Poster Session 1


Variational Selective Autoencoder: Learning from Partially-Observed Heterogeneous Data

Yu Gong · Hossein Hajimirsadeghi · Jiawei He · Thibaut Durand · Greg Mori

Learning from heterogeneous data poses challenges such as combining data from various sources and of different types. Meanwhile, heterogeneous data are often associated with missingness in real-world applications due to heterogeneity and noise of input sources. In this work, we propose the variational selective autoencoder (VSAE), a general framework to learn representations from partially-observed heterogeneous data. VSAE learns the latent dependencies in heterogeneous data by modeling the joint distribution of observed data, unobserved data, and the imputation mask which represents how the data are missing. It results in a unified model for various downstream tasks including data generation and imputation. Evaluation on both low-dimensional and high-dimensional heterogeneous datasets for these two tasks shows improvement over state-of-the-art models.

Does Invariant Risk Minimization Capture Invariance?

Pritish Kamath · Akilesh Tangella · Danica Sutherland · Nathan Srebro

We show that the Invariant Risk Minimization (IRM) formulation of Arjovsky et al. (2019) can fail to capture "natural" invariances, at least when used in its practical "linear" form, and even on very simple problems which directly follow the motivating examples for IRM. This can lead to worse generalization on new environments, even when compared to unconstrained ERM. The issue stems from a significant gap between the linear variant (as in their concrete method IRMv1) and the full non-linear IRM formulation. Additionally, even when capturing the "right" invariances, we show that it is possible for IRM to learn a sub-optimal predictor, due to the loss function not being invariant across environments. The issues arise even when measuring invariance on the population distributions, but are exacerbated by the fact that IRM is extremely fragile to sampling.

Comparing the Value of Labeled and Unlabeled Data in Method-of-Moments Latent Variable Estimation

Mayee Chen · Benjamin Cohen-Wang · Stephen Mussmann · Frederic Sala · Christopher Re

Labeling data for modern machine learning is expensive and time-consuming. Latent variable models can be used to infer labels from weaker, easier-to-acquire sources operating on unlabeled data. Such models can also be trained using labeled data, presenting a key question: should a user invest in few labeled or many unlabeled points? We answer this via a framework centered on model misspecification in method-of-moments latent variable estimation. Our core result is a bias-variance decomposition of the generalization error, which shows that the unlabeled-only approach incurs additional bias under misspecification. We then introduce a correction that provably removes this bias in certain cases. We apply our decomposition framework to three scenarios---well-specified, misspecified, and corrected models---to 1) choose between labeled and unlabeled data and 2) learn from their combination. We observe theoretically and with synthetic experiments that for well-specified models, labeled points are worth a constant factor more than unlabeled points. With misspecification, however, their relative value is higher due to the additional bias but can be reduced with correction. We also apply our approach to study real-world weak supervision techniques for dataset construction.

Simultaneously Reconciled Quantile Forecasting of Hierarchically Related Time Series

Xing Han · Sambarta Dasgupta · Joydeep Ghosh

Many real-life applications involve simultaneously forecasting multiple time series that are hierarchically related via aggregation or disaggregation operations. For instance, commercial organizations often want to forecast inventories simultaneously at store, city, and state levels for resource planning purposes. In such applications, it is important that the forecasts, in addition to being reasonably accurate, are also consistent w.r.t one another. Although forecasting such hierarchical time series has been pursued by economists and data scientists, the current state-of-the-art models use strong assumptions, e.g., all forecasts being unbiased estimates, noise distribution being Gaussian. Besides, state-of-the-art models have not harnessed the power of modern nonlinear models, especially ones based on deep learning. In this paper, we propose using a flexible nonlinear model that optimizes quantile regression loss coupled with suitable regularization terms to maintain the consistency of forecasts across hierarchies. The theoretical framework introduced herein can be applied to any forecasting model with an underlying differentiable loss function. A proof of optimality of our proposed method is also provided. Simulation studies over a range of datasets highlight the efficacy of our approach.

A Dynamical View on Optimization Algorithms of Overparameterized Neural Networks

Zhiqi Bu · Shiyun Xu · Kan Chen

When equipped with efficient optimization algorithms, the over-parameterized neural networks have demonstrated high level of performance even though the loss function is non-convex and non-smooth. While many works have been focusing on understanding the loss dynamics by training neural networks with the gradient descent (GD), in this work, we consider a broad class of optimization algorithms that are commonly used in practice. For example, we show from a dynamical system perspective that the Heavy Ball (HB) method can converge to global minimum on mean squared error (MSE) at a linear rate (similar to GD); however, the Nesterov accelerated gradient descent (NAG) may only converge to global minimum sublinearly.

Our results rely on the connection between neural tangent kernel (NTK) and finitely-wide over-parameterized neural networks with ReLU activation, which leads to analyzing the limiting ordinary differential equations (ODE) for optimization algorithms. We show that, optimizing the non-convex loss over the weights corresponds to optimizing some strongly convex loss over the prediction error. As a consequence, we can leverage the classical convex optimization theory to understand the convergence behavior of neural networks. We believe our approach can also be extended to other optimization algorithms and network architectures.

Inductive Mutual Information Estimation: A Convex Maximum-Entropy Copula Approach

Yves-Laurent Kom Samo

We propose a novel estimator of the mutual information between two ordinal vectors $x$ and $y$. Our approach is inductive (as opposed to deductive) in that it depends on the data generating distribution solely through some nonparametric properties revealing associations in the data, and does not require having enough data to fully characterize the true joint distributions $P_{x, y}$. Specifically, our approach consists of (i) noting that $I\left(y; x\right) = I\left(u_y; u_x\right)$ where $u_y$ and $u_x$ are the \emph{copula-uniform dual representations} of $y$ and $x$ (i.e. their images under the probability integral transform), and (ii) estimating the copula entropies $h\left(u_y\right)$, $h\left(u_x\right)$ and $h\left(u_y, u_x\right)$ by solving a maximum-entropy problem over the space of copula densities under a constraint of the type $\bm{\alpha}_m = E\left[\phi_m(u_y, u_x)\right]$. We prove that, so long as the constraint is feasible, this problem admits a unique solution, it is in the exponential family, and it can be learned by solving a convex optimization problem. The resulting estimator, which we denote MIND, is marginal-invariant, always non-negative, unbounded for any sample size $n$, consistent, has MSE rate $O(1/n)$, and is more data-efficient than competing approaches.

Bayesian Active Learning by Soft Mean Objective Cost of Uncertainty

Guang Zhao · Edward Dougherty · Byung-Jun Yoon · Francis J. Alexander · Xiaoning Qian

To achieve label efficiency for training supervised learning models, pool-based active learning sequentially selects samples from a set of candidates as queries to label by optimizing an acquisition function. One category of existing methods adopts one-step-look-ahead strategies based on acquisition functions tailored with the learning objectives, for example based on the expected loss reduction (ELR) or the mean objective cost of uncertainty (MOCU) proposed recently. These active learning methods are optimal with the maximum classification error reduction when one considers a single query. However, it is well-known that there is no performance guarantee in the long run for these myopic methods. In this paper, we show that these methods are not guaranteed to converge to the optimal classifier of the true model because MOCU is not strictly concave. Moreover, we suggest a strictly concave approximation of MOCU---Soft MOCU---that can be used to define an acquisition function to guide Bayesian active learning with theoretical convergence guarantee. For training Bayesian classifiers with both synthetic and real-world data, our experiments demonstrate the superior performance of active learning by Soft MOCU compared to other existing methods.

Best Student Paper
Matérn Gaussian Processes on Graphs

Viacheslav Borovitskiy · Iskander Azangulov · Alexander Terenin · Peter Mostowsky · Marc Deisenroth · Nicolas Durrande

Gaussian processes are a versatile framework for learning unknown functions in a manner that permits one to utilize prior information about their properties. Although many different Gaussian process models are readily available when the input space is Euclidean, the choice is much more limited for Gaussian processes whose input space is an undirected graph. In this work, we leverage the stochastic partial differential equation characterization of Matérn Gaussian processes—a widely-used model class in the Euclidean setting—to study their analog for undirected graphs. We show that the resulting Gaussian processes inherit various attractive properties of their Euclidean and Riemannian analogs and provide techniques that allow them to be trained using standard methods, such as inducing points. This enables graph Matérn Gaussian processes to be employed in mini-batch and non-conjugate settings, thereby making them more accessible to practitioners and easier to deploy within larger learning frameworks.

A Hybrid Approximation to the Marginal Likelihood

Eric Chuu · Debdeep Pati · Anirban Bhattacharya

Computing the marginal likelihood or evidence is one of the core challenges in Bayesian analysis. While there are many established methods for estimating this quantity, they predominantly rely on using a large number of posterior samples obtained from a Markov Chain Monte Carlo (MCMC) algorithm. As the dimension of the parameter space increases, however, many of these methods become prohibitively slow and potentially inaccurate. In this paper, we propose a novel method in which we use the MCMC samples to learn a high probability partition of the parameter space and then form a deterministic approximation over each of these partition sets. This two-step procedure, which constitutes both a probabilistic and a deterministic component, is termed a Hybrid approximation to the marginal likelihood. We demonstrate its versatility in a plethora of examples with varying dimension and sample size, and we also highlight the Hybrid approximation's effectiveness in situations where there is either a limited number or only approximate MCMC samples available.

Fundamental Limits of Ridge-Regularized Empirical Risk Minimization in High Dimensions

Hossein Taheri · Ramtin Pedarsani · Christos Thrampoulidis

Despite the popularity of Empirical Risk Minimization (ERM) algorithms, a theory that explains their statistical properties in modern high-dimensional regimes is only recently emerging. We characterize for the first time the fundamental limits on the statistical accuracy of convex ridge-regularized ERM for inference in high-dimensional generalized linear models. For a stylized setting with Gaussian features and problem dimensions that grow large at a proportional rate, we start with sharp performance characterizations and then derive tight lower bounds on the estimation and prediction error. Our bounds provably hold over a wide class of loss functions, and, for any value of the regularization parameter and of the sampling ratio. Our precise analysis has several attributes. First, it leads to a recipe for optimally tuning the loss function and the regularization parameter. Second, it allows to precisely quantify the sub-optimality of popular heuristic choices, such as optimally-tuned least-squares. Third, we use the bounds to precisely assess the merits of ridge-regularization as a function of the sampling ratio. Our bounds are expressed in terms of the Fisher Information of random variables that are simple functions of the data distribution, thus making ties to corresponding bounds in classical statistics.

Hogwild! over Distributed Local Data Sets with Linearly Increasing Mini-Batch Sizes

Nhuong Nguyen · Toan Nguyen · PHUONG HA NGUYEN · Quoc Tran-Dinh · Lam Nguyen · Marten van Dijk

Hogwild! implements asynchronous Stochastic Gradient Descent (SGD) where multiple threads in parallel access a common repository containing training data, perform SGD iterations and update shared state that represents a jointly learned (global) model. We consider big data analysis where training data is distributed among local data sets in a heterogeneous way -- and we wish to move SGD computations to local compute nodes where local data resides. The results of these local SGD computations are aggregated by a central ``aggregator'' which mimics Hogwild!. We show how local compute nodes can start choosing small mini-batch sizes which increase to larger ones in order to reduce communication cost (round interaction with the aggregator). We improve state-of-the-art literature and show O(K^{0.5}) communication rounds for heterogeneous data for strongly convex problems, where K is the total number of gradient computations across all local compute nodes. For our scheme, we prove a tight and novel non-trivial convergence analysis for strongly convex problems for heterogeneous data which does not use the bounded gradient assumption as seen in many existing publications. The tightness is a consequence of our proofs for lower and upper bounds of the convergence rate, which show a constant factor difference. We show experimental results for plain convex and non-convex problems for biased (i.e., heterogeneous) and unbiased local data sets.

Detection and Defense of Topological Adversarial Attacks on Graphs

Yingxue Zhang · Florence Regol · Soumyasundar Pal · Sakif Khan · Liheng Ma · Mark Coates

Graph neural network (GNN) models achieve superior performance when classifying nodes in graph-structured data. Given that state-of-the-art GNNs share many similarities with their CNN cousins and that CNNs suffer adversarial vulnerabilities, there has also been interest in exploring analogous vulnerabilities in GNNs. Indeed, recent work has demonstrated that node classification performance of several graph models, including the popular graph convolution network (GCN) model, can be severely degraded through adversarial perturbations to the graph structure and the node features. In this work, we take a first step towards detecting adversarial attacks against graph models. We first propose a straightforward single node threshold test for detecting nodes subject to targeted attacks. Subsequently, we describe a kernel-based two-sample test for detecting whether a given subset of nodes within a graph has been maliciously corrupted. The efficacy of our algorithms is established via thorough experiments using commonly used node classification benchmark datasets. We also illustrate the potential practical benefit of our detection method by demonstrating its application to a real-world Bitcoin transaction network.

Efficient Interpolation of Density Estimators

Paxton Turner · Jingbo Liu · Philippe Rigollet

We study the problem of space and time efficient evaluation of a nonparametric estimator that approximates an unknown density. In the regime where consistent estimation is possible, we use a piecewise multivariate polynomial interpolation scheme to give a computationally efficient construction that converts the original estimator to a new estimator that can be queried efficiently and has low space requirements, all without adversely deteriorating the original approximation quality. Our result gives a new statistical perspective on the problem of fast evaluation of kernel density estimators in the presence of underlying smoothness. As a corollary, we give a succinct derivation of a classical result of Kolmogorov---Tikhomirov on the metric entropy of Holder classes of smooth functions.

On the Convergence of Gradient Descent in GANs: MMD GAN As a Gradient Flow

Youssef Mroueh · Truyen Nguyen

We consider the maximum mean discrepancy MMD GAN problem and propose a parametric kernelized gradient flow that mimics the min-max game in gradient regularized MMD GAN. We show that this flow provides a descent direction minimizing the MMD on a statistical manifold of probability distributions. We then derive an explicit condition which ensures that gradient descent on the parameter space of the generator in gradient regularized MMD GAN is globally convergent to the target distribution. Under this condition , we give non asymptotic convergence results for MMD GAN. Another contribution of this paper is the introduction of a dynamic formulation of a regularization of MMD and demonstrating that the parametric kernelized descent for MMD is the gradient flow of this functional with respect to the new Riemannian structure. Our obtained theoretical result allows ones to treat gradient flows for quite general functionals and thus has potential applications to other types of variational inferences on a statistical manifold beyond GANs. Finally, numerical experiments suggest that our parametric kernelized gradient flow stabilizes GAN training and guarantees convergence.

Deep Spectral Ranking

ILKAY YILDIZ · Jennifer Dy · Deniz Erdogmus · Susan Ostmo · J. Peter Campbell · Michael F. Chiang · Stratis Ioannidis

Learning from ranking observations arises in many domains, and siamese deep neural networks have shown excellent inference performance in this setting. However, SGD does not scale well, as an epoch grows exponentially with the ranking observation size. We show that a spectral algorithm can be combined with deep learning methods to significantly accelerate training. We combine a spectral estimate of Plackett-Luce ranking scores with a deep model via the Alternating Directions Method of Multipliers with a Kullback-Leibler proximal penalty. Compared to a state-of-the-art siamese network, our algorithms are up to 175 times faster and attain better predictions by up to 26% Top-1 Accuracy and 6% Kendall-Tau correlation over five real-life ranking datasets.

Designing Transportable Experiments Under S-admissability

My Phan · David Arbour · Drew Dimmery · Anup Rao

We consider the problem of designing a randomized experiment on a source population to estimate the Average Treatment Effect (ATE) on a target population. We propose a novel approach which explicitly considers the target when designing the experiment on the source. Under the covariate shift assumption, we design an unbiased importance-weighted estimator for the target population's ATE. To reduce the variance of our estimator, we design a covariate balance condition (Target Balance) between the treatment and control groups based on the target population. We show that Target Balance achieves a higher variance reduction asymptotically than methods that do not consider the target population during the design phase. Our experiments illustrate that Target Balance reduces the variance even for small sample sizes.

Semi-Supervised Learning with Meta-Gradient

Taihong Xiao · Xin-Yu Zhang · Haolin Jia · Ming-Ming Cheng · Ming-Hsuan Yang

In this work, we propose a simple yet effective meta-learning algorithm in semi-supervised learning. We notice that most existing consistency-based approaches suffer from overfitting and limited model generalization ability, especially when training with only a small number of labeled data. To alleviate this issue, we propose a learn-to-generalize regularization term by utilizing the label information and optimize the problem in a meta-learning fashion. Specifically, we seek the pseudo labels of the unlabeled data so that the model can generalize well on the labeled data, which is formulated as a nested optimization problem. We address this problem using the meta-gradient that bridges between the pseudo label and the regularization term. In addition, we introduce a simple first-order approximation to avoid computing higher-order derivatives and provide theoretic convergence analysis. Extensive evaluations on the SVHN, CIFAR, and ImageNet datasets demonstrate that the proposed algorithm performs favorably against state-of-the-art methods.

Have We Learned to Explain?: How Interpretability Methods Can Learn to Encode Predictions in their Interpretations.

Neil Jethani · Mukund Sudarshan · Yindalon Aphinyanaphongs · Rajesh Ranganath

While the need for interpretable machine learning has been established, many common approaches are slow, lack fidelity, or hard to evaluate. Amortized explanation methods reduce the cost of providing interpretations by learning a global selector model that returns feature importances for a single instance of data. The selector model is trained to optimize the fidelity of the interpretations, as evaluated by a predictor model for the target. Popular methods learn the selector and predictor model in concert, which we show allows predictions to be encoded within interpretations. We introduce EVAL-X as a method to quantitatively evaluate interpretations and REAL-X as an amortized explanation method, which learn a predictor model that approximates the true data generating distribution given any subset of the input. We show EVAL-X can detect when predictions are encoded in interpretations and show the advantages of REAL-X through quantitative and radiologist evaluation.

Sample Complexity Bounds for Two Timescale Value-based Reinforcement Learning Algorithms

Tengyu Xu · Yingbin Liang

Two timescale stochastic approximation (SA) has been widely used in value-based reinforcement learning algorithms. In the policy evaluation setting, it can model the linear and nonlinear temporal difference learning with gradient correction (TDC) algorithms as linear SA and nonlinear SA, respectively. In the policy optimization setting, two timescale nonlinear SA can also model the greedy gradient-Q (Greedy-GQ) algorithm. In previous studies, the non-asymptotic analysis of linear TDC and Greedy-GQ has been studied in the Markovian setting, with single-sample update at each iteration. For the nonlinear TDC algorithm, only the asymptotic convergence has been established. In this paper, we study the non-asymptotic convergence rate of two time-scale linear and nonlinear TDC and Greedy-GQ under Markovian sampling and with mini-batch data for each update. For linear TDC, we provide a novel non-asymptotic analysis and our sample complexity result achieves the complexity $\mathcal{O}(\epsilon^{-1}\log(1/\epsilon))$. For nonlinear TDC and Greedy-GQ, we show that both algorithms attain $\epsilon$-accurate stationary solution with sample complexity $\mathcal{O}(\epsilon^{-2})$. It is the first time that non-asymptotic convergence result has been established for nonlinear TDC and our result for Greedy-GQ outperforms previous result orderwisely by a factor of $\mathcal{O}(\epsilon^{-1}\log(1/\epsilon))$.

Reinforcement Learning for Mean Field Games with Strategic Complementarities

Kiyeob Lee · Desik Rengarajan · Dileep Kalathil · Srinivas Shakkottai

Mean Field Games (MFG) are the class of games with a very large number of agents and the standard equilibrium concept is a Mean Field Equilibrium (MFE). Algorithms for learning MFE in dynamic MFGs are unknown in general. Our focus is on an important subclass that possess a monotonicity property called Strategic Complementarities (MFG-SC). We introduce a natural refinement to the equilibrium concept that we call Trembling-Hand-Perfect MFE (T-MFE), which allows agents to employ a measure of randomization while accounting for the impact of such randomization on their payoffs. We propose a simple algorithm for computing T-MFE under a known model. We also introduce a model-free and a model-based approach to learning T-MFE and provide sample complexities of both algorithms. We also develop a fully online learning scheme that obviates the need for a simulator. Finally, we empirically evaluate the performance of the proposed algorithms via examples motivated by real-world applications.

Maximizing Agreements for Ranking, Clustering and Hierarchical Clustering via MAX-CUT

Vaggos Chatziafratis · Mohammad Mahdian · Sara Ahmadian

In this paper, we study a number of well-known combinatorial optimization problems that fit in the following paradigm: the input is a collection of (potentially inconsistent) local relationships between the elements of a ground set (e.g., pairwise comparisons, similar/dissimilar pairs, or ancestry structure of triples of points), and the goal is to aggregate this information into a global structure (e.g., a ranking, a clustering, or a hierarchical clustering) in a way that maximizes agreement with the input. Well-studied problems such as rank aggregation, correlation clustering, and hierarchical clustering with triplet constraints fall in this class of problems.

We study these problems on stochastic instances with a hidden embedded ground truth solution. Our main algorithmic contribution is a unified technique that uses the maximum cut problem in graphs to approximately solve these problems. Using this technique, we can often get approximation guarantees in the stochastic setting that are better than the known worst case inapproximability bounds for the corresponding problem. On the negative side, we improve the worst case inapproximability bound on several hierarchical clustering formulations through a reduction to related ranking problems.

Training a Single Bandit Arm

Eren Ozbay · Vijay Kamble

In several applications of the stochastic multi-armed bandit problem, the traditional objective of maximizing the expected sum of rewards obtained can be inappropriate. Motivated by the problem of optimizing job assignments to train novice workers of unknown quality in labor platforms, we consider a new objective in the classical setup. Instead of maximizing the expected total reward from $T$ pulls, we consider the vector of cumulative rewards earned from the $K$ arms at the end of $T$ pulls, and aim to maximize the expected value of the highest cumulative reward across the $K$ arms. This corresponds to the objective of training a single, highly skilled worker using a limited supply of training jobs. For this new objective, we show that any policy must incur an instance-dependent asymptotic regret of $\Omega(\log T)$ (with a higher instance-dependent constant compared to the traditional objective) and an instance-independent regret of $\Omega(K^{1/3}T^{2/3})$. We then design an explore-then-commit policy, featuring exploration based on appropriately tuned confidence bounds on the mean reward and an adaptive stopping criterion, which adapts to the problem difficulty and achieves these bounds (up to logarithmic factors). Our numerical experiments demonstrate the efficacy of this policy compared to several natural alternatives in practical parameter regimes.

Sketch based Memory for Neural Networks

Rina Panigrahy · Xin Wang · Manzil Zaheer

Deep learning has shown tremendous success on a variety of problems. However, unlike traditional computational paradigm, most neural networks do not have access to a memory, which might be hampering its ability to scale to large data structures such as graphs, lookup-tables, databases. We propose a neural architecture where sketch based memory is integrated into a neural network in a uniform manner at every layer. This architecture supplements a neural layer by information accessed from the memory before feeding it to the next layer, thereby significantly expanding the capacity of the network to solve larger problem instances. We show theoretically that problems involving key-value lookup that are traditionally stored in standard databases can now be solved using neural networks augmented by our memory architecture. We also show that our memory layer can be viewed as a kernel function. We show benefits on diverse problems such as long tail image classification, language model, large graph multi hop traversal, etc. arguing that they are all build upon the classical key-value lookup problem (or the variant where the keys may be fuzzy).

Provably Efficient Actor-Critic for Risk-Sensitive and Robust Adversarial RL: A Linear-Quadratic Case

Yufeng Zhang · Zhuoran Yang · Zhaoran Wang

Risk-sensitivity plays a central role in artificial intelligence safety. In this paper, we study the global convergence of the actor-critic algorithm for risk-sensitive reinforcement learning (RSRL) with exponential utility, which remains challenging for policy optimization as it lacks the linearity needed to formulate policy gradient. To bypass such an issue of nonlinearity, we resort to the equivalence between RSRL and robust adversarial reinforcement learning (RARL), which is formulated as a zero-sum Markov game with a hypothetical adversary. In particular, the Nash equilibrium (NE) of such a game yields the optimal policy for RSRL, which is provably robust. We focus on a simple yet fundamental setting known as linear-quadratic (LQ) game. To attain the optimal policy, we develop a nested natural actor-critic algorithm, which provably converges to the NE of the LQ game at a sublinear rate, thus solving both RSRL and RARL. To the best knowledge, the proposed nested actor-critic algorithm appears to be the first model-free policy optimization algorithm that provably attains the optimal policy for RSRL and RARL in the LQ setting, which sheds light on more general settings.

List Learning with Attribute Noise

Mahdi Cheraghchi · Elena Grigorescu · Brendan Juba · Karl Wimmer · Ning Xie

We introduce and study the model of list learning with attribute noise. Learning with attribute noise was introduced by Shackelford and Volper (COLT, 1988) as a variant of PAC learning, in which the algorithm has access to noisy examples and uncorrupted labels, and the goal is to recover an accurate hypothesis. Sloan (COLT, 1988) and Goldman and Sloan (Algorithmica, 1995) discovered information-theoretic limits to learning in this model, which have impeded further progress. In this article we extend the model to that of list learning, drawing inspiration from the list-decoding model in coding theory, and its recent variant studied in the context of learning. On the positive side, we show that sparse conjunctions can be efficiently list learned under some assumptions on the underlying ground-truth distribution. On the negative side, our results show that even in the list-learning model, efficient learning of parities and majorities is not possible regardless of the representation used.

No-Regret Algorithms for Private Gaussian Process Bandit Optimization

Abhimanyu Dubey

The widespread proliferation of data-driven decision-making has ushered in a recent interest in the design of privacy-preserving algorithms. In this paper, we consider the ubiquitous problem of gaussian process (GP) bandit optimization from the lens of privacy-preserving statistics. We propose a solution for differentially private GP bandit optimization that combines uniform kernel approximation with random perturbations, providing a generic framework to create differentially-private (DP) Gaussian process bandit algorithms. For two specific DP settings - joint and local differential privacy, we provide algorithms based on efficient quadrature Fourier feature approximators, that are computationally efficient and provably no-regret for a class of stationary kernel functions. In contrast to previous work, our algorithms maintain differential privacy throughout the optimization procedure and critically do not rely on the sample path for prediction, making them scalable and straightforward to release as well.

The Minecraft Kernel: Modelling correlated Gaussian Processes in the Fourier domain

Fergus Simpson · Alexis Boukouvalas · Vaclav Cadek · Elvijs Sarkans · Nicolas Durrande

In the univariate setting, using the kernel spectral representation is an appealing approach for generating stationary covariance functions. However, performing the same task for multiple-output Gaussian processes is substantially more challenging. We demonstrate that current approaches to modelling cross-covariances with a spectral mixture kernel possess a critical blind spot. Pairs of highly correlated (or highly anti-correlated) processes are not reproducible, aside from the special case when their spectral densities are of identical shape. We present a solution to this issue by replacing the conventional Gaussian components of a spectral mixture with block components of finite bandwidth (i.e. rectangular step functions). The proposed family of kernel represents the first multi-output generalisation of the spectral mixture kernel that can approximate any stationary multi-output kernel to arbitrary precision.

Non-Stationary Off-Policy Optimization

Joey Hong · Branislav Kveton · Manzil Zaheer · Yinlam Chow · Amr Ahmed

Off-policy learning is a framework for evaluating and optimizing policies without deploying them, from data collected by another policy. Real-world environments are typically non-stationary and the offline learned policies should adapt to these changes. To address this challenge, we study the novel problem of off-policy optimization in piecewise-stationary contextual bandits. Our proposed solution has two phases. In the offline learning phase, we partition logged data into categorical latent states and learn a near-optimal sub-policy for each state. In the online deployment phase, we adaptively switch between the learned sub-policies based on their performance. This approach is practical and analyzable, and we provide guarantees on both the quality of off-policy optimization and the regret during online deployment. To show the effectiveness of our approach, we compare it to state-of-the-art baselines on both synthetic and real-world datasets. Our approach outperforms methods that act only on observed context.

Learning the Truth From Only One Side of the Story

Heinrich Jiang · Qijia Jiang · Aldo Pacchiano

Learning under one-sided feedback (i.e., where we only observe the labels for examples we predicted positively on) is a fundamental problem in machine learning -- applications include lending and recommendation systems. Despite this, there has been surprisingly little progress made in ways to mitigate the effects of the sampling bias that arises. We focus on generalized linear models and show that without adjusting for this sampling bias, the model may converge suboptimally or even fail to converge to the optimal solution. We propose an adaptive approach that comes with theoretical guarantees and show that it outperforms several existing methods empirically. Our method leverages variance estimation techniques to efficiently learn under uncertainty, offering a more principled alternative compared to existing approaches.

Generalization of Quasi-Newton Methods: Application to Robust Symmetric Multisecant Updates

Damien Scieur · Lewis Liu · Thomas Pumir · Nicolas Boumal

Quasi-Newton (qN) techniques approximate the Newton step by estimating the Hessian using the so-called secant equations. Some of these methods compute the Hessian using several secant equations but produce non-symmetric updates. Other quasi-Newton schemes, such as BFGS, enforce symmetry but cannot satisfy more than one secant equation. We propose a new type of quasi-Newton symmetric update using several secant equations in a least-squares sense. Our approach generalizes and unifies the design of quasi-Newton updates and satisfies provable robustness guarantees.

Counterfactual Representation Learning with Balancing Weights

Serge Assaad · Shuxi Zeng · Chenyang Tao · Shounak Datta · Nikhil Mehta · Ricardo Henao · Fan Li · Lawrence Carin Duke

A key to causal inference with observational data is achieving balance in predictive features associated with each treatment type. Recent literature has explored representation learning to achieve this goal. In this work, we discuss the pitfalls of these strategies – such as a steep trade-off between achieving balance and predictive power – and present a remedy via the integration of balancing weights in causal learning. Specifically, we theoretically link balance to the quality of propensity estimation, emphasize the importance of identifying a proper target population, and elaborate on the complementary roles of feature balancing and weight adjustments. Using these concepts, we then develop an algorithm for flexible, scalable and accurate estimation of causal effects. Finally, we show how the learned weighted representations may serve to facilitate alternative causal learning procedures with appealing statistical features. We conduct an extensive set of experiments on both synthetic examples and standard benchmarks, and report encouraging results relative to state-of-the-art baselines.

An Efficient Algorithm For Generalized Linear Bandit: Online Stochastic Gradient Descent and Thompson Sampling

QIN DING · Cho-Jui Hsieh · James Sharpnack

We consider the contextual bandit problem, where a player sequentially makes decisions based on past observations to maximize the cumulative reward. Although many algorithms have been proposed for contextual bandit, most of them rely on finding the maximum likelihood estimator at each iteration, which requires $O(t)$ time at the $t$-th iteration and are memory inefficient. A natural way to resolve this problem is to apply online stochastic gradient descent (SGD) so that the per-step time and memory complexity can be reduced to constant with respect to $t$, but a contextual bandit policy based on online SGD updates that balances exploration and exploitation has remained elusive. In this work, we show that online SGD can be applied to the generalized linear bandit problem. The proposed SGD-TS algorithm, which uses a single-step SGD update to exploit past information and uses Thompson Sampling for exploration, achieves $\tilde{O}(\sqrt{dT})$ regret with the total time complexity that scales linearly in $T$ and $d$, where $T$ is the total number of rounds and $d$ is the number of features. Experimental results show that SGD-TS consistently outperforms existing algorithms on both synthetic and real datasets.

Regression Discontinuity Design under Self-selection

Sida Peng · Yang Ning

Regression Discontinuity (RD) design is commonly used to estimate the causal effect of a policy. Existing RD relies on the continuity assumption of potential outcomes. However, self selection leads to different distributions of covariates on two sides of the policy intervention, which violates this assumption. The standard RD estimators are no longer applicable in such setting. We show that the direct causal effect can still be recovered under a class of weighted average treatment effects. We propose a set of estimators through a weighted local linear regression framework and prove the consistency and asymptotic normality of the estimators. We apply our method to a novel data set from Microsoft Bing on Generalized Second Price (GSP) auction and show that by placing the advertisement on the second ranked position can increase the click-ability by 1.91%.

Multitask Bandit Learning Through Heterogeneous Feedback Aggregation

Zhi Wang · Chicheng Zhang · Manish Kumar Singh · Laurel Riek · Kamalika Chaudhuri

In many real-world applications, multiple agents seek to learn how to perform highly related yet slightly different tasks in an online bandit learning protocol. We formulate this problem as the $\epsilon$-multi-player multi-armed bandit problem, in which a set of players concurrently interact with a set of arms, and for each arm, the reward distributions for all players are similar but not necessarily identical. We develop an upper confidence bound-based algorithm, RobustAgg($\epsilon$), that adaptively aggregates rewards collected by different players. In the setting where an upper bound on the pairwise dissimilarities of reward distributions between players is known, we achieve instance-dependent regret guarantees that depend on the amenability of information sharing across players. We complement these upper bounds with nearly matching lower bounds. In the setting where pairwise dissimilarities are unknown, we provide a lower bound, as well as an algorithm that trades off minimax regret guarantees for adaptivity to unknown similarity structure.

Problem-Complexity Adaptive Model Selection for Stochastic Linear Bandits

Avishek Ghosh · Abishek Sankararaman · Ramchandran Kannan

We consider the problem of model selection for two popular stochastic linear bandit settings, and propose algorithms that adapts to the unknown problem complexity. In the first setting, we consider the $K$ armed mixture bandits, where the mean reward of arm $i \in [K]$ is $\mu_i+ \langle \alpha_{i,t},\theta^* \rangle $, with $\alpha_{i,t} \in \mathbb{R}^d$ being the known context vector and $\mu_i \in [-1,1]$ and $\theta^*$ are unknown parameters. We define $\|\theta^*\|$ as the problem complexity and consider a sequence of nested hypothesis classes, each positing a different upper bound on $\|\theta^*\|$. Exploiting this, we propose Adaptive Linear Bandit (ALB), a novel phase based algorithm that adapts to the true problem complexity, $\|\theta^*\|$. We show that ALB achieves regret scaling of $\widetilde{O}(\|\theta^*\|\sqrt{T})$, where $\|\theta^*\|$ is apriori unknown. As a corollary, when $\theta^*=0$, ALB recovers the minimax regret for the simple bandit algorithm without such knowledge of $\theta^*$. ALB is the first algorithm that uses parameter norm as model section criteria for linear bandits. Prior state of art algorithms achieve a regret of $\widetilde{O}(L\sqrt{T})$, where $L$ is the upper bound on $\|\theta^*\|$, fed as an input to the problem. In the second setting, we consider the standard linear bandit problem (with possibly an infinite number of arms) where the sparsity of $\theta^*$, denoted by $d^* \leq d$, is unknown to the algorithm. Defining $d^*$ as the problem complexity (similar to Foster et. al '19), we show that ALB achieves $\widetilde{O}(d^*\sqrt{T})$ regret, matching that of an oracle who knew the true sparsity level. This methodology is then extended to the case of finitely many arms and similar results are proven. We further verify through synthetic and real-data experiments that the performance gains are fundamental and not artifacts of mathematical bounds. In particular, we show $1.5-3$x drop in cumulative regret over non-adaptive algorithms.

Cluster Trellis: Data Structures & Algorithms for Exact Inference in Hierarchical Clustering

Sebastian Macaluso · Craig Greenberg · Nicholas Monath · Ji Ah Lee · Patrick Flaherty · Kyle Cranmer · Andrew McGregor · Andrew McCallum

Hierarchical clustering is a fundamental task often used to discover meaningful structures in data. Due to the combinatorial number of possible hierarchical clusterings, approximate algorithms are typically used for inference. In contrast to existing methods, we present novel dynamic-programming algorithms for exact inference in hierarchical clustering based on a novel trellis data structure, and we prove that we can exactly compute the partition function, maximum likelihood hierarchy, and marginal probabilities of sub-hierarchies and clusters. Our algorithms scale in time and space proportional to the powerset of N elements, which is super-exponentially more efficient than explicitly considering each of the (2N − 3)!! possible hierarchies. Also, for larger datasets where our exact algorithms become infeasible, we introduce an approximate algorithm based on a sparse trellis that out- performs greedy and beam search baselines.

Dynamic Cutset Networks

Chiradeep Roy · Tahrima Rahman · Hailiang Dong · Nicholas Ruozzi · Vibhav Gogate

Tractable probabilistic models (TPMs) are appealing because they admit polynomial-time inference for a wide variety of queries. In this work, we extend the cutset network (CN) framework, a powerful sub-class of TPMs that often outperforms probabilistic graphical models in terms of prediction accuracy, to the temporal domain. This extension, dubbed dynamic cutset networks (DCNs), uses a CN to model the prior distribution and a conditional CN to model the transition distribution. We show that although exact inference is intractable when arbitrary conditional CNs are used, particle filtering is efficient. To ensure tractability of exact inference, we introduce a novel constrained conditional model called AND/OR conditional cutset networks and show that under certain conditions exact inference is linear in the size of the corresponding constrained DCN. Experiments on several sequential datasets demonstrate the efficacy of our framework.

A Theoretical Analysis of Catastrophic Forgetting through the NTK Overlap Matrix

Thang Doan · Mehdi Abbana Bennani · Bogdan Mazoure · Guillaume Rabusseau · Pierre Alquier

Continual learning (CL) is a setting in which an agent has to learn from an incoming stream of data during its entire lifetime. Although major advances have been made in the field, one recurring problem which remains unsolved is that of Catastrophic Forgetting (CF). While the issue has been extensively studied empirically, little attention has been paid from a theoretical angle. In this paper, we show that the impact of CF increases as two tasks increasingly align. We introduce a measure of task similarity called the NTK overlap matrix which is at the core of CF. We analyze common projected gradient algorithms and demonstrate how they mitigate forgetting. Then, we propose a variant of Orthogonal Gradient Descent (OGD) which leverages structure of the data through Principal Component Analysis (PCA). Experiments support our theoretical findings and show how our method can help reduce CF on classical CL datasets.

Learning-to-Rank with Partitioned Preference: Fast Estimation for the Plackett-Luce Model

Jiaqi Ma · Xinyang Yi · Weijing Tang · Zhe Zhao · Lichan Hong · Ed Chi · Qiaozhu Mei

We consider the problem of listwise learning-to-rank (LTR) on data with \textit{partitioned preference}, where a set of items are sliced into ordered and disjoint partitions, but the ranking of items within a partition is unknown. The Plackett-Luce (PL) model has been widely used in listwise LTR methods. However, given $N$ items with $M$ partitions, calculating the likelihood of data with partitioned preference under the PL model has a time complexity of $O(N+S!)$, where $S$ is the maximum size of the top $M-1$ partitions. This computational challenge restrains existing PL-based listwise LTR methods to only a special case of partitioned preference, \textit{top-$K$ ranking}, where the exact order of the top $K$ items is known. In this paper, we exploit a random utility model formulation of the PL model and propose an efficient approach through numerical integration for calculating the likelihood. This numerical approach reduces the aforementioned time complexity to $O(N+MS)$, which allows training deep-neural-network-based ranking models with a large output space. We demonstrate that the proposed method outperforms well-known LTR baselines and remains scalable through both simulation experiments and applications to real-world eXtreme Multi-Label (XML) classification tasks. The proposed method also achieves state-of-the-art performance on XML datasets with relatively large numbers of labels per sample.

Nonparametric Variable Screening with Optimal Decision Stumps

Jason Klusowski · Peter M Tian

Decision trees and their ensembles are endowed with a rich set of diagnostic tools for ranking and screening variables in a predictive model. Despite the widespread use of tree based variable importance measures, pinning down their theoretical properties has been challenging and therefore largely unexplored. To address this gap between theory and practice, we derive finite sample performance guarantees for variable selection in nonparametric models using a single-level CART decision tree (a decision stump). Under standard operating assumptions in variable screening literature, we find that the marginal signal strength of each variable and ambient dimensionality can be considerably weaker and higher, respectively, than state-of-the-art nonparametric variable selection methods. Furthermore, unlike previous marginal screening methods that estimate each marginal projection via a truncated basis expansion, the fitted model used here is a simple, parsimonious decision stump, thereby eliminating the need for tuning the number of basis terms. Thus, surprisingly, even though decision stumps are highly inaccurate for estimation purposes, they can still be used to perform consistent model selection.

Principal Subspace Estimation Under Information Diffusion

Fan Zhou · Ping Li · Zhixin Zhou

Let $\mathbf{A} = \mathbf{L}_0 + \mathbf{S}_0$, where $\mathbf{L}_0 \in \mathbb{R}^{d\times d}$ is low rank and $\mathbf{S}_0$ is a perturbation matrix. We study the principal subspace estimation of $\mathbf{L}_0$ through observations $\mathbf{y}_j = f(\mathbf{A})\mathbf{x}_j$, $j=1,\dots,n$, where $f:\mathbb{R}\rightarrow \mathbb{R}$ is an unknown polynomial and $\mathbf{x}_j$'s are i.i.d. random input signals. Such models are widely used in graph signal processing to model information diffusion dynamics over networks with applications in network topology inference and data analysis. We develop an estimation procedure based on nuclear norm penalization, and establish upper bounds on the principal subspace estimation error when $\mathbf{A}$ is the adjacency matrix of a random graph generated by $\mathbf{L}_0$. Our theory shows that when the signal strength is strong enough, the exact rank of $\mathbf{L}_0$ can be recovered. By applying our results to blind community detection, we show that consistency of spectral clustering can be achieved for some popular stochastic block models. Together with the experimental results, our theory show that there is a fundamental limit of using the principal components obtained from diffused graph signals which is commonly adapted in current practice. Finally, under some structured perturbation $\mathbf{S}_0$, we build the connection between this model with spiked covariance model and develop a new estimation procedure. We show that such estimators can be optimal under the minimax paradigm.

Learning with Gradient Descent and Weakly Convex Losses

Dominic Richards · Mike Rabbat

We study the learning performance of gradient descent when the empirical risk is weakly convex, namely, the smallest negative eigenvalue of the empirical risk's Hessian is bounded in magnitude. By showing that this eigenvalue can control the stability of gradient descent, generalisation error bounds are proven that hold under a wider range of step sizes compared to previous work. Out of sample guarantees are then achieved by decomposing the test error into generalisation, optimisation and approximation errors, each of which can be bounded and traded off with respect to algorithmic parameters, sample size and magnitude of this eigenvalue. In the case of a two layer neural network, we demonstrate that the empirical risk can satisfy a notion of local weak convexity, specifically, the Hessian's smallest eigenvalue during training can be controlled by the normalisation of the layers, i.e., network scaling. This allows test error guarantees to then be achieved when the population risk minimiser satisfies a complexity assumption. By trading off the network complexity and scaling, insights are gained into the implicit bias of neural network scaling, which are further supported by experimental findings.

Active Online Learning with Hidden Shifting Domains

Yining Chen · Haipeng Luo · Tengyu Ma · Chicheng Zhang

Online machine learning systems need to adapt to domain shifts. Meanwhile, acquiring label at every timestep is expensive. We propose a surprisingly simple algorithm that adaptively balances its regret and its number of label queries in settings where the data streams are from a mixture of hidden domains. For online linear regression with oblivious adversaries, we provide a tight tradeoff that depends on the durations and dimensionalities of the hidden domains. Our algorithm can adaptively deal with interleaving spans of inputs from different domains. We also generalize our results to non-linear regression for hypothesis classes with bounded eluder dimension and adaptive adversaries. Experiments on synthetic and realistic datasets demonstrate that our algorithm achieves lower regret than uniform queries and greedy queries with equal labeling budget.

Differentially Private Online Submodular Maximization

Sebastian Perez-Salazar · Rachel Cummings

In this work we consider the problem of online submodular maximization under a cardinality constraint with differential privacy (DP). A stream of T submodular functions over a common finite ground set U arrives online, and at each time-step the decision maker must choose at most k elements of U before observing the function. The decision maker obtains a profit equal to the function evaluated on the chosen set and aims to learn a sequence of sets that achieves low expected regret. In the full-information setting, we develop an $(\varepsilon,\delta)$-DP algorithm with expected (1-1/e)-regret bound of $O( \frac{k^2\log |U|\sqrt{T \log k/\delta}}{\varepsilon} )$. This algorithm contains k ordered experts that learn the best marginal increments for each item over the whole time horizon while maintaining privacy of the functions. In the bandit setting, we provide an $(\varepsilon,\delta+ O(e^{-T^{1/3}}))$-DP algorithm with expected (1-1/e)-regret bound of $O( \frac{\sqrt{\log k/\delta}}{\varepsilon} (k (|U| \log |U|)^{1/3})^2 T^{2/3} )$. One challenge for privacy in this setting is that the payoff and feedback of expert i depends on the actions taken by her i-1 predecessors. This particular type of information leakage is not covered by post-processing, and new analysis is required. Our techniques for maintaining privacy with feedforward may be of independent interest.

Fast Statistical Leverage Score Approximation in Kernel Ridge Regression

Yifan Chen · Yun Yang

Nyström approximation is a fast randomized method that rapidly solves kernel ridge regression (KRR) problems through sub-sampling the n-by-n empirical kernel matrix appearing in the objective function. However, the performance of such a sub-sampling method heavily relies on correctly estimating the statistical leverage scores for forming the sampling distribution, which can be as costly as solving the original KRR. In this work, we propose a linear time (modulo poly-log terms) algorithm to accurately approximate the statistical leverage scores in the stationary-kernel-based KRR with theoretical guarantees. Particularly, by analyzing the first-order condition of the KRR objective, we derive an analytic formula, which depends on both the input distribution and the spectral density of stationary kernels, for capturing the non-uniformity of the statistical leverage scores. Numerical experiments demonstrate that with the same prediction accuracy our method is orders of magnitude more efficient than existing methods in selecting the representative sub-samples in the Nyström approximation.

Distributionally Robust Optimization for Deep Kernel Multiple Instance Learning

Hitesh Sapkota · Yiming Ying · Feng Chen · Qi Yu

Multiple Instance Learning (MIL) provides a promising solution to many real-world problems, where labels are only available at the bag level but missing for instances due to a high labeling cost. As a powerful Bayesian non-parametric model, Gaussian Processes (GP) have been extended from classical supervised learning to MIL settings, aiming to identify the most likely positive (or least negative) instance from a positive (or negative) bag using only the bag-level labels. However, solely focusing on a single instance in a bag makes the model less robust to outliers or multi-modal scenarios, where a single bag contains a diverse set of positive instances. We propose a general GP mixture framework that simultaneously considers multiple instances through a latent mixture model. By adding a top-k constraint, the framework is equivalent to choosing the top-k most positive instances, making it more robust to outliers and multimodal scenarios. We further introduce a Distributionally Robust Optimization (DRO) constraint that removes the limitation of specifying a fix k value. To ensure the prediction power over high-dimensional data (e.g., videos and images) that are common in MIL, we augment the GP kernel with fixed basis functions by using a deep neural network to learn adaptive basis functions so that the covariance structure of high-dimensional data can be accurately captured. Experiments are conducted on highly challenging real-world video anomaly detection tasks to demonstrate the effectiveness of the proposed model.

Finding First-Order Nash Equilibria of Zero-Sum Games with the Regularized Nikaido-Isoda Function

Ioannis Tsaknakis · Mingyi Hong

Efficiently finding First-order Nash Equilibria (FNE) in zero-sum games can be challenging, even in a two-player setting. This work proposes an algorithm for finding the FNEs of a two-player zero-sum game, in which the local cost functions can be non-convex, and the players only have access to local stochastic gradients. The proposed approach is based on reformulating the problem of interest as minimizing the Regularized Nikaido-Isoda (RNI) function. We show that the global minima of the RNI correspond to the set of FNEs, and that for certain classes of non-convex games the RNI minimization problem becomes convex. Moreover, we introduce a first-order (stochastic) optimization method, and establish its convergence to a neighborhood of a stationary solution of the RNI objective. The key in the analysis is to properly control the bias between the local stochastic gradient and the true one. Although the RNI function has been used in analyzing convex games, to our knowledge, this is the first time that the properties of the RNI formulation have been exploited to find FNEs for non-convex games in a stochastic setting.

Efficient Methods for Structured Nonconvex-Nonconcave Min-Max Optimization

Jelena Diakonikolas · Constantinos Daskalakis · Michael Jordan

The use of min-max optimization in the adversarial training of deep neural network classifiers, and the training of generative adversarial networks has motivated the study of nonconvex-nonconcave optimization objectives, which frequently arise in these applications. Unfortunately, recent results have established that even approximate first-order stationary points of such objectives are intractable, even under smoothness conditions, motivating the study of min-max objectives with additional structure. We introduce a new class of structured nonconvex-nonconcave min-max optimization problems, proposing a generalization of the extragradient algorithm which provably converges to a stationary point. The algorithm applies not only to Euclidean spaces, but also to general $\ell_p$-normed finite-dimensional real vector spaces. We also discuss its stability under stochastic oracles and provide bounds on its sample complexity. Our iteration complexity and sample complexity bounds either match or improve the best known bounds for the same or less general nonconvex-nonconcave settings, such as those that satisfy variational coherence or in which a weak solution to the associated variational inequality problem is assumed to exist.

On the Linear Convergence of Policy Gradient Methods for Finite MDPs

Jalaj Bhandari · Daniel Russo

We revisit the finite time analysis of policy gradient methods in the one of the simplest settings: finite state and action MDPs with a policy class consisting of all stochastic policies and with exact gradient evaluations. There has been some recent work viewing this setting as an instance of smooth non-linear optimization problems, to show sub-linear convergence rates with small step-sizes. Here, we take a completely different perspective based on illuminating connections with policy iteration, to show how many variants of policy gradient algorithms succeed with large step-sizes and attain a linear rate of convergence.

Influence Decompositions For Neural Network Attribution

Kyle Reing · Greg Ver Steeg · Aram Galstyan

Methods of neural network attribution have emerged out of a necessity for explanation and accountability in the predictions of black-box neural models. Most approaches use a variation of sensitivity analysis, where individual input variables are perturbed and the downstream effects on some output metric are measured. We demonstrate that a number of critical functional properties are not revealed when only considering lower-order perturbations. Motivated by these shortcomings, we propose a general framework for decomposing the orders of influence that a collection of input variables has on an output classification. These orders are based on the cardinality of input subsets which are perturbed to yield a change in classification. This decomposition can be naturally applied to attribute which input variables rely on higher-order coordination to impact the classification decision. We demonstrate that our approach correctly identifies higher-order attribution on a number of synthetic examples. Additionally, we showcase the differences between attribution in our approach and existing approaches on benchmark networks for MNIST and ImageNet.

Learning GPLVM with arbitrary kernels using the unscented transformation

Daniel Augusto de Souza · Diego Mesquita · João Paulo Gomes · César Lincoln Mattos

Gaussian Process Latent Variable Model (GPLVM) is a flexible framework to handle uncertain inputs in Gaussian Processes (GPs) and incorporate GPs as components of larger graphical models. Nonetheless, the standard GPLVM variational inference approach is tractable only for a narrow family of kernel functions. The most popular implementations of GPLVM circumvent this limitation using quadrature methods, which may become a computational bottleneck even for relatively low dimensions. For instance, the widely employed Gauss-Hermite quadrature has exponential complexity on the number of dimensions. In this work, we propose using the unscented transformation instead. Overall, this method presents comparable, if not better, performance than off-the-shelf solutions to GPLVM, and its computational complexity scales only linearly on dimension. In contrast to Monte Carlo methods, our approach is deterministic and works well with quasi-Newton methods, such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. We illustrate the applicability of our method with experiments on dimensionality reduction and multistep-ahead prediction with uncertainty propagation.

On the Privacy Properties of GAN-generated Samples

Zinan Lin · Vyas Sekar · Giulia Fanti

The privacy implications of generative adversarial networks (GANs) are a topic of great interest, leading to several recent algorithms for training GANs with privacy guarantees. By drawing connections to the generalization properties of GANs, we prove that under some assumptions, GAN-generated samples inherently satisfy some (weak) privacy guarantees. First, we show that if a GAN is trained on m samples and used to generate n samples, the generated samples are (epsilon, delta)-differentially-private for (epsilon, delta) pairs where delta scales as O(n/m). We show that under some special conditions, this upper bound is tight. Next, we study the robustness of GAN-generated samples to membership inference attacks. We model membership inference as a hypothesis test in which the adversary must determine whether a given sample was drawn from the training dataset or from the underlying data distribution. We show that this adversary can achieve an area under the ROC curve that scales no better than O(m^{-1/4}).

Differentially Private Weighted Sampling

Edith Cohen · Ofir Geri · Tamas Sarlos · Uri Stemmer

Common datasets have the form of elements with keys (e.g., transactions and products) and the goal is to perform analytics on the aggregated form of key and frequency pairs. A weighted sample of keys by (a function of) frequency is a highly versatile summary that provides a sparse set of representative keys and supports approximate evaluations of query statistics. We propose private weighted sampling (PWS): A method that sanitizes a weighted sample as to ensure element-level differential privacy, while retaining its utility to the maximum extent possible. PWS maximizes the reporting probabilities of keys and estimation quality of a broad family of statistics. PWS improves over the state of the art even for the well-studied special case of private histograms, when no sampling is performed. We empirically observe significant performance gains of 20%-300% increase in key reporting for common Zipfian frequency distributions and accurate estimation with x2-8 lower frequencies. PWS is applied as a post-processing of a non-private sample, without requiring the original data. Therefore, it can be a seamless addition to existing implementations, such as those optimizes for distributed or streamed data. We believe that due to practicality and performance, PWS may become a method of choice in applications where privacy is desired.

Convergence and Accuracy Trade-Offs in Federated Learning and Meta-Learning

Zachary Charles · Jakub Konečný

We study a family of algorithms, which we refer to as local update methods, generalizing many federated and meta-learning algorithms. We prove that for quadratic models, local update methods are equivalent to first-order optimization on a surrogate loss we exactly characterize. Moreover, fundamental algorithmic choices (such as learning rates) explicitly govern a trade-off between the condition number of the surrogate loss and its alignment with the true loss. We derive novel convergence rates showcasing these trade-offs and highlight their importance in communication-limited settings. Using these insights, we are able to compare local update methods based on their convergence/accuracy trade-off, not just their convergence to critical points of the empirical loss. Our results shed new light on a broad range of phenomena, including the efficacy of server momentum in federated learning and the impact of proximal client updates.

Misspecification in Prediction Problems and Robustness via Improper Learning

Annie Marsden · John Duchi · Gregory Valiant

We study probabilistic prediction games when the underlying model is misspecified, investigating the consequences of predicting using an incorrect parametric model. We show that for a broad class of loss functions and parametric families of distributions, the regret of playing a ``proper'' predictor---one from the putative model class---relative to the best predictor in the same model class has lower bound scaling at least as $\sqrt{\gamma n}$, where $\gamma$ is a measure of the model misspecification to the true distribution in terms of total variation distance. In contrast, using an aggregation-based (improper) learner, one can obtain regret $d \log n$ for any underlying generating distribution, where $d$ is the dimension of the parameter; we exhibit instances in which this is unimprovable even over the family of all learners that may play distributions in the convex hull of the parametric family. These results suggest that simple strategies for aggregating multiple learners together should be more robust, and several experiments conform to this hypothesis.

Sharp Analysis of a Simple Model for Random Forests

Jason Klusowski

Random forests have become an important tool for improving accuracy in regression and classification problems since their inception by Leo Breiman in 2001. In this paper, we revisit a historically important random forest model, called centered random forests, originally proposed by Breiman in 2004 and later studied by G\'erard Biau in 2012, where a feature is selected at random and the splits occurs at the midpoint of the node along the chosen feature. If the regression function is $d$-dimensional and Lipschitz, we show that, given access to $n$ observations, the mean-squared prediction error is $O((n(\log n)^{(d-1)/2})^{-\frac{1}{d\log2+1}})$. This positively answers an outstanding question of Biau about whether the rate of convergence for this random forest model could be improved beyond $O(n^{-\frac{1}{d(4/3)\log2+1}})$. Furthermore, by a refined analysis of the approximation and estimation errors for linear models, we show that our new rate cannot be improved in general. Finally, we generalize our analysis and improve current prediction error bounds for another random forest model, called median random forests, in which each tree is constructed from subsampled data and the splits are performed at the empirical median along a chosen feature.

On Learning Continuous Pairwise Markov Random Fields

Abhin Shah · Devavrat Shah · Gregory Wornell

We consider learning a sparse pairwise Markov Random Field (MRF) with continuous-valued variables from i.i.d samples. We adapt the algorithm of Vuffray et al. (2019) to this setting and provide finite-sample analysis revealing sample complexity scaling logarithmically with the number of variables, as in the discrete and Gaussian settings. Our approach is applicable to a large class of pairwise MRFs with continuous variables and also has desirable asymptotic properties, including consistency and normality under mild conditions. Further, we establish that the population version of the optimization criterion employed in Vuffray et al. (2019) can be interpreted as local maximum likelihood estimation (MLE). As part of our analysis, we introduce a robust variation of sparse linear regression a` la Lasso, which may be of interest in its own right.

Non-asymptotic Performance Guarantees for Neural Estimation of f-Divergences

Sreejith Sreekumar · Zhengxin Zhang · Ziv Goldfeld

Statistical distances (SDs), which quantify the dissimilarity between probability distributions, are central to machine learning and statistics. A modern method for estimating such distances from data relies on parametrizing a variational form by a neural network (NN) and optimizing it. These estimators are abundantly used in practice, but corresponding performance guarantees are partial and call for further exploration. In particular, there seems to be a fundamental tradeoff between the two sources of error involved: approximation and estimation. While the former needs the NN class to be rich and expressive, the latter relies on controlling complexity. This paper explores this tradeoff by means of non-asymptotic error bounds, focusing on three popular choices of SDs---Kullback-Leibler divergence, chi-squared divergence, and squared Hellinger distance. Our analysis relies on non-asymptotic function approximation theorems and tools from empirical process theory. Numerical results validating the theory are also provided.

Sampling in Combinatorial Spaces with SurVAE Flow Augmented MCMC

Priyank Jaini · Didrik Nielsen · Max Welling

Hybrid Monte Carlo is a powerful Markov Chain Monte Carlo method for sampling from complex continuous distributions. However, a major limitation of HMC is its inability to be applied to discrete domains due to the lack of gradient signal. In this work, we introduce a new approach based on augmenting monte carlo methods with SurVAE Flow to sample from discrete distributions using a combination of neural transport methods like normalizing flows, variational dequantization, and the Metropolis-Hastings rule. Our method first learns a continuous embedding of the discrete space using a surjective map and subsequently learns a bijective transformation from the continuous space to an approximately Gaussian distributed latent variable. Sampling proceeds by simulating MCMC chains in the latent space and mapping these samples to the target discrete space via the learned transformations. We demonstrate the efficacy of our algorithm on a range of examples from statistics, computational physics, and, machine learning, and observe improvements compared to alternative algorithms.

Alternating Direction Method of Multipliers for Quantization

Tianjian Huang · Prajwal Singhania · Maziar Sanjabi · Pabitra Mitra · Meisam Razaviyayn

Quantization of the parameters of machine learning models, such as deep neural networks, requires solving constrained optimization problems, where the constraint set is formed by the Cartesian product of many simple discrete sets. For such optimization problems, we study the performance of the Alternating Direction Method of Multipliers for Quantization (ADMM-Q) algorithm, which is a variant of the widely-used ADMM method applied to our discrete optimization problem. We establish the convergence of the iterates of ADMM-Q to certain stationary points. To the best of our knowledge, this is the first analysis of an ADMM-type method for problems with discrete variables/constraints. Based on our theoretical insights, we develop a few variants of ADMM-Q that can handle inexact update rules, and have improved performance via the use of "soft projection" and "injecting randomness to the algorithm". We empirically evaluate the efficacy of our proposed approaches.

Convergence of Gaussian-smoothed optimal transport distance with sub-gamma distributions and dependent samples

Yixing Zhang · Xiuyuan Cheng · Galen Reeves

The Gaussian-smoothed optimal transport (GOT) framework, recently proposed by Goldfeld et al., scales to high dimensions in estimation and provides an alternative to entropy regularization. This paper provides convergence guarantees for estimating the GOT distance under more general settings. For the Gaussian-smoothed $p$-Wasserstein distance in $d$ dimensions, our results require only the existence of a moment greater than $d + 2p$. For the special case of sub-gamma distributions, we quantify the dependence on the dimension $d$ and establish a phase transition with respect to the scale parameter. We also prove convergence for dependent samples, only requiring a condition on the pairwise dependence of the samples measured by the covariance of the feature map of a kernel space. A key step in our analysis is to show that the GOT distance is dominated by a family of kernel maximum mean discrepancy (MMD) distances with a kernel that depends on the cost function as well as the amount of Gaussian smoothing. This insight provides further interpretability for the GOT framework and also introduces a class of kernel MMD distances with desirable properties. The theoretical results are supported by numerical experiments.The Gaussian-smoothed optimal transport (GOT) framework, recently proposed by Goldfeld et al., scales to high dimensions in estimation and provides an alternative to entropy regularization. This paper provides convergence guarantees for estimating the GOT distance under more general settings. For the Gaussian-smoothed $p$-Wasserstein distance in $d$ dimensions, our results require only the existence of a moment greater than $d + 2p$. For the special case of sub-gamma distributions, we quantify the dependence on the dimension $d$ and establish a phase transition with respect to the scale parameter. We also prove convergence for dependent samples, only requiring a condition on the pairwise dependence of the samples measured by the covariance of the feature map of a kernel space. A key step in our analysis is to show that the GOT distance is dominated by a family of kernel maximum mean discrepancy (MMD) distances with a kernel that depends on the cost function as well as the amount of Gaussian smoothing. This insight provides further interpretability for the GOT framework and also introduces a class of kernel MMD distances with desirable properties. The theoretical results are supported by numerical experiments.

Near-Optimal Provable Uniform Convergence in Offline Policy Evaluation for Reinforcement Learning

Ming Yin · Yu Bai · Yu-Xiang Wang

The problem of \emph{Offline Policy Evaluation} (OPE) in Reinforcement Learning (RL) is a critical step towards applying RL in real life applications. Existing work on OPE mostly focus on evaluating a \emph{fixed} target policy $\pi$, which does not provide useful bounds for offline policy learning as $\pi$ will then be data-dependent. We address this problem by \emph{simultaneously} evaluating all policies in a policy class $\Pi$ --- uniform convergence in OPE --- and obtain nearly optimal error bounds for a number of global / local policy classes. Our results imply that the model-based planning achieves an optimal episode complexity of $\widetilde{O}(H^3/d_m\epsilon^2)$ in identifying an $\epsilon$-optimal policy under the \emph{time-inhomogeneous episodic} MDP model ($H$ is the planning horizon, $d_m$ is a quantity that reflects the exploration of the logging policy $\mu$). To the best of our knowledge, this is the first time the optimal rate is shown to be possible for the offline RL setting and the paper is the first that systematically investigates the uniform convergence in OPE.

Online Model Selection for Reinforcement Learning with Function Approximation

Jonathan Lee · Aldo Pacchiano · Vidya Muthukumar · Weihao Kong · Emma Brunskill

Deep reinforcement learning has achieved impressive successes yet often requires a very large amount of interaction data. This result is perhaps unsurprising, as using complicated function approximation often requires more data to fit, and early theoretical results on linear Markov decision processes provide regret bounds that scale with the dimension of the linear approximation. Ideally, we would like to automatically identify the minimal dimension of the approximation that is sufficient to encode an optimal policy. Towards this end, we consider the problem of model selection in RL with function approximation, given a set of candidate RL algorithms with known regret guarantees. The learner's goal is to adapt to the complexity of the optimal algorithm without knowing it a priori. We present a meta-algorithm that successively rejects increasingly complex models using a simple statistical test. Given at least one candidate that satisfies realizability, we prove the meta-algorithm adapts to the optimal complexity with regret that is only marginally suboptimal in the number of episodes and number of candidate algorithms. The dimension and horizon dependencies remain optimal with respect to the best candidate, and our meta-algorithmic approach is flexible to incorporate multiple candidate algorithms and models. Finally, we show that the meta-algorithm automatically admits significantly improved instance-dependent regret bounds that depend on the gaps between the maximal values attainable by the candidates.

Animal pose estimation from video data with a hierarchical von Mises-Fisher-Gaussian model

Libby Zhang · Tim Dunn · Jesse Marshall · Bence Olveczky · Scott Linderman

Animal pose estimation from video data is an important step in many biological studies, but current methods struggle in complex environments where occlusions are common and training data is scarce. Recent work has demonstrated improved accuracy with deep neural networks, but these methods often do not incorporate prior distributions that could improve localization. Here we present GIMBAL: a hierarchical von Mises-Fisher-Gaussian model that improves upon deep networks' estimates by leveraging spatiotemporal constraints. The spatial constraints come from the animal's skeleton, which induces a curved manifold of keypoint configurations. The temporal constraints come from the postural dynamics, which govern how angles between keypoints change over time. Importantly, the conditional conjugacy of the model permits simple and efficient Bayesian inference algorithms. We assess the model on a unique experimental dataset with video of a freely-behaving rodent from multiple viewpoints and ground-truth motion capture data for 20 keypoints. GIMBAL extends existing techniques, and in doing so offers more accurate estimates of keypoint positions, especially in challenging contexts.

Density of States Estimation for Out of Distribution Detection

Warren Morningstar · Cusuh Ham · Andrew Gallagher · Balaji Lakshminarayanan · Alex Alemi · Joshua Dillon

Perhaps surprisingly, recent studies have shown probabilistic model likelihoods have poor specificity for out-of-distribution (OOD) detection and often assign higher likelihoods to OOD data than in-distribution data. To ameliorate this issue we propose DoSE, the density of states estimator. Drawing on the statistical physics notion of density of states,'' the DoSE decision rule avoids direct comparison of model probabilities, and instead utilizes theprobability of the model probability,'' or indeed the frequency of any reasonable statistic. The frequency is calculated using nonparametric density estimators (e.g., KDE and one-class SVM) which measure the typicality of various model statistics given the training data and from which we can flag test points with low typicality as anomalous. Unlike many other methods, DoSE requires neither labeled data nor OOD examples. DoSE is modular and can be trivially applied to any existing, trained model. We demonstrate DoSE's state-of-the-art performance against other unsupervised OOD detectors on previously established ``hard'' benchmarks.

A constrained risk inequality for general losses

John Duchi · Feng Ruan

We provide a general constrained risk inequality that applies to arbitrary non-decreasing losses, extending a result of Brown and Low [\emph{Ann.~Stat.~1996}]. Given two distributions $P_0$ and $P_1$, we find a lower bound for the risk of estimating a parameter $\theta(P_1)$ under $P_1$ given an upper bound on the risk of estimating the parameter $\theta(P_0)$ under $P_0$. The inequality is a useful pedagogical tool, as its proof relies only on the Cauchy-Schwartz inequality, it applies to general losses, and it transparently gives risk lower bounds on super-efficient and adaptive estimators.

Accumulations of Projections—A Unified Framework for Random Sketches in Kernel Ridge Regression

Yifan Chen · Yun Yang

Building a sketch of an n-by-n empirical kernel matrix is a common approach to accelerate the computation of many kernel methods. In this paper, we propose a unified framework of constructing sketching methods in kernel ridge regression (KRR), which views the sketching matrix S as an accumulation of m rescaled sub-sampling matrices with independent columns. Our framework incorporates two commonly used sketching methods, sub-sampling sketches (known as the Nyström method) and sub-Gaussian sketches, as special cases with m=1 and m=infinity respectively. Under the new framework, we provide a unified error analysis of sketching approximation and show that our accumulation scheme improves the low accuracy of sub-sampling sketches when certain incoherence characteristic is high, and accelerates the more accurate but computationally heavier sub-Gaussian sketches. By optimally choosing the number m of accumulations, we show that a best trade-off between computational efficiency and statistical accuracy can be achieved. In practice, the sketching method can be as efficiently implemented as the sub-sampling sketches, as only minor extra matrix additions are needed. Our empirical evaluations also demonstrate that the proposed method may attain the accuracy close to sub-Gaussian sketches, while is as efficient as sub-sampling-based sketches.

Active Learning under Label Shift

Eric Zhao · Anqi Liu · Animashree Anandkumar · Yisong Yue

We address the problem of active learning under label shift: when the class proportions of source and target domains differ. We introduce a "medial distribution" to incorporate a tradeoff between importance weighting and class-balanced sampling and propose their combined usage in active learning. Our method is known as Mediated Active Learning under Label Shift (MALLS). It balances the bias from class-balanced sampling and the variance from importance weighting. We prove sample complexity and generalization guarantees for MALLS which show active learning reduces asymptotic sample complexity even under arbitrary label shift. We empirically demonstrate MALLS scales to high-dimensional datasets and can reduce the sample complexity of active learning by 60% in deep active learning tasks.

Competing AI: How does competition feedback affect machine learning?

Tony Ginart · Eva Zhang · Yongchan Kwon · James Zou

This papers studies how competition affects machine learning (ML) predictors. As ML becomes more ubiquitous, it is often deployed by companies to compete over customers. For example, digital platforms like Yelp use ML to predict user preference and make recommendations. A service that is more often queried by users, perhaps because it more accurately anticipates user preferences, is also more likely to obtain additional user data (e.g. in the form of a Yelp review). Thus, competing predictors cause feedback loops whereby a predictor's performance impacts what training data it receives and biases its predictions over time. We introduce a flexible model of competing ML predictors that enables both rapid experimentation and theoretical tractability. We show with empirical and mathematical analysis that competition causes predictors to specialize for specific sub-populations at the cost of worse performance over the general population. We further analyze the impact of predictor specialization on the overall prediction quality experienced by users. We show that having too few or too many competing predictors in a market can hurt the overall prediction quality. Our theory is complemented by experiments on several real datasets using popular learning algorithms, such as neural networks and nearest neighbor methods.

Random Coordinate Underdamped Langevin Monte Carlo

Zhiyan Ding · Qin Li · Jianfeng Lu · Stephen Wright

The Underdamped Langevin Monte Carlo (ULMC) is a popular Markov chain Monte Carlo sampling method. It requires the computation of the full gradient of the log-density at each iteration, an expensive operation if the dimension of the problem is high. We propose a sampling method called Random Coordinate ULMC (RC-ULMC), which selects a single coordinate at each iteration to be updated and leaves the other coordinates untouched. We investigate the computational complexity of RC-ULMC and compare it with the classical ULMC for strongly log-concave probability distributions. We show that RC-ULMC is always cheaper than the classical ULMC, with a significant cost reduction when the problem is highly skewed and high dimensional. Our complexity bound for RC-ULMC is also tight in terms of dimension dependence.

Differentiable Greedy Algorithm for Monotone Submodular Maximization: Guarantees, Gradient Estimators, and Applications

Shinsaku Sakaue

Motivated by, e.g., sensitivity analysis and end-to-end learning, the demand for differentiable optimization algorithms has been increasing. This paper presents a theoretically guaranteed differentiable greedy algorithm for monotone submodular function maximization. We smooth the greedy algorithm via randomization, and prove that it almost recovers original approximation guarantees in expectation for the cases of cardinality and $\kappa$-extendible system constraints. We then present how to efficiently compute gradient estimators of any expected output-dependent quantities. We demonstrate the usefulness of our method by instantiating it for various applications.

Differentiable Causal Discovery Under Unmeasured Confounding

Rohit Bhattacharya · Tushar Nagarajan · Daniel Malinsky · Ilya Shpitser

The data drawn from biological, economic, and social systems are often confounded due to the presence of unmeasured variables. Prior work in causal discovery has focused on discrete search procedures for selecting acyclic directed mixed graphs (ADMGs), specifically ancestral ADMGs, that encode ordinary conditional independence constraints among the observed variables of the system. However, confounded systems also exhibit more general equality restrictions that cannot be represented via these graphs, placing a limit on the kinds of structures that can be learned using ancestral ADMGs. In this work, we derive differentiable algebraic constraints that fully characterize the space of ancestral ADMGs, as well as more general classes of ADMGs, arid ADMGs and bow-free ADMGs, that capture all equality restrictions on the observed variables. We use these constraints to cast causal discovery as a continuous optimization problem and design differentiable procedures to find the best fitting ADMG when the data comes from a confounded linear system of equations with correlated errors. We demonstrate the efficacy of our method through simulations and application to a protein expression dataset. Code implementing our methods is open-source and publicly available at and will be incorporated into the Ananke package.

Algorithms for Fairness in Sequential Decision Making

Min Wen · Osbert Bastani · Ufuk Topcu

It has recently been shown that if feedback effects of decisions are ignored, then imposing fairness constraints such as demographic parity or equality of opportunity can actually exacerbate unfairness. We propose to address this challenge by modeling feedback effects as Markov decision processes (MDPs). First, we propose analogs of fairness properties for the MDP setting. Second, we propose algorithms for learning fair decision-making policies for MDPs. Finally, we demonstrate the need to account for dynamical effects using simulations on a loan applicant MDP.

Minimax Optimal Regression over Sobolev Spaces via Laplacian Regularization on Neighborhood Graphs

Alden Green · Sivaraman Balakrishnan · Ryan Tibshirani

In this paper we study the statistical properties of Laplacian smoothing, a graph-based approach to nonparametric regression. Under standard regularity conditions, we establish upper bounds on the error of the Laplacian smoothing estimator \smash{$\widehat{f}$}, and a goodness-of-fit test also based on \smash{$\widehat{f}$}. These upper bounds match the minimax optimal estimation and testing rates of convergence over the first-order Sobolev class $H^1(\mathcal{X})$, for $\mathcal{X} \subseteq \mathbb{R}^d$ and $1 \leq d < 4$; in the estimation problem, for $d = 4$, they are optimal modulo a $\log n$ factor. Additionally, we prove that Laplacian smoothing is manifold-adaptive: if $\mathcal{X} \subseteq \mathbb{R}^d$ is an $m$-dimensional manifold with $m < d$, then the error rate of Laplacian smoothing (in either estimation or testing) depends only on $m$, in the same way it would if $\mathcal{X}$ were a full-dimensional set in $\mathbb{R}^m$.

The Base Measure Problem and its Solution

Alexey Radul · Boris Alexeev

Probabilistic programming systems generally compute with probability density functions, leaving the base measure of each such function implicit. This mostly works, but creates problems when densities with respect to different base measures are accidentally combined or compared. Mistakes also happen when computing volume corrections for continuous changes of variables, which in general depend on the support measure. We motivate and clarify the problem in the context of a composable library of probability distributions and bijective transformations. We solve the problem by standardizing on Hausdorff measure as a base, and deriving formulas for comparing and combining mixed-dimension densities, as well as updating densities with respect to Hausdorff measure under diffeomorphic transformations. We also propose a software architecture that implements these formulas efficiently in the common case. We hope that by adopting our solution, probabilistic programming systems can become more robust and general, and make a broader class of models accessible to practitioners.

Sample Elicitation

Jiaheng Wei · Zuyue Fu · Yang Liu · Xingyu Li · Zhuoran Yang · Zhaoran Wang

It is important to collect credible training samples $(x,y)$ for building data-intensive learning systems (e.g., a deep learning system). Asking people to report complex distribution $p(x)$, though theoretically viable, is challenging in practice. This is primarily due to the cognitive loads required for human agents to form the report of this highly complicated information. While classical elicitation mechanisms apply to eliciting a complex and generative (and continuous) distribution $p(x)$, we are interested in eliciting samples $x_i \sim p(x)$ from agents directly. We coin the above problem sample elicitation. This paper introduces a deep learning aided method to incentivize credible sample contributions from self-interested and rational agents. We show that with an accurate estimation of a certain $f$-divergence function we can achieve approximate incentive compatibility in eliciting truthful samples. We then present an efficient estimator with theoretical guarantees via studying the variational forms of the $f$-divergence function. We also show a connection between this sample elicitation problem and $f$-GAN, and how this connection can help reconstruct an estimator of the distribution based on collected samples. Experiments on synthetic data, MNIST, and CIFAR-10 datasets demonstrate that our mechanism elicits truthful samples. Our implementation is available at

Bayesian Coresets: Revisiting the Nonconvex Optimization Perspective

Yibo Zhang · Rajiv Khanna · Anastasios Kyrillidis · Sanmi Koyejo

Bayesian coresets have emerged as a promising approach for implementing scalable Bayesian inference. The Bayesian coreset problem involves selecting a (weighted) subset of the data samples, such that the posterior inference using the selected subset closely approximates the posterior inference using the full dataset. This manuscript revisits Bayesian coresets through the lens of sparsity constrained optimization. Leveraging recent advances in accelerated optimization methods, we propose and analyze a novel algorithm for coreset selection. We provide explicit convergence rate guarantees and present an empirical evaluation on a variety of benchmark datasets to highlight our proposed algorithm's superior performance compared to state-of-the-art on speed and accuracy.

The Sample Complexity of Meta Sparse Regression

Zhanyu Wang · Jean Honorio

This paper addresses the meta-learning problem in sparse linear regression with infinite tasks. We assume that the learner can access several similar tasks. The goal of the learner is to transfer knowledge from the prior tasks to a similar but novel task. For $p$ parameters, size of the support set $k$, and $l$ samples per task, we show that $T \in O((k \log (p-k)) / l)$ tasks are sufficient in order to recover the common support of all tasks. With the recovered support, we can greatly reduce the sample complexity for estimating the parameter of the novel task, i.e., $l \in O(1)$ with respect to $T$ and $p$. We also prove that our rates are minimax optimal. A key difference between meta-learning and the classical multi-task learning, is that meta-learning focuses only on the recovery of the parameters of the novel task, while multi-task learning estimates the parameter of all tasks, which requires $l$ to grow with $T$. Instead, our efficient meta-learning estimator allows for $l$ to be constant with respect to $T$ (i.e., few-shot learning).

Nonlinear Projection Based Gradient Estimation for Query Efficient Blackbox Attacks

Huichen Li · Linyi Li · Xiaojun Xu · Xiaolu Zhang · Shuang Yang · Bo Li

Gradient estimation and vector space projection have been studied as two distinct topics. We aim to bridge the gap between the two by investigating how to efficiently estimate gradient based on a projected low-dimensional space. We first provide lower and upper bounds for gradient estimation under both linear and nonlinear projections, and outline checkable sufficient conditions under which one is better than the other. Moreover, we analyze the query complexity for the projection-based gradient estimation and present a sufficient condition for query-efficient estimators. Built upon our theoretic analysis, we propose a novel query-efficient Nonlinear Gradient Projection-based Boundary Blackbox Attack (NonLinear-BA). We conduct extensive experiments on four image datasets: ImageNet, CelebA, CIFAR-10, and MNIST, and show the superiority of the proposed methods compared with the state-of-the-art baselines. In particular, we show that the projection-based boundary blackbox attacks are able to achieve much smaller magnitude of perturbations with 100% attack success rate based on efficient queries. Both linear and nonlinear projections demonstrate their advantages under different conditions. We also evaluate NonLinear-BA against the commercial online API MEGVII Face++, and demonstrate the high blackbox attack performance both quantitatively and qualitatively. The code is publicly available at

Right Decisions from Wrong Predictions: A Mechanism Design Alternative to Individual Calibration

Shengjia Zhao · Stefano Ermon

Decision makers often need to rely on imperfect probabilistic forecasts. While average performance metrics are typically available, it is difficult to assess the quality of individual forecasts and the corresponding utilities. To convey confidence about individual predictions to decision-makers, we propose a compensation mechanism ensuring that the forecasted utility matches the actually accrued utility. While a naive scheme to compensate decision-makers for prediction errors can be exploited and might not be sustainable in the long run, we propose a mechanism based on fair bets and online learning that provably cannot be exploited. We demonstrate an application showing how passengers could confidently optimize individual travel plans based on flight delay probabilities estimated by an airline.

SONIA: A Symmetric Blockwise Truncated Optimization Algorithm

Majid Jahani · MohammadReza Nazari · Rachael Tappenden · Albert Berahas · Martin Takac

This work presents a new optimization algorithm for empirical risk minimization. The algorithm bridges the gap between first- and second-order methods by computing a search direction that uses a second-order-type update in one subspace, coupled with a scaled steepest descent step in the orthogonal complement. To this end, partial curvature information is incorporated to help with ill-conditioning, while simultaneously allowing the algorithm to scale to the large problem dimensions often encountered in machine learning applications. Theoretical results are presented to confirm that the algorithm converges to a stationary point in both the strongly convex and nonconvex cases. A stochastic variant of the algorithm is also presented, along with corresponding theoretical guarantees. Numerical results confirm the strengths of the new approach on standard machine learning problems.

Online Robust Control of Nonlinear Systems with Large Uncertainty

Dimitar Ho · Hoang Le · John Doyle · Yisong Yue

Robust control is a core approach for controlling systems with performance guarantees that are robust to modeling error, and is widely used in real-world systems. However, current robust control approaches can only handle small system uncertainty, and thus require significant effort in system identification prior to controller design. We present an online approach that robustly controls a nonlinear system under large model uncertainty. Our approach is based on decomposing the problem into two sub-problems, robust control design'' (which assumes small model uncertainty) andchasing consistent models'', which can be solved using existing tools from control theory and online learning, respectively. We provide a learning convergence analysis that yields a finite mistake bound on the number of times performance requirements are not met and can provide strong safety guarantees, by bounding the worst-case state deviation. To the best of our knowledge, this is the first approach for online robust control of nonlinear systems with such learning theoretic and safety guarantees. We also show how to instantiate this framework for general robotic systems, demonstrating the practicality of our approach.

Shapley Flow: A Graph-based Approach to Interpreting Model Predictions

Jiaxuan Wang · Jenna Wiens · Scott Lundberg

Many existing approaches for estimating feature importance are problematic because they ignore or hide dependencies among features. A causal graph, which encodes the relationships among input variables, can aid in assigning feature importance. However, current approaches that assign credit to nodes in the causal graph fail to explain the entire graph. In light of these limitations, we propose Shapley Flow, a novel approach to interpreting machine learning models. It considers the entire causal graph, and assigns credit to edges instead of treating nodes as the fundamental unit of credit assignment. Shapley Flow is the unique solution to a generalization of the Shapley value axioms for directed acyclic graphs. We demonstrate the benefit of using Shapley Flow to reason about the impact of a model's input on its output. In addition to maintaining insights from existing approaches, Shapley Flow extends the flat, set-based, view prevalent in game theory based explanation methods to a deeper, graph-based, view. This graph-based view enables users to understand the flow of importance through a system, and reason about potential interventions.

Improving Classifier Confidence using Lossy Label-Invariant Transformations

Sooyong Jang · Insup Lee · James Weimer

Providing reliable model uncertainty estimates is imperative to enabling robust decision making by autonomous agents and humans alike. While recently there have been significant advances in confidence calibration for trained models, examples with poor calibration persist in most calibrated models. Consequently, multiple techniques have been proposed that leverage label-invariant transformations of the input (i.e., an input manifold) to improve worst-case confidence calibration. However, manifold-based confidence calibration techniques generally do not scale and/or require expensive retraining when applied to models with large input spaces (e.g., ImageNet). In this paper, we present the recursive lossy label-invariant calibration (ReCal) technique that leverages label-invariant transformations of the input that induce a loss of discriminatory information to recursively group (and calibrate) inputs – without requiring model retraining. We show that ReCal outperforms other calibration methods on multiple datasets, especially, on large-scale datasets such as ImageNet.

Associative Convolutional Layers

Hamed Omidvar · Vahideh Akhlaghi · Hao Su · Massimo Franceschetti · Rajesh Gupta

We provide a general and easy to implement method for reducing the number of parameters of Convolutional Neural Networks (CNNs) during the training and inference phases. We introduce a simple trainable auxiliary neural network which can generate approximate versions of ``slices'' of the sets of convolutional filters of any CNN architecture from a low dimensional ``code'' space. These slices are then concatenated to form the sets of filters in the CNN architecture. The auxiliary neural network, which we call “Convolutional Slice Generator” (CSG), is unique to the network and provides the association among its convolutional layers. We apply our method to various CNN architectures including ResNet, DenseNet, MobileNet and ShuffleNet. Experiments on CIFAR-10 and ImageNet-1000, without any hyper-parameter tuning, show that our approach reduces the network parameters by approximately $2\times$ while the reduction in accuracy is confined to within one percent and sometimes the accuracy even improves after compression. Interestingly, through our experiments, we show that even when the CSG takes random binary values for its weights that are not learned, still acceptable performances are achieved. To show that our approach generalizes to other tasks, we apply it to an image segmentation architecture, Deeplab V3, on the Pascal VOC 2012 dataset. Results show that without any parameter tuning, there is $\approx 2.3\times$ parameter reduction and the mean Intersection over Union (mIoU) drops by $\approx 3\%$. Finally, we provide comparisons with several related methods showing the superiority of our method in terms of accuracy.

An Optimal Reduction of TV-Denoising to Adaptive Online Learning

Dheeraj Baby · Xuandong Zhao · Yu-Xiang Wang

We consider the problem of estimating a function from $n$ noisy samples whose discrete Total Variation (TV) is bounded by $C_n$. We reveal a deep connection to the seemingly disparate problem of \emph{Strongly Adaptive} online learning [Daniely et al 2015] and provide an $O(n \log n)$ time algorithm that attains the near minimax optimal rate of $\tilde O (n^{1/3}C_n^{2/3})$ under squared error loss. The resulting algorithm runs online and optimally \emph{adapts} to the \emph{unknown} smoothness parameter $C_n$. This leads to a new and more versatile alternative to wavelets-based methods for (1) adaptively estimating TV bounded functions; (2) online forecasting of TV bounded trends in time series.

TenIPS: Inverse Propensity Sampling for Tensor Completion

Chengrun Yang · Lijun Ding · Ziyang Wu · Madeleine Udell

Tensors are widely used to represent multiway arrays of data. The recovery of missing entries in a tensor has been extensively studied, generally under the assumption that entries are missing completely at random (MCAR). However, in most practical settings, observations are missing not at random (MNAR): the probability that a given entry is observed (also called the propensity) may depend on other entries in the tensor or even on the value of the missing entry. In this paper, we study the problem of completing a partially observed tensor with MNAR observations, without prior information about the propensities. To complete the tensor, we assume that both the original tensor and the tensor of propensities have low multilinear rank. The algorithm first estimates the propensities using a convex relaxation and then predicts missing values using a higher-order SVD approach, reweighting the observed tensor by the inverse propensities. We provide finite-sample error bounds on the resulting complete tensor. Numerical experiments demonstrate the effectiveness of our approach.

Revisiting Model-Agnostic Private Learning: Faster Rates and Active Learning

Chong Liu · Yuqing Zhu · Kamalika Chaudhuri · Yu-Xiang Wang

The Private Aggregation of Teacher Ensembles (PATE) framework is one of the most promising recent approaches in differentially private learning. Existing theoretical analysis shows that PATE consistently learns any VC-classes in the realizable setting, but falls short in explaining its success in more general cases where the error rate of the optimal classifier is bounded away from zero. We fill in this gap by introducing the Tsybakov Noise Condition (TNC) and establish stronger and more interpretable learning bounds. These bounds provide new insights into when PATE works and improve over existing results even in the narrower realizable setting. We also investigate the compelling idea of using active learning for saving privacy budget. The novel components in the proofs include a more refined analysis of the majority voting classifier --- which could be of independent interest --- and an observation that the synthetic ``student'' learning problem is nearly realizable by construction under the Tsybakov noise condition.

Accelerating Metropolis-Hastings with Lightweight Inference Compilation

Feynman Liang · Nimar Arora · Nazanin Tehrani · Yucen Li · Michael Tingley · Erik Meijer

In order to construct accurate proposers for Metropolis-Hastings Markov Chain Monte Carlo, we integrate ideas from probabilistic graphical models and neural networks in an open-source framework we call Lightweight Inference Compilation (LIC). LIC implements amortized inference within an open-universe declarative probabilistic programming language (PPL). Graph neural networks are used to parameterize proposal distributions as functions of Markov blankets, which during ``compilation'' are optimized to approximate single-site Gibbs sampling distributions. Unlike prior work in inference compilation (IC), LIC forgoes importance sampling of linear execution traces in favor of operating directly on Bayesian networks. Through using a declarative PPL, the Markov blankets of nodes (which may be non-static) are queried at inference-time to produce proposers Experimental results show LIC can produce proposers which have less parameters, greater robustness to nuisance random variables, and improved posterior sampling in a Bayesian logistic regression and n-schools inference application.

Fast Adaptation with Linearized Neural Networks

Wesley Maddox · Shuai Tang · Pablo Moreno · Andrew Gordon Wilson · Andreas Damianou

The inductive biases of trained neural networks are difficult to understand and, consequently, to adapt to new settings. We study the inductive biases of linearizations of neural networks, which we show to be surprisingly good summaries of the full network functions. Inspired by this finding, we propose a technique for embedding these inductive biases into Gaussian processes through a kernel designed from the Jacobian of the network. In this setting, domain adaptation takes the form of interpretable posterior inference, with accompanying uncertainty estimation. This inference is analytic and free of local optima issues found in standard techniques such as fine-tuning neural network weights to a new task. We develop significant computational speed-ups based on matrix multiplies, including a novel implementation for scalable Fisher vector products. Our experiments on both image classification and regression demonstrate the promise and convenience of this framework for transfer learning, compared to neural network fine-tuning.