AISTATS 2021 Papers

In a low-rank linear bandit problem, the reward of an action (represented by a matrix of size $d_1 \times d_2$) is the inner product between the action and an unknown low-rank matrix $\Theta^*$. We propose an algorithm based on a novel combination of online-to-confidence-set conversion \citep{abbasi2012online} and the exponentially weighted average forecaster constructed by a covering of low-rank matrices. In $T$ rounds, our algorithm achieves $\widetilde{O}((d_1+d_2)^{3/2}\sqrt{rT})$ regret that improves upon the standard linear bandit regret bound of $\widetilde{O}(d_1d_2\sqrt{T})$ when the rank of $\Theta^*$: $r \ll \min\{d_1,d_2\}$. We also extend our algorithmic approach to the generalized linear setting to get an algorithm which enjoys a similar bound under regularity conditions on the link function. To get around the computational intractability of covering based approaches, we propose an efficient algorithm by extending the "Explore-Subspace-Then-Refine" algorithm of \citet{jun2019bilinear}. Our efficient algorithm achieves $\widetilde{O}((d_1+d_2)^{3/2}\sqrt{rT})$ regret under a mild condition on the action set $\mathcal{X}$ and the $r$-th singular value of $\Theta^*$. Our upper bounds match the conjectured lower bound of \cite{jun2019bilinear} for a subclass of low-rank linear bandit problems. Further, we show that existing lower bounds for the sparse linear bandit problem strongly suggest that our regret bounds are unimprovable. To complement our theoretical contributions, we also conduct experiments to demonstrate that our algorithm can greatly outperform the performance of the standard linear bandit approach when $\Theta^*$ is low-rank.

Longitudinal datasets measured repeatedly over time from individual subjects, arise in many biomedical, psychological, social, and other studies. A common approach to analyse high-dimensional data that contains missing values is to learn a low-dimensional representation using variational autoencoders (VAEs). However, standard VAEs assume that the learnt representations are i.i.d., and fail to capture the correlations between the data samples. We propose the Longitudinal VAE (L-VAE), that uses a multi-output additive Gaussian process (GP) prior to extend the VAE’s capability to learn structured low-dimensional representations imposed by auxiliary covariate information, and derive a new KL divergence upper bound for such GPs. Our approach can simultaneously accommodate both time-varying shared and random effects, produce structured low-dimensional representations, disentangle effects of individual covariates or their interactions, and achieve highly accurate predictive performance. We compare our model against previous methods on synthetic as well as clinical datasets, and demonstrate the state-of-the-art performance in data imputation, reconstruction, and long-term prediction tasks.

We propose a new reinforcement learning algorithm derived from a regularized linear-programming formulation of optimal control in MDPs. The method is closely related to the classic Relative Entropy Policy Search (REPS) algorithm of Peters et al. (2010), with the key difference that our method introduces a Q-function that enables efficient exact model-free implementation. The main feature of our algorithm (called QREPS) is a convex loss function for policy evaluation that serves as a theoretically sound alternative to the widely used squared Bellman error. We provide a practical saddle-point optimization method for minimizing this loss function and provide an error-propagation analysis that relates the quality of the individual updates to the performance of the output policy. Finally, we demonstrate the effectiveness of our method on a range of benchmark problems.

We address the challenge of learning factored policies in cooperative MARL scenarios. In particular, we consider the situation in which a team of agents collaborates to optimize a common cost. The goal is to obtain factored policies that determine the individual behavior of each agent so that the resulting joint policy is optimal. The main contribution of this work is the introduction of Logical Team Q-learning (LTQL). LTQL does not rely on assumptions about the environment and hence is generally applicable to any collaborative MARL scenario. We derive LTQL as a stochastic approximation to a dynamic programming method we introduce in this work. We conclude the paper by providing experiments (both in the tabular and deep settings) that illustrate the claims.

Providing meaningful privacy to users of location based services is particularly challenging when multiple locations are revealed in a short period of time. This is primarily due to the tremendous degree of dependence that can be anticipated between points. We propose a Rényi divergence based privacy framework for bounding expected privacy loss for conditionally dependent data. Additionally, we demonstrate an algorithm for achieving this privacy under Gaussian process conditional priors. This framework both exemplifies why conditionally dependent data is so challenging to protect and offers a strategy for preserving privacy to within a fixed radius for sensitive locations in a user’s trace.

This paper addresses the problem of localizing change points in high-dimensional linear regression models with piecewise constant regression coefficients. We develop a dynamic programming approach to estimate the locations of the change points whose performance improves upon the current state-of-the-art, even as the dimension, the sparsity of the regression coefficients, the temporal spacing between two consecutive change points, and the magnitude of the difference of two consecutive regression coefficient vectors are allowed to vary with the sample size. Furthermore, we devise a computationally-efficient refinement procedure that provably reduces the localization error of preliminary estimates of the change points. We demonstrate minimax lower bounds on the localization error that nearly match the upper bound on the localization error of our methodology and show that the signal-to-noise condition we impose is essentially the weakest possible based on information-theoretic arguments. Extensive numerical results support our theoretical findings, and experiments on real air quality data reveal change points supported by historical information not used by the algorithm.

Local SGD is a promising approach to overcome the communication overhead in distributed learning by reducing the synchronization frequency among worker nodes. Despite the recent theoretical advances of local SGD in empirical risk minimization, the efficiency of its counterpart in minimax optimization remains unexplored. Motivated by large scale minimax learning problems, such as adversarial robust learning and GANs, we propose local Stochastic Gradient Descent Ascent (local SGDA), where the primal and dual variables can be trained locally and averaged periodically to significantly reduce the number of communications. We show that local SGDA can provably optimize distributed minimax problems in both homogeneous and heterogeneous data with reduced number of communications and establish convergence rates under strongly-convex-strongly-concave and nonconvex-strongly-concave settings. In addition, we propose a novel variant, dubbed as local SGDA+, to solve nonconvex-nonconcave problems. We also give corroborating empirical evidence on different distributed minimax problems.

We present a unified framework for analyzing local SGD methods in the convex and strongly convex regimes for distributed/federated training of supervised machine learning models. We recover several known methods as a special case of our general framework, including Local SGD/FedAvg, SCAFFOLD, and several variants of SGD not originally designed for federated learning. Our framework covers both the identical and heterogeneous data settings, supports both random and deterministic number of local steps, and can work with a wide array of local stochastic gradient estimators, including shifted estimators which are able to adjust the fixed points of local iterations for faster convergence. As an application of our framework, we develop multiple novel FL optimizers which are superior to existing methods. In particular, we develop the first linearly converging local SGD method which does not require any data homogeneity or other strong assumptions.

This work addresses adversarial robustness in deep learning by considering deep networks with stochastic local winner-takes-all (LWTA) activations. This type of network units result in sparse representations from each model layer, as the units are organized in blocks where only one unit generates a non-zero output. The main operating principle of the introduced units lies on stochastic arguments, as the network performs posterior sampling over competing units to select the winner. We combine these LWTA arguments with tools from the field of Bayesian non-parametrics, specifically the stick-breaking construction of the Indian Buffet Process, to allow for inferring the sub-part of each layer that is essential for modeling the data at hand. Then, inference is performed by means of stochastic variational Bayes. We perform a thorough experimental evaluation of our model using benchmark datasets. As we show, our method achieves high robustness to adversarial perturbations, with state-of-the-art performance in powerful adversarial attack schemes.

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.

One goal in Bayesian machine learning is to encode prior knowledge into prior distributions, to model data efficiently. We consider prior knowledge from systems of linear partial differential equations together with their boundary conditions. We construct multi-output Gaussian process priors with realizations in the solution set of such systems, in particular only such solutions can be represented by Gaussian process regression. The construction is fully algorithmic via Gröbner bases and it does not employ any approximation. It builds these priors combining two parametrizations via a pullback: the first parametrizes the solutions for the system of differential equations and the second parametrizes all functions adhering to the boundary conditions.

Recently, invariant risk minimization (IRM) (Arjovsky et al. 2019) was proposed as a promising solution to address out-of-distribution (OOD) generalization. In Ahuja et al. (2020), it was shown that solving for the Nash equilibria of a new class of “ensemble-games” is equivalent to solving IRM. In this work, we extend the framework in Ahuja et al. (2020) for linear regressions by projecting the ensemble-game on an $\ell_{\infty}$ ball. We show that such projections help achieve non-trivial out-of-distribution guarantees despite not achieving perfect invariance. For linear models with confounders, we prove that Nash equilibria of these games are closer to the ideal OOD solutions than the standard empirical risk minimization (ERM) and we also provide learning algorithms that provably converge to these Nash Equilibria. Empirical comparisons of the proposed approach with the state-of-the-art show consistent gains in achieving OOD solutions in several settings involving anti-causal variables and confounders.

There is an increasing use of algorithms to inform decisions in many settings, from student evaluations, college admissions, to credit scoring. These decisions are made by applying a decision rule to individual’s observed features. Given the impacts of these decisions on individuals, decision makers are increasingly required to be transparent on their decision making to offer the “right to explanation.” Meanwhile, being transparent also invites potential manipulations, also known as gaming, that the individuals can utilize the knowledge to strategically alter their features in order to receive a more beneficial decision. In this work, we study the problem of \emph{robust} decision-making under strategic behavior. Prior works often assume that the decision maker has full knowledge of individuals’ cost structure for manipulations. We study the robust variant that relaxes this assumption: The decision maker does not have full knowledge but knows only a subset of the individuals’ available actions and associated costs. To approach this non-quantifiable uncertainty, we define robustness based on the worst-case guarantee of a decision, over all possible actions (including actions unknown to the decision maker) individuals might take. A decision rule is called \emph{robust optimal} if its worst case performance is (weakly) better than that of all other decision rules. Our main contributions are two-fold. First, we provide a crisp characterization of the above robust optimality: For any decision rules under mild conditions that are robust optimal, there exists a linear decision rule that is equally robust optimal. Second, we explore the computational problem of searching for the robust optimal decision rule and interestingly, we demonstrate the problem is closely related to distributionally robust optimization. We believe our results promotes the use of simple linear decisions with uncertain individual manipulations.

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.

We study learning algorithms that seek to minimize the conditional value-at-risk (CVaR), when all the learner knows is that the losses (and gradients) incurred may be heavy-tailed. We begin by studying a general-purpose estimator of CVaR for potentially heavy-tailed random variables, which is easy to implement in practice, and requires nothing more than finite variance and a distribution function that does not change too fast or slow around just the quantile of interest. With this estimator in hand, we then derive a new learning algorithm which robustly chooses among candidates produced by stochastic gradient-driven sub-processes, obtain excess CVaR bounds, and finally complement the theory with a regression application.

Due to the over-parameterization nature, neural networks are a powerful tool for nonlinear function approximation. In order to achieve good generalization on unseen data, a suitable inductive bias is of great importance for neural networks. One of the most straightforward ways is to regularize the neural network with some additional objectives. L2 regularization serves as a standard regularization for neural networks. Despite its popularity, it essentially regularizes one dimension of the individual neuron, which is not strong enough to control the capacity of highly over-parameterized neural networks. Motivated by this, hyperspherical uniformity is proposed as a novel family of relational regularizations that impact the interaction among neurons. We consider several geometrically distinct ways to achieve hyperspherical uniformity. The effectiveness of hyperspherical uniformity is justified by theoretical insights and empirical evaluations.

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.

Adversarial training provides a principled approach for training robust neural networks. From an optimization perspective, adversarial training is essentially solving a bilevel optimization problem. The leader problem is trying to learn a robust classifier, while the follower maximization is trying to generate adversarial samples. Unfortunately, such a bilevel problem is difficult to solve due to its highly complicated structure. This work proposes a new adversarial training method based on a generic learning-to-learn (L2L) framework. Specifically, instead of applying existing hand-designed algorithms for the inner problem, we learn an optimizer, which is parametrized as a convolutional neural network. At the same time, a robust classifier is learned to defense the adversarial attack generated by the learned optimizer. Experiments over CIFAR-10 and CIFAR-100 datasets demonstrate that L2L outperforms existing adversarial training methods in both classification accuracy and computational efficiency. Moreover, our L2L framework can be extended to generative adversarial imitation learning and stabilize the training.

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.