arXiv subjects

Edgar Dobriban

Publications and source records attributed to Edgar Dobriban.

At least 73 records · Page 4Linked to original sources

arXiv2023

SE(3)-Equivariant Attention Networks for Shape Reconstruction in Function Space

We propose a method for 3D shape reconstruction from unoriented point clouds. Our method consists of a novel SE(3)-equivariant coordinate-based network (TF-ONet), that parametrizes the occupancy field of the shape and respects the inherent symmetries of the problem. In contrast to previous shape reconstruction methods that align the input to a regular grid, we operate directly on the irregular point cloud. Our architecture leverages equivariant attention layers that operate on local tokens. This mechanism enables local shape modelling, a crucial property for scalability to large scenes. Given an unoriented, sparse, noisy point cloud as input, we produce equivariant features for each point. These serve as keys and values for the subsequent equivariant cross-attention blocks that parametrize the occupancy field. By querying an arbitrary point in space, we predict its occupancy score. We show that our method outperforms previous SO(3)-equivariant methods, as well as non-equivariant methods trained on SO(3)-augmented datasets. More importantly, local modelling together with SE(3)-equivariance create an ideal setting for SE(3) scene reconstruction. We show that by training only on single, aligned objects and without any pre-segmentation, we can reconstruct novel scenes containing arbitrarily many objects in random poses without any performance loss.

cs.CV↗

arXiv2022

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

Long horizon robot learning tasks with sparse rewards 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 \emph{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 two orders of magnitude. Videos of trained agents and our code can be found at: https://sites.google.com/view/stickymittens

cs.LG↗

arXiv2022

Distributed linear regression by averaging

Distributed statistical learning problems arise commonly when dealing with large datasets. In this setup, datasets are partitioned over machines, which compute locally, and communicate short messages. Communication is often the bottleneck. In this paper, we study one-step and iterative weighted parameter averaging in statistical linear models under data parallelism. We do linear regression on each machine, send the results to a central server, and take a weighted average of the parameters. Optionally, we iterate, sending back the weighted average and doing local ridge regressions centered at it. How does this work compared to doing linear regression on the full data? Here we study the performance loss in estimation, test error, and confidence interval length in high dimensions, where the number of parameters is comparable to the training data size. We find the performance loss in one-step weighted averaging, and also give results for iterative averaging. We also find that different problems are affected differently by the distributed framework. Estimation error and confidence interval length increase a lot, while prediction error increases much less. We rely on recent results from random matrix theory, where we develop a new calculus of deterministic equivalents as a tool of broader interest.

math.ST↗

arXiv2022

PAC Prediction Sets for Meta-Learning

Uncertainty quantification is a key component of machine learning models targeted at safety-critical systems such as in healthcare or autonomous vehicles. We study this problem in the context of meta learning, where the goal is to quickly adapt a predictor to new tasks. In particular, we propose a novel algorithm to construct \emph{PAC prediction sets}, which capture uncertainty via sets of labels, that can be adapted to new tasks with only a few training examples. These prediction sets satisfy an extension of the typical PAC guarantee to the meta learning setting; in particular, the PAC guarantee holds with high probability over future tasks. We demonstrate the efficacy of our approach on four datasets across three application domains: mini-ImageNet and CIFAR10-C in the visual domain, FewRel in the language domain, and the CDC Heart Dataset in the medical domain. In particular, our prediction sets satisfy the PAC guarantee while having smaller size compared to other baselines that also satisfy this guarantee.

cs.LG↗

arXiv2022

Collaborative Learning of Discrete Distributions under Heterogeneity and Communication Constraints

In modern machine learning, users often have to collaborate to learn the distribution of the data. Communication can be a significant bottleneck. Prior work has studied homogeneous users -- i.e., whose data follow the same discrete distribution -- and has provided optimal communication-efficient methods for estimating that distribution. However, these methods rely heavily on homogeneity, and are less applicable in the common case when users' discrete distributions are heterogeneous. Here we consider a natural and tractable model of heterogeneity, where users' discrete distributions only vary sparsely, on a small number of entries. We propose a novel two-stage method named SHIFT: First, the users collaborate by communicating with the server to learn a central distribution; relying on methods from robust statistics. Then, the learned central distribution is fine-tuned to estimate their respective individual distribution. We show that SHIFT is minimax optimal in our model of heterogeneity and under communication constraints. Further, we provide experimental results using both synthetic data and $n$-gram frequency estimation in the text domain, which corroborate its efficiency.

stat.ML↗

arXiv2022

Pursuit of a Discriminative Representation for Multiple Subspaces via Sequential Games

We consider the problem of learning discriminative representations for data in a high-dimensional space with distribution supported on or around multiple low-dimensional linear subspaces. That is, we wish to compute a linear injective map of the data such that the features lie on multiple orthogonal subspaces. Instead of treating this learning problem using multiple PCAs, we cast it as a sequential game using the closed-loop transcription (CTRL) framework recently proposed for learning discriminative and generative representations for general low-dimensional submanifolds. We prove that the equilibrium solutions to the game indeed give correct representations. Our approach unifies classical methods of learning subspaces with modern deep learning practice, by showing that subspace learning problems may be provably solved using the modern toolkit of representation learning. In addition, our work provides the first theoretical justification for the CTRL framework, in the important case of linear subspaces. We support our theoretical findings with compelling empirical evidence. We also generalize the sequential game formulation to more general representation learning problems. Our code, including methods for easy reproduction of experimental results, is publically available on GitHub.

stat.ML↗

arXiv2022

Implicit Regularization and Convergence for Weight Normalization

Normalization methods such as batch [Ioffe and Szegedy, 2015], weight [Salimansand Kingma, 2016], instance [Ulyanov et al., 2016], and layer normalization [Baet al., 2016] have been widely used in modern machine learning. Here, we study the weight normalization (WN) method [Salimans and Kingma, 2016] and a variant called reparametrized projected gradient descent (rPGD) for overparametrized least-squares regression. WN and rPGD reparametrize the weights with a scale g and a unit vector w and thus the objective function becomes non-convex. We show that this non-convex formulation has beneficial regularization effects compared to gradient descent on the original objective. These methods adaptively regularize the weights and converge close to the minimum l2 norm solution, even for initializations far from zero. For certain stepsizes of g and w , we show that they can converge close to the minimum norm solution. This is different from the behavior of gradient descent, which converges to the minimum norm solution only when started at a point in the range space of the feature matrix, and is thus more sensitive to initialization.

cs.LG↗

arXiv2022

Memory Classifiers: Two-stage Classification for Robustness in Machine Learning

The performance of machine learning models can significantly degrade under distribution shifts of the data. We propose a new method for classification which can improve robustness to distribution shifts, by combining expert knowledge about the ``high-level" structure of the data with standard classifiers. Specifically, we introduce two-stage classifiers called memory classifiers. First, these identify prototypical data points -- memories -- to cluster the training data. This step is based on features designed with expert guidance; for instance, for image data they can be extracted using digital image processing algorithms. Then, within each cluster, we learn local classifiers based on finer discriminating features, via standard models like deep neural networks. We establish generalization bounds for memory classifiers. We illustrate in experiments that they can improve generalization and robustness to distribution shifts on image datasets. We show improvements which push beyond standard data augmentation techniques.

cs.LG↗

arXiv2022

Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces

We introduce a unified framework for group equivariant networks on homogeneous spaces derived from a Fourier perspective. We consider tensor-valued feature fields, before and after a convolutional layer. We present a unified derivation of kernels via the Fourier domain by leveraging the sparsity of Fourier coefficients of the lifted feature fields. The sparsity emerges when the stabilizer subgroup of the homogeneous space is a compact Lie group. We further introduce a nonlinear activation, via an elementwise nonlinearity on the regular representation after lifting and projecting back to the field through an equivariant convolution. We show that other methods treating features as the Fourier coefficients in the stabilizer subgroup are special cases of our activation. Experiments on $SO(3)$ and $SE(3)$ show state-of-the-art performance in spherical vector field regression, point cloud classification, and molecular completion.

cs.CV↗

arXiv2022

PAC-Wrap: Semi-Supervised PAC Anomaly Detection

Anomaly detection is essential for preventing hazardous outcomes for safety-critical applications like autonomous driving. Given their safety-criticality, these applications benefit from provable bounds on various errors in anomaly detection. To achieve this goal in the semi-supervised setting, we propose to provide Probably Approximately Correct (PAC) guarantees on the false negative and false positive detection rates for anomaly detection algorithms. Our method (PAC-Wrap) can wrap around virtually any existing semi-supervised and unsupervised anomaly detection method, endowing it with rigorous guarantees. Our experiments with various anomaly detectors and datasets indicate that PAC-Wrap is broadly effective.

cs.LG↗

arXiv2022

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

Methods for learning from data depend on various types of tuning parameters, such as penalization strength or step size. Since performance can depend strongly on these parameters, it is important to compare classes of estimators-by considering prescribed finite sets of tuning parameters-not just particularly tuned methods. In this work, we investigate classes of methods via the relative performance of the best method in the class. We consider the central problem of linear regression-with a random isotropic ground truth-and investigate the estimation performance of two fundamental methods, gradient descent and ridge regression. We unveil the following phenomena. (1) For general designs, constant stepsize gradient descent outperforms ridge regression when the eigenvalues of the empirical data covariance matrix 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 show that ridge regression outperforms gradient descent. (2) For orthogonal designs, we compute the exact minimax optimal class of estimators (achieving min-max-min optimality), showing it is equivalent to gradient descent with decaying learning rate. We find the sub-optimality of ridge regression and gradient descent with constant step size. Our results highlight that statistical performance can depend strongly on tuning parameters. In particular, while optimally tuned ridge regression is the best estimator in our setting, it can be outperformed by gradient descent by an arbitrary/unbounded amount when both methods are only tuned over finitely many regularization parameters.

math.ST↗

arXiv2022

Fair Bayes-Optimal Classifiers Under Predictive Parity

Increasing concerns about disparate effects of AI have motivated a great deal of work on fair machine learning. Existing works mainly focus on independence- and separation-based measures (e.g., demographic parity, equality of opportunity, equalized odds), while sufficiency-based measures such as predictive parity are much less studied. This paper considers predictive parity, which requires equalizing the probability of success given a positive prediction among different protected groups. We prove that, if the overall performances of different groups vary only moderately, all fair Bayes-optimal classifiers under predictive parity are group-wise thresholding rules. Perhaps surprisingly, this may not hold if group performance levels vary widely; in this case we find that predictive parity among protected groups may lead to within-group unfairness. We then propose an algorithm we call FairBayes-DPP, aiming to ensure predictive parity when our condition is satisfied. FairBayes-DPP is an adaptive thresholding algorithm that aims to achieve predictive parity, while also seeking to maximize test accuracy. We provide supporting experiments conducted on synthetic and empirical data.

stat.ML↗

arXiv2022

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. We demonstrate the effectiveness of our approach on covariate shifts based on DomainNet and ImageNet. Our algorithm satisfies the PAC constraint, and gives prediction sets with the smallest average normalized size among approaches that always satisfy the PAC constraint.

cs.LG↗

arXiv2022

Provable tradeoffs in adversarially robust classification

It is well known that machine learning methods can be vulnerable to adversarially-chosen perturbations of their inputs. Despite significant progress in the area, foundational open problems remain. In this paper, we address several key questions. We derive exact and approximate Bayes-optimal robust classifiers for the important setting of two- and three-class Gaussian classification problems with arbitrary imbalance, for $\ell_2$ and $\ell_\infty$ adversaries. In contrast to classical Bayes-optimal classifiers, determining the optimal decisions here cannot be made pointwise and new theoretical approaches are needed. We develop and leverage new tools, including recent breakthroughs from probability theory on robust isoperimetry, which, to our knowledge, have not yet been used in the area. Our results reveal fundamental tradeoffs between standard and robust accuracy that grow when data is imbalanced. We also show further results, including an analysis of classification calibration for convex losses in certain models, and finite sample rates for the robust risk.

cs.LG↗

arXiv2022

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.

cs.LG↗

arXiv2021

Consistency of invariance-based randomization tests

Invariance-based randomization tests -- such as permutation tests, rotation tests, or sign changes -- are an important and widely used class of statistical methods. They allow drawing inferences under weak assumptions on the data distribution. Most work focuses on their type I error control properties, while their consistency properties are much less understood. We develop a general framework and a set of results on the consistency of invariance-based randomization tests in signal-plus-noise models. Our framework is grounded in the deep mathematical area of representation theory. We allow the transforms to be general compact topological groups, such as rotation groups, acting by general linear group representations. We study test statistics with a generalized sub-additivity property. We apply our framework to a number of fundamental and highly important problems in statistics, including sparse vector detection, testing for low-rank matrices in noise, sparse detection in linear regression, and two-sample testing. Comparing with minimax lower bounds, we find perhaps surprisingly that in some cases, randomization tests detect signals at the minimax optimal rate.

math.ST↗

arXiv2021

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.

cs.LG↗

arXiv2021

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 $(ε,δ)$-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 $(ε,δ)$-unbiased for $(A^\top A)^{-1}$ with a sketch of size $m=O(d+\sqrt d/ε)$. This implies that for $m=O(d)$, the inversion bias of this estimator is $O(1/\sqrt d)$, which is much smaller than the $Θ(1)$ 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 $ε$ inversion bias for sketch size $m=O(d\log d+\sqrt d/ε)$ 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.

cs.DS↗