, we show that popular explanations for the generalization gap instead predict the variance to be monotonic, which leaves an unresolved mystery. We show that the same unimodal variance appears in a simple high-dimensional logistic regression problem, as well as for randomized smoothing. Overall, our results highlight the power of bias-variance decompositions in modern settings--by providing two measurements instead of one, they can rule out some theories and clarify others.

Sparse sketches with small inversion bias

For a tall $n\times d$ matrix $A$ and a random $m\times n$ sketching matrix $S$, the sketched estimate of the inverse covariance matrix $(A^\top A)^{-1}$ is typically biased: $E[(\tilde A^\top\tilde A)^{-1}]\ne(A^\top A)^{-1}$, where $\tilde A=SA$. This phenomenon, which we call inversion bias, arises, e.g., in statistics and distributed optimization, when averaging multiple independently constructed estimates of quantities that depend on the inverse covariance. We develop a framework for analyzing inversion bias, based on our proposed concept of an $(\epsilon,\delta)$-unbiased estimator for random matrices. We show that when the sketching matrix $S$ is dense and has i.i.d. sub-gaussian entries, then after simple rescaling, the estimator $(\frac m{m-d}\tilde A^\top\tilde A)^{-1}$ is $(\epsilon,\delta)$-unbiased for $(A^\top A)^{-1}$ with a sketch of size $m=O(d+\sqrt d/\epsilon)$. This implies that for $m=O(d)$, the inversion bias of this estimator is $O(1/\sqrt d)$, which is much smaller than the $\Theta(1)$ approximation error obtained as a consequence of the subspace embedding guarantee for sub-gaussian sketches. We then propose a new sketching technique, called LEverage Score Sparsified (LESS) embeddings, which uses ideas from both data-oblivious sparse embeddings as well as data-aware leverage-based row sampling methods, to get $\epsilon$ inversion bias for sketch size $m=O(d\log d+\sqrt d/\epsilon)$ in time $O(\text{nnz}(A)\log n+md^2)$, where nnz is the number of non-zeros. The key techniques enabling our analysis include an extension of a classical inequality of Bai and Silverstein for random quadratic forms, which we call the Restricted Bai-Silverstein inequality; and anti-concentration of the Binomial distribution via the Paley-Zygmund inequality, which we use to prove a lower bound showing that leverage score sampling sketches generally do not achieve small inversion bias.

What causes the test error? Going beyond bias-variance via ANOVA

Modern machine learning methods are often overparametrized, allowing adaptation to the data at a fine level. This can seem puzzling; in the worst case, such models do not need to generalize. This puzzle inspired a great amount of work, arguing when overparametrization reduces test error, in a phenomenon called "double descent". Recent work aimed to understand in greater depth why overparametrization is helpful for generalization. This leads to discovering the unimodality of variance as a function of the level of parametrization, and to decomposing the variance into that arising from label noise, initialization, and randomness in the training data to understand the sources of the error. In this work we develop a deeper understanding of this area. Specifically, we propose using the analysis of variance (ANOVA) to decompose the variance in the test error in a symmetric way, for studying the generalization performance of certain two-layer linear and non-linear networks. The advantage of the analysis of variance is that it reveals the effects of initialization, label noise, and training data more clearly than prior approaches. Moreover, we also study the monotonicity and unimodality of the variance components. While prior work studied the unimodality of the overall variance, we study the properties of each term in variance decomposition. One key insight is that in typical settings, the interaction between training samples and initialization can dominate the variance; surprisingly being larger than their marginal effect. Also, we characterize "phase transitions" where the variance changes from unimodal to monotone. On a technical level, we leverage advanced deterministic equivalent techniques for Haar random matrices, that---to our knowledge---have not yet been used in the area. We also verify our results in numerical simulations and on empirical data examples.