Poster Session 3


Top-m identification for linear bandits

Clémence Réda · Emilie Kaufmann · Andrée Delahaye-Duriez

Motivated by an application to drug repurposing, we propose the first algorithms to tackle the identification of the m ≥ 1 arms with largest means in a linear bandit model, in the fixed-confidence setting. These algorithms belong to the generic family of Gap-Index Focused Algorithms (GIFA) that we introduce for Top-m identification in linear bandits. We propose a unified analysis of these algorithms, which shows how the use of contexts might decrease the sample complexity. We further validate these algorithms empirically on simulated data and on a simple drug repurposing task.

Quantum Tensor Networks, Stochastic Processes, and Weighted Automata

Sandesh Adhikary · Siddarth Srinivasan · Jacob E Miller · Guillaume Rabusseau · Byron Boots

Modeling joint probability distributions over sequences has been studied from many perspectives. The physics community developed matrix product states, a tensor-train decomposition for probabilistic modeling, motivated by the need to tractably model many-body systems. But similar models have also been studied in the stochastic processes and weighted automata literature, with little work on how these bodies of work relate to each other. We address this gap by showing how stationary or uniform versions of popular quantum tensor network models have equivalent representations in the stochastic processes and weighted automata literature, in the limit of infinitely long sequences. We demonstrate several equivalence results between models used in these three communities: (i) uniform variants of matrix product states, Born machines and locally purified states from the quantum tensor networks literature, (ii) predictive state representations, hidden Markov models, norm-observable operator models and hidden quantum Markov models from the stochastic process literature, and (iii) stochastic weighted automata, probabilistic automata and quadratic automata from the formal languages literature. Such connections may open the door for results and methods developed in one area to be applied in another.

Multi-Armed Bandits with Cost Subsidy

Deeksha Sinha · Karthik Abinav Sankararaman · Abbas Kazerouni · Vashist Avadhanula

In this paper, we consider a novel variant of the multi-armed bandit (MAB) problem, MAB with cost subsidy, which models many real-life applications where the learning agent has to pay to select an arm and is concerned about optimizing cumulative costs and rewards. We present two applications, intelligent SMS routing problem and ad audience optimization problem faced by several businesses (especially online platforms), and show how our problem uniquely captures key features of these applications. We show that naive generalizations of existing MAB algorithms like Upper Confidence Bound and Thompson Sampling do not perform well for this problem. We then establish a fundamental lower bound on the performance of any online learning algorithm for this problem, highlighting the hardness of our problem in comparison to the classical MAB problem. We also present a simple variant of explore-then-commit and establish near-optimal regret bounds for this algorithm. Lastly, we perform extensive numerical simulations to understand the behavior of a suite of algorithms for various instances and recommend a practical guide to employ different algorithms.

Logical Team Q-learning: An approach towards factored policies in cooperative MARL

Lucas Cassano · Ali H. Sayed

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.

Online k-means Clustering

Vincent Cohen-Addad · Benjamin Guedj · Varun Kanade · Guy Rom

We study the problem of learning a clustering of an online set of points. The specific formulation we use is the k-means objective: At each time step the algorithm has to maintain a set of k candidate centers and the loss incurred by the algorithm is the squared distance between the new point and the closest center. The goal is to minimize regret with respect to the best solution to the k-means objective in hindsight. We show that provided the data lies in a bounded region, learning is possible, namely an implementation of the Multiplicative Weights Update Algorithm (MWUA) using a discretized grid achieves a regret bound of $\tilde{O}(\sqrt{T})$ in expectation. We also present an online-to-offline reduction that shows that an efficient no-regret online algorithm (despite being allowed to choose a different set of candidate centres at each round) implies an offline efficient algorithm for the k-means problem, which is known to be NP-hard. In light of this hardness, we consider the slightly weaker requirement of comparing regret with respect to $(1 + \epsilon)OPT$ and present a no-regret algorithm with runtime $O\left(T \mathrm{poly}(\log(T),k,d,1/\epsilon)^{O(kd)}\right)$. Our algorithm is based on maintaining a set of points of bounded size which is a coreset that helps identifying the \emph{relevant} regions of the space for running an adaptive, more efficient, variant of the MWUA. We show that simpler online algorithms, such as \emph{Follow The Leader} (FTL), fail to produce sublinear regret in the worst case. We also report preliminary experiments with synthetic and real-world data. Our theoretical results answer an open question of Dasgupta (2008).

Aggregating Incomplete and Noisy Rankings

Dimitris Fotakis · Alkis Kalavasis · Konstantinos Stavropoulos

We consider the problem of learning the true ordering of a set of alternatives from largely incomplete and noisy rankings. We introduce a natural generalization of both the Mallows model, a popular model of ranking distributions, and the extensively studied model of ranking from pairwise comparisons. Our selective Mallows model outputs a noisy ranking on any given subset of alternatives, based on an underlying Mallows distribution. Assuming a sequence of subsets where each pair of alternatives appears frequently enough, we obtain strong asymptotically tight upper and lower bounds on the sample complexity of learning the underlying complete central ranking and the (identities and the) ranking of the top k alternatives from selective Mallows rankings. Moreover, building on the work of (Braverman and Mossel, 2009), we show how to efficiently compute the maximum likelihood complete ranking from selective Mallows rankings.

Online probabilistic label trees

Marek Wydmuch · Kalina Jasinska-Kobus · Devanathan Thiruvenkatachari · Krzysztof Dembczynski

We introduce online probabilistic label trees (OPLTs), an algorithm that trains a label tree classifier in a fully online manner without any prior knowledge about the number of training instances, their features and labels. OPLTs are characterized by low time and space complexity as well as strong theoretical guarantees. They can be used for online multi-label and multi-class classification, including the very challenging scenarios of one- or few-shot learning. We demonstrate the attractiveness of OPLTs in a wide empirical study on several instances of the tasks mentioned above.

Graphical Normalizing Flows

Antoine Wehenkel · Gilles Louppe

Normalizing flows model complex probability distributions by combining a base distribution with a series of bijective neural networks. State-of-the-art architectures rely on coupling and autoregressive transformations to lift up invertible functions from scalars to vectors. In this work, we revisit these transformations as probabilistic graphical models, showing they reduce to Bayesian networks with a pre-defined topology and a learnable density at each node. From this new perspective, we propose the graphical normalizing flow, a new invertible transformation with either a prescribed or a learnable graphical structure. This model provides a promising way to inject domain knowledge into normalizing flows while preserving both the interpretability of Bayesian networks and the representation capacity of normalizing flows. We show that graphical conditioners discover relevant graph structure when we cannot hypothesize it. In addition, we analyze the effect of $\ell_1$-penalization on the recovered structure and on the quality of the resulting density estimation. Finally, we show that graphical conditioners lead to competitive white box density estimators. Our implementation is available at \url{}.

Scalable Gaussian Process Variational Autoencoders

Metod Jazbec · Matt Ashman · Vincent Fortuin · Michael Pearce · Stephan Mandt · Gunnar Rätsch

Conventional variational autoencoders fail in modeling correlations between data points due to their use of factorized priors. Amortized Gaussian process inference through GP-VAEs has led to significant improvements in this regard, but is still inhibited by the intrinsic complexity of exact GP inference. We improve the scalability of these methods through principled sparse inference approaches. We propose a new scalable GP-VAE model that outperforms existing approaches in terms of runtime and memory footprint, is easy to implement, and allows for joint end-to-end optimization of all components.

Kernel Distributionally Robust Optimization: Generalized Duality Theorem and Stochastic Approximation

Jia-Jie Zhu · Wittawat Jitkrittum · Moritz Diehl · Bernhard Schölkopf

We propose kernel distributionally robust optimization (Kernel DRO) using insights from the robust optimization theory and functional analysis. Our method uses reproducing kernel Hilbert spaces (RKHS) to construct a wide range of convex ambiguity sets, which can be generalized to sets based on integral probability metrics and finite-order moment bounds. This perspective unifies multiple existing robust and stochastic optimization methods. We prove a theorem that generalizes the classical duality in the mathematical problem of moments. Enabled by this theorem, we reformulate the maximization with respect to measures in DRO into the dual program that searches for RKHS functions. Using universal RKHSs, the theorem applies to a broad class of loss functions, lifting common limitations such as polynomial losses and knowledge of the Lipschitz constant. We then establish a connection between DRO and stochastic optimization with expectation constraints. Finally, we propose practical algorithms based on both batch convex solvers and stochastic functional gradient, which apply to general optimization and machine learning tasks.

Improved Complexity Bounds in Wasserstein Barycenter Problem

Darina Dvinskikh · Daniil Tiapkin

In this paper, we focus on computational aspects of the Wasserstein barycenter problem. We propose two algorithms to compute Wasserstein barycenters of $m$ discrete measures of size $n$ with accuracy $\e$. The first algorithm, based on mirror prox with a specific norm, meets the complexity of celebrated accelerated iterative Bregman projections (IBP), namely $\widetilde O(mn^2\sqrt n/\e)$, however, with no limitations in contrast to the (accelerated) IBP, which is numerically unstable under small regularization parameter. The second algorithm, based on area-convexity and dual extrapolation, improves the previously best-known convergence rates for the Wasserstein barycenter problem enjoying $\widetilde O(mn^2/\e)$ complexity.

On the convergence of the Metropolis algorithm with fixed-order updates for multivariate binary probability distributions

Kai Brügge · Asja Fischer · Christian Igel

The Metropolis algorithm is arguably the most fundamental Markov chain Monte Carlo (MCMC) method. But the algorithm is not guaranteed to converge to the desired distribution in the case of multivariate binary distributions (e.g., Ising models or stochastic neural networks such as Boltzmann machines) if the variables (sites or neurons) are updated in a fixed order, a setting commonly used in practice. The reason is that the corresponding Markov chain may not be irreducible. We propose a modified Metropolis transition operator that behaves almost always identically to the standard Metropolis operator and prove that it ensures irreducibility and convergence to the limiting distribution in the multivariate binary case with fixed-order updates. The result provides an explanation for the behaviour of Metropolis MCMC in that setting and closes a long-standing theoretical gap. We experimentally studied the standard and modified Metropolis operator for models where they actually behave differently. If the standard algorithm also converges, the modified operator exhibits similar (if not better) performance in terms of convergence speed.

Generating Interpretable Counterfactual Explanations By Implicit Minimisation of Epistemic and Aleatoric Uncertainties

Lisa Schut · Oscar Key · Rory Mc Grath · Luca Costabello · Bogdan Sacaleanu · medb corcoran · Yarin Gal

Counterfactual explanations (CEs) are a practical tool for demonstrating why machine learning classifiers make particular decisions. For CEs to be useful, it is important that they are easy for users to interpret. Existing methods for generating interpretable CEs rely on auxiliary generative models, which may not be suitable for complex datasets, and incur engineering overhead. We introduce a simple and fast method for generating interpretable CEs in a white-box setting without an auxiliary model, by using the predictive uncertainty of the classifier. Our experiments show that our proposed algorithm generates more interpretable CEs, according to IM1 scores (Van Looveren et al., 2019), than existing methods. Additionally, our approach allows us to estimate the uncertainty of a CE, which may be important in safety-critical applications, such as those in the medical domain.

Anderson acceleration of coordinate descent

Quentin Bertrand · Mathurin Massias

Acceleration of first order methods is mainly obtained via inertia à la Nesterov, or via nonlinear extrapolation. The latter has known a recent surge of interest, with successful applications to gradient and proximal gradient techniques. On multiple Machine Learning problems, coordinate descent achieves performance significantly superior to full-gradient methods. Speeding up coordinate descent in practice is not easy: inertially accelerated versions of coordinate descent are theoretically accelerated, but might not always lead to practical speed-ups. We propose an accelerated version of coordinate descent using extrapolation, showing considerable speed up in practice, compared to inertial accelerated coordinate descent and extrapolated (proximal) gradient descent. Experiments on least squares, Lasso, elastic net and logistic regression validate the approach.

Neural Enhanced Belief Propagation on Factor Graphs

Víctor Garcia Satorras · Max Welling

A graphical model is a structured representation of locally dependent random variables. A traditional method to reason over these random variables is to perform inference using belief propagation. When provided with the true data generating process, belief propagation can infer the optimal posterior probability estimates in tree structured factor graphs. However, in many cases we may only have access to a poor approximation of the data generating process, or we may face loops in the factor graph, leading to suboptimal estimates. In this work we first extend graph neural networks to factor graphs (FG-GNN). We then propose a new hybrid model that runs conjointly a FG-GNN with belief propagation. The FG-GNN receives as input messages from belief propagation at every inference iteration and outputs a corrected version of them. As a result, we obtain a more accurate algorithm that combines the benefits of both belief propagation and graph neural networks. We apply our ideas to error correction decoding tasks, and we show that our algorithm can outperform belief propagation for LDPC codes on bursty channels.

Deep Generative Missingness Pattern-Set Mixture Models

Sahra Ghalebikesabi · Rob Cornish · Chris Holmes · Luke Kelly

We propose a variational autoencoder architecture to model both ignorable and nonignorable missing data using pattern-set mixtures as proposed by Little (1993). Our model explicitly learns to cluster the missing data into missingness pattern sets based on the observed data and missingness masks. Underpinning our approach is the assumption that the data distribution under missingness is probabilistically semi-supervised by samples from the observed data distribution. Our setup trades off the characteristics of ignorable and nonignorable missingness and can thus be applied to data of both types. We evaluate our method on a wide range of data sets with different types of missingness and achieve state-of-the-art imputation performance. Our model outperforms many common imputation algorithms, especially when the amount of missing data is high and the missingness mechanism is nonignorable.

Collaborative Classification from Noisy Labels

Lucas Maystre · Nagarjuna Kumarappan · Judith Bütepage · Mounia Lalmas

We consider a setting where users interact with a collection of N items on an online platform. We are given class labels possibly corrupted by noise, and we seek to recover the true class of each item. We postulate a simple probabilistic model of the interactions between users and items, based on the assumption that users interact with classes in different proportions. We then develop a message-passing algorithm that decodes the noisy class labels efficiently. Under suitable assumptions, our method provably recovers all items' true classes in the large N limit, even when the interaction graph remains sparse. Empirically, we show that our approach is effective on several practical applications, including predicting the location of businesses, the category of consumer goods, and the language of audio content.

Sequential Random Sampling Revisited: Hidden Shuffle Method

Michael Shekelyan · Graham Cormode

Random sampling (without replacement) is ubiquitously employed to obtain a representative subset of the data. Unlike common methods, sequential methods report samples in ascending order of index without keeping track of previous samples. This enables lightweight iterators that can jump directly from one sampled position to the next. Previously, sequential methods focused on drawing from the distribution of gap sizes, which requires intricate algorithms that are difficult to validate and can be slow in the worst-case. This can be avoided by a new method, the Hidden Shuffle. The name mirrors the fact that although the algorithm does not resemble shuffling, its correctness can be proven by conceptualising the sampling process as a random shuffle. The Hidden Shuffle algorithm stores just a handful of values, can be implemented in few lines of code, offers strong worst-case guarantees and is shown to be faster than state-of-the-art methods while using comparably few random variates.

Local Competition and Stochasticity for Adversarial Robustness in Deep Learning

Konstantinos Panagiotis Panousis · Sotirios Chatzis · Antonios Alexos · Sergios Theodoridis

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.

On the proliferation of support vectors in high dimensions

Daniel Hsu · Vidya Muthukumar · Ji Xu

The support vector machine (SVM) is a well-established classification method whose name refers to the particular training examples, called support vectors, that determine the maximum margin separating hyperplane. The SVM classifier is known to enjoy good generalization properties when the number of support vectors is small compared to the number of training examples. However, recent research has shown that in sufficiently high-dimensional linear classification problems, the SVM can generalize well despite a proliferation of support vectors where all training examples are support vectors. In this paper, we identify new deterministic equivalences for this phenomenon of support vector proliferation, and use them to (1) substantially broaden the conditions under which the phenomenon occurs in high-dimensional settings, and (2) prove a nearly matching converse result.

Adaptive wavelet pooling for convolutional neural networks

Moritz Wolter · Jochen Garcke

Convolutional neural networks (CNN)s have become the go-to choice for most image and video processing tasks. Most CNN architectures rely on pooling layers to reduce the resolution along spatial dimensions. The reduction allows subsequent deep convolution layers to operate with greater efficiency. This paper introduces adaptive wavelet pooling layers, which employ fast wavelet transforms (FWT) to reduce the feature resolution. The FWT decomposes the input features into multiple scales reducing the feature dimensions by removing the fine-scale subbands. Our approach adds extra flexibility through wavelet-basis function optimization and coefficient weighting at different scales. The adaptive wavelet layers integrate directly into well-known CNNs like the LeNet, Alexnet, or Densenet architectures. Using these networks, we validate our approach and find competitive performance on the MNIST, CIFAR10, and SVHN (street view house numbers) data-sets.

Fully Gap-Dependent Bounds for Multinomial Logit Bandit

Jiaqi Yang

We study the multinomial logit (MNL) bandit problem, where at each time step, the seller offers an assortment of size at most $K$ from a pool of $N$ items, and the buyer purchases an item from the assortment according to a MNL choice model. The objective is to learn the model parameters and maximize the expected revenue. We present (i) an algorithm that identifies the optimal assortment $S^*$ within $\widetilde{O}(\sum_{i = 1}^N \Delta_i^{-2})$ time steps with high probability, and (ii) an algorithm that incurs $O(\sum_{i \notin S^*} K\Delta_i^{-1} \log T)$ regret in $T$ time steps. To our knowledge, our algorithms are the \emph{first} to achieve gap-dependent bounds that \emph{fully} depends on the suboptimality gaps of \emph{all} items. Our technical contributions include an algorithmic framework that relates the MNL-bandit problem to a variant of the top-$K$ arm identification problem in multi-armed bandits, a generalized epoch-based offering procedure, and a layer-based adaptive estimation procedure.

Inference in Stochastic Epidemic Models via Multinomial Approximations

Nick Whiteley · Lorenzo Rimella

We introduce a new method for inference in stochastic epidemic models which uses recursive multinomial approximations to integrate over unobserved variables and thus circumvent likelihood intractability. The method is applicable to a class of discrete-time, finite-population compartmental models with partial, randomly under-reported or missing count observations. In contrast to state-of-the-art alternatives such as Approximate Bayesian Computation techniques, no forward simulation of the model is required and there are no tuning parameters. Evaluating the approximate marginal likelihood of model parameters is achieved through a computationally simple filtering recursion. The accuracy of the approximation is demonstrated through analysis of real and simulated data using a model of the 1995 Ebola outbreak in the Democratic Republic of Congo. We show how the method can be embedded within a Sequential Monte Carlo approach to estimating the time-varying reproduction number of COVID-19 in Wuhan, China, recently published by Kucharski et al. (2020).

Hyperparameter Transfer Learning with Adaptive Complexity

Samuel Horváth · Aaron Klein · Peter Richtarik · Cedric Archambeau

Bayesian optimization (BO) is a data-efficient approach to automatically tune the hyperparameters of machine learning models. In practice, one frequently has to solve similar hyperparameter tuning problems sequentially. For example, one might have to tune a type of neural network learned across a series of different classification problems. Recent work on multi-task BO exploits knowledge gained from previous hyperparameter tuning tasks to speed up a new tuning task. However, previous approaches do not account for the fact that BO is a sequential decision making procedure. Hence, there is in general a mismatch between the number of evaluations collected in the current tuning task compared to the number of evaluations accumulated in all previously completed tasks. In this work, we enable multi-task BO to compensate for this mismatch, such that the transfer learning procedure is able to handle different data regimes in a principled way. We propose a new multi-task BO method that learns a set of ordered, non-linear basis functions of increasing complexity via nested drop-out and automatic relevance determination. Experiments on a variety of hyperparameter tuning problems show that our method improves the sample efficiency of recently published multi-task BO methods.

On Multilevel Monte Carlo Unbiased Gradient Estimation for Deep Latent Variable Models

Yuyang Shi · Rob Cornish

Standard variational schemes for training deep latent variable models rely on biased gradient estimates of the target objective. Techniques based on the Evidence Lower Bound (ELBO), and tighter variants obtained via importance sampling, produce biased gradient estimates of the true log-likelihood. The family of Reweighted Wake-Sleep (RWS) methods further relies on a biased estimator of the inference objective, which biases training of the encoder also. In this work, we show how Multilevel Monte Carlo (MLMC) can provide a natural framework for debiasing these methods with two different estimators. We prove rigorously that this approach yields unbiased gradient estimators with finite variance under reasonable conditions. Furthermore, we investigate methods that can reduce variance and ensure finite variance in practice. Finally, we show empirically that the proposed unbiased estimators outperform IWAE and other debiasing method on a variety of applications at the same expected cost.

Linearly Constrained Gaussian Processes with Boundary Conditions

Markus Lange-Hegermann

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.

Equitable and Optimal Transport with Multiple Agents

Meyer Scetbon · Laurent Meunier · Jamal Atif · Marco Cuturi

We introduce an extension of the Optimal Transport problem when multiple costs are involved. Considering each cost as an agent, we aim to share equally between agents the work of transporting one distribution to another. To do so, we minimize the transportation cost of the agent who works the most. Another point of view is when the goal is to partition equitably goods between agents according to their heterogeneous preferences. Here we aim to maximize the utility of the least advantaged agent. This is a fair division problem. Like Optimal Transport, the problem can be cast as a linear optimization problem. When there is only one agent, we recover the Optimal Transport problem. When two agents are considered, we are able to recover Integral Probability Metrics defined by $\alpha$-Hölder functions, which include the widely-known Dudley metric. To the best of our knowledge, this is the first time a link is given between the Dudley metric and Optimal Transport. We provide an entropic regularization of that problem which leads to an alternative algorithm faster than the standard linear program.

Learning Complexity of Simulated Annealing

Avrim Blum · Chen Dan · Saeed Seddighin

Simulated annealing is an effective and general means of optimization. It is in fact inspired by metallurgy, where the temperature of a material determines its behavior in thermodynamics. Likewise, in simulated annealing, the actions that the algorithm takes depend entirely on the value of a variable which captures the notion of temperature. Typically, simulated annealing starts with a high temperature, which makes the algorithm pretty unpredictable, and gradually cools the temperature down to become more stable. A key component that plays a crucial role in the performance of simulated annealing is the criteria under which the temperature changes namely, the cooling schedule. Motivated by this, we study the following question in this work: "Given enough samples to the instances of a specific class of optimization problems, can we design optimal (or approximately optimal) cooling schedules that minimize the runtime or maximize the success rate of the algorithm on average when the underlying problem is drawn uniformly at random from the same class?" We provide positive results both in terms of sample complexity and simulation complexity. For sample complexity, we show that O~(m^1/2) samples suffice to find an approximately optimal cooling schedule of length m. We complement this result by giving a lower bound of Ω~(m^1/3) on the sample complexity of any learning algorithm that provides an almost optimal cooling schedule. These results are general and rely on no assumption. For simulation complexity, however, we make additional assumptions to measure the success rate of an algorithm. To this end, we introduce the monotone stationary graph that models the performance of simulated annealing. Based on this model, we present polynomial time algorithms with provable guarantees for the learning problem.

Dirichlet Pruning for Convolutional Neural Networks

Kamil Adamczewski · Mijung Park

We introduce Dirichlet pruning, a novel post-processing technique to transform a large neural network model into a compressed one. Dirichlet pruning is a form of structured pruning which assigns the Dirichlet distribution over each layer's channels in convolutional layers (or neurons in fully-connected layers), and learns the parameters of the distribution over these units using variational inference. The learnt parameters allow us to informatively and intuitively remove unimportant units, resulting in a compact architecture containing only crucial features for a task at hand. This method yields low GPU footprint, as the number of parameters is linear in the number of channels (or neurons) and training requires as little as one epoch to converge. We perform extensive experiments, in particular on larger architectures such as VGG and WideResNet (94\% and 72\% compression rate, respectively) where our method achieves the state-of-the-art compression performance and provides interpretable features as a by-product.

A Variational Inference Approach to Learning Multivariate Wold Processes

Jalal Etesami · William Trouleau · Negar Kiyavash · Matthias Grossglauser · Patrick Thiran

Temporal point-processes are often used for mathematical modeling of sequences of discrete events with asynchronous timestamps. We focus on a class of temporal point-process models called multivariate Wold processes (MWP). These processes are well suited to model real-world communication dynamics. Statistical inference on such processes often requires learning their corresponding parameters using a set of observed timestamps. In this work, we relax some of the restrictive modeling assumptions made in the state-of-the-art and introduce a Bayesian approach for inferring the parameters of MWP. We develop a computationally efficient variational inference algorithm that allows scaling up the approach to high-dimensional processes and long sequences of observations. Our experimental results on both synthetic and real-world datasets show that our proposed algorithm outperforms existing methods.

Learn to Expect the Unexpected: Probably Approximately Correct Domain Generalization

Vikas Garg · Adam Tauman Kalai · Katrina Ligett · Steven Wu

Domain generalization is the problem of machine learning when the training data and the test data come from different ``domains'' (data distributions). We propose an elementary theoretical model of the domain generalization problem, introducing the concept of a meta-distribution over domains. In our model, the training data available to a learning algorithm consist of multiple datasets, each from a single domain, drawn in turn from the meta-distribution. We show that our model can capture a rich range of learning phenomena specific to domain generalization for three different settings: learning with Massart noise, learning decision trees, and feature selection. We demonstrate approaches that leverage domain generalization to reduce computational or data requirements in each of these settings. Experiments demonstrate that our feature selection algorithm indeed ignores spurious correlations and improves generalization.

Wasserstein Random Forests and Applications in Heterogeneous Treatment Effects

Qiming Du · Gérard Biau · Francois Petit · Raphaël Porcher

We present new insights into causal inference in the context of Heterogeneous Treatment Effects by proposing natural variants of Random Forests to estimate the key conditional distributions. To achieve this, we recast Breiman’s original splitting criterion in terms of Wasserstein distances between empirical measures. This reformulation indicates that Random Forests are well adapted to estimate conditional distributions and provides a natural extension of the algorithm to multi- variate outputs. Following the philosophy of Breiman’s construction, we propose some variants of the splitting rule that are well-suited to the conditional distribution estimation problem. Some preliminary theoretical connections are established along with various numerical experiments, which show how our approach may help to conduct more transparent causal inference in complex situations.

Deep Neural Networks Are Congestion Games: From Loss Landscape to Wardrop Equilibrium and Beyond

Nina Vesseron · Ievgen Redko · Charlotte Laclau

The theoretical analysis of deep neural networks (DNN) is arguably among the most challenging research directions in machine learning (ML) right now, as it requires from scientists to lay novel statistical learning foundations to explain their behaviour in practice. While some success has been achieved recently in this endeavour, the question on whether DNNs can be analyzed using the tools from other scientific fields outside the ML community has not received the attention it may well have deserved. In this paper, we explore the interplay between DNNs and game theory (GT), and show how one can benefit from the classic readily available results from the latter when analyzing the former. In particular, we consider the widely studied class of congestion games, and illustrate their intrinsic relatedness to both linear and non-linear DNNs and to the properties of their loss surface. Beyond retrieving the state-of-the-art results from the literature, we argue that our work provides a very promising novel tool for analyzing the DNNs and support this claim by proposing concrete open problems that can advance significantly our understanding of DNNs when solved.

Improving predictions of Bayesian neural nets via local linearization

Alexander Immer · Maciej Korzepa · Matthias Bauer

The generalized Gauss-Newton (GGN) approximation is often used to make practical Bayesian deep learning approaches scalable by replacing a second order derivative with a product of first order derivatives. In this paper we argue that the GGN approximation should be understood as a local linearization of the underlying Bayesian neural network (BNN), which turns the BNN into a generalized linear model (GLM). Because we use this linearized model for posterior inference, we should also predict using this modified model instead of the original one. We refer to this modified predictive as "GLM predictive" and show that it effectively resolves common underfitting problems of the Laplace approximation. It extends previous results in this vein to general likelihoods and has an equivalent Gaussian process formulation, which enables alternative inference schemes for BNNs in function space. We demonstrate the effectiveness of our approach on several standard classification datasets as well as on out-of-distribution detection. We provide an implementation at

All of the Fairness for Edge Prediction with Optimal Transport

Charlotte Laclau · Ievgen Redko · Manvi Choudhary · Christine Largeron

Machine learning and data mining algorithms have been increasingly used recently to support decision-making systems in many areas of high societal importance such as healthcare, education, or security. While being very efficient in their predictive abilities, the deployed algorithms sometimes tend to learn an inductive model with a discriminative bias due to the presence of this latter in the learning sample. This problem gave rise to a new field of algorithmic fairness where the goal is to correct the discriminative bias introduced by a certain attribute in order to decorrelate it from the model's output. In this paper, we study the problem of fairness for the task of edge prediction in graphs, a largely underinvestigated scenario compared to a more popular setting of fair classification. To this end, we formulate the problem of fair edge prediction, analyze it theoretically, and propose an embedding-agnostic repairing procedure for the adjacency matrix of an arbitrary graph with a trade-off between the group and individual fairness. We experimentally show the versatility of our approach and its capacity to provide explicit control over different notions of fairness and prediction accuracy.

Deep Fourier Kernel for Self-Attentive Point Processes

Shixiang Zhu · Minghe Zhang · Ruyi Ding · Yao Xie

We present a novel attention-based model for discrete event data to capture complex non-linear temporal dependence structures. We borrow the idea from the attention mechanism and incorporate it into the point processes' conditional intensity function. We further introduce a novel score function using Fourier kernel embedding, whose spectrum is represented using neural networks, which drastically differs from the traditional dot-product kernel and can capture a more complex similarity structure. We establish our approach's theoretical properties and demonstrate our approach's competitive performance compared to the state-of-the-art for synthetic and real data.

Optimal Quantisation of Probability Measures Using Maximum Mean Discrepancy

Onur Teymur · Jackson Gorham · Marina Riabiz · Chris Oates

Several researchers have proposed minimisation of maximum mean discrepancy (MMD) as a method to quantise probability measures, i.e., to approximate a distribution by a representative point set. We consider sequential algorithms that greedily minimise MMD over a discrete candidate set. We propose a novel non-myopic algorithm and, in order to both improve statistical efficiency and reduce computational cost, we investigate a variant that applies this technique to a mini-batch of the candidate set at each iteration. When the candidate points are sampled from the target, the consistency of these new algorithms—and their mini-batch variants—is established. We demonstrate the algorithms on a range of important computational problems, including optimisation of nodes in Bayesian cubature and the thinning of Markov chain output.

High-Dimensional Multi-Task Averaging and Application to Kernel Mean Embedding

Hannah Marienwald · Jean-Baptiste Fermanian · Gilles Blanchard

We propose an improved estimator for the multi-task averaging problem, whose goal is the joint estimation of the means of multiple distributions using separate, independent data sets. The naive approach is to take the empirical mean of each data set individually, whereas the proposed method exploits similarities between tasks, without any related information being known in advance. First, for each data set, similar or neighboring means are determined from the data by multiple testing. Then each naive estimator is shrunk towards the local average of its neighbors. We prove theoretically that this approach provides a reduction in mean squared error. This improvement can be significant when the dimension of the input space is large; demonstrating a ``blessing of dimensionality'' phenomenon. An application of this approach is the estimation of multiple kernel mean embeddings, which plays an important role in many modern applications. The theoretical results are verified on artificial and real world data.

Predictive Complexity Priors

Eric Nalisnick · Jonathan Gordon · Jose Miguel Hernandez-Lobato

Specifying a Bayesian prior is notoriously difficult for complex models such as neural networks. Reasoning about parameters is made challenging by the high-dimensionality and over-parameterization of the space. Priors that seem benign and uninformative can have unintuitive and detrimental effects on a model's predictions. For this reason, we propose predictive complexity priors: a functional prior that is defined by comparing the model's predictions to those of a reference model. Although originally defined on the model outputs, we transfer the prior to the model parameters via a change of variables. The traditional Bayesian workflow can then proceed as usual. We apply our predictive complexity prior to high-dimensional regression, reasoning over neural network depth, and sharing of statistical strength for few-shot learning.

A Bayesian nonparametric approach to count-min sketch under power-law data streams

Emanuele Dolera · Stefano Favaro · Stefano Peluchetti

The count-min sketch (CMS) is a randomized data structure that provides estimates of tokens' frequencies in a large data stream using a compressed representation of the data by random hashing. In this paper, we rely on a recent Bayesian nonparametric (BNP) view on the CMS to develop a novel learning-augmented CMS under power-law data streams. We assume that tokens in the stream are drawn from an unknown discrete distribution, which is endowed with a normalized inverse Gaussian process (NIGP) prior. Then, using distributional properties of the NIGP, we compute the posterior distribution of a token's frequency in the stream, given the hashed data, and in turn corresponding BNP estimates. Applications to synthetic and real data show that our approach achieves a remarkable performance in the estimation of low-frequency tokens. This is known to be a desirable feature in the context of natural language processing, where it is indeed common in the context of the power-law behaviour of the data.

Approximately Solving Mean Field Games via Entropy-Regularized Deep Reinforcement Learning

Kai Cui · Heinz Koeppl

The recent mean field game (MFG) formalism facilitates otherwise intractable computation of approximate Nash equilibria in many-agent settings. In this paper, we consider discrete-time finite MFGs subject to finite-horizon objectives. We show that all discrete-time finite MFGs with non-constant fixed point operators fail to be contractive as typically assumed in existing MFG literature, barring convergence via fixed point iteration. Instead, we incorporate entropy-regularization and Boltzmann policies into the fixed point iteration. As a result, we obtain provable convergence to approximate fixed points where existing methods fail, and reach the original goal of approximate Nash equilibria. All proposed methods are evaluated with respect to their exploitability, on both instructive examples with tractable exact solutions and high-dimensional problems where exact methods become intractable. In high-dimensional scenarios, we apply established deep reinforcement learning methods and empirically combine fictitious play with our approximations.

On Riemannian Stochastic Approximation Schemes with Fixed Step-Size

Alain Durmus · Pablo Jiménez · Eric Moulines · Salem SAID

This paper studies fixed step-size stochastic approximation (SA) schemes, including stochastic gradient schemes, in a Riemannian framework. It is motivated by several applications, where geodesics can be computed explicitly, and their use accelerates crude Euclidean methods. A fixed step-size scheme defines a family of time-homogeneous Markov chains, parametrized by the step-size. Here, using this formulation, non-asymptotic performance bounds are derived, under Lyapunov conditions. Then, for any step-size, the corresponding Markov chain is proved to admit a unique stationary distribution, and to be geometrically ergodic. This result gives rise to a family of stationary distributions indexed by the step-size, which is further shown to converge to a Dirac measure, concentrated at the solution of the problem at hand, as the step-size goes to $0$. Finally, the asymptotic rate of this convergence is established, through an asymptotic expansion of the bias, and a central limit theorem.

Free-rider Attacks on Model Aggregation in Federated Learning

Yann Fraboni · Richard Vidal · Marco Lorenzi

Free-rider attacks against federated learning consist in dissimulating participation to the federated learning process with the goal of obtaining the final aggregated model without actually contributing with any data. This kind of attacks are critical in sensitive applications of federated learning when data is scarce and the model has high commercial value. We introduce here the first theoretical and experimental analysis of free-rider attacks on federated learning schemes based on iterative parameters aggregation, such as FedAvg or FedProx, and provide formal guarantees for these attacks to converge to the aggregated models of the fair participants. We first show that a straightforward implementation of this attack can be simply achieved by not updating the local parameters during the iterative federated optimization. As this attack can be detected by adopting simple countermeasures at the server level, we subsequently study more complex disguising schemes based on stochastic updates of the free-rider parameters. We demonstrate the proposed strategies on a number of experimental scenarios, in both iid and non-iid settings. We conclude by providing recommendations to avoid free-rider attacks in real world applications of federated learning, especially in sensitive domains where security of data and models is critical.

Unconstrained MAP Inference, Exponentiated Determinantal Point Processes, and Exponential Inapproximability

Naoto Ohsaka

We study the computational complexity of two hard problems on determinantal point processes (DPPs). One is maximum a posteriori (MAP) inference, i.e., to find a principal submatrix having the maximum determinant. The other is probabilistic inference on exponentiated DPPs (E-DPPs), which can sharpen or weaken the diversity preference of DPPs with an exponent parameter $p$. We prove the following complexity-theoretic hardness results that explain the difficulty in approximating unconstrained MAP inference and the normalizing constant for E-DPPs. (1) Unconstrained MAP inference for an $n \times n$ matrix is NP-hard to approximate within a $2^{\beta n}$-factor, where $\beta = 10^{-10^{13}}$. This result improves upon a $(9/8-\epsilon)$-factor inapproximability given by Kulesza and Taskar (2012). (2) The normalizing constant for E-DPPs of any (fixed) constant exponent $p \geq \beta^{-1} = 10^{10^{13}}$ is NP-hard to approximate within a $2^{\beta pn}$-factor. This gives a(nother) negative answer to open questions posed by Kulesza and Taskar (2012); Ohsaka and Matsuoka (2020).

On the Importance of Hyperparameter Optimization for Model-based Reinforcement Learning

Baohe Zhang · Raghu Rajan · Luis Pineda · Nathan Lambert · André Biedenkapp · Kurtland Chua · Frank Hutter · Roberto Calandra

Model-based Reinforcement Learning (MBRL) is a promising framework for learning control in a data-efficient manner. MBRL algorithms can be fairly complex due to the separate dynamics modeling and the subsequent planning algorithm, and as a result, they often possess tens of hyperparameters and architectural choices. For this reason, MBRL typically requires significant human expertise before it can be applied to new problems and domains. To alleviate this problem, we propose to use automatic hyperparameter optimization (HPO). We demonstrate that this problem can be tackled effectively with automated HPO, which we demonstrate to yield significantly improved performance compared to human experts. In addition, we show that tuning of several MBRL hyperparameters dynamically, i.e. during the training itself, further improves the performance compared to using static hyperparameters which are kept fix for the whole training. Finally, our experiments provide valuable insights into the effects of several hyperparameters, such as plan horizon or learning rate and their influence on the stability of training and resulting rewards.

Fast Learning in Reproducing Kernel Krein Spaces via Signed Measures

Fanghui Liu · Xiaolin Huang · Yingyi Chen · Johan Suykens

In this paper, we attempt to solve a long-lasting open question for non-positive definite (non-PD) kernels in machine learning community: can a given non-PD kernel be decomposed into the difference of two PD kernels (termed as positive decomposition)? We cast this question as a distribution view by introducing the signed measure, which transforms positive decomposition to measure decomposition: a series of non-PD kernels can be associated with the linear combination of specific finite Borel measures. In this manner, our distribution-based framework provides a sufficient and necessary condition to answer this open question. Specifically, this solution is also computationally implementable in practice to scale non-PD kernels in large sample cases, which allows us to devise the first random features algorithm to obtain an unbiased estimator. Experimental results on several benchmark datasets verify the effectiveness of our algorithm over the existing methods.

Regularization Matters: A Nonparametric Perspective on Overparametrized Neural Network

Tianyang Hu · Wenjia Wang · Cong Lin · Guang Cheng

Overparametrized neural networks trained by gradient descent (GD) can provably overfit any training data. However, the generalization guarantee may not hold for noisy data. From a nonparametric perspective, this paper studies how well overparametrized neural networks can recover the true target function in the presence of random noises. We establish a lower bound on the L2 estimation error with respect to the GD iteration, which is away from zero without a delicate choice of early stopping. In turn, through a comprehensive analysis of L2-regularized GD trajectories, we prove that for overparametrized one-hidden-layer ReLU neural network with the L2 regularization: (1) the output is close to that of the kernel ridge regression with the corresponding neural tangent kernel; (2) minimax optimal rate of the L2 estimation error is achieved. Numerical experiments confirm our theory and further demonstrate that the L2 regularization approach improves the training robustness and works for a wider range of neural networks.

On the number of linear functions composing deep neural network: Towards a refined definition of neural networks complexity

Yuuki Takai · Akiyoshi Sannai · Matthieu Cordonnier

The classical approach to measure the expressive power of deep neural networks with piecewise linear activations is based on counting their maximum number of linear regions. This complexity measure is quite relevant to understand general properties of the expressivity of neural networks such as the benefit of depth over width. Nevertheless, it appears limited when it comes to comparing the expressivity of different network architectures. This lack becomes particularly prominent when considering permutation-invariant networks, due to the symmetrical redundancy among the linear regions. To tackle this, we propose a refined definition of piecewise linear function complexity: instead of counting the number of linear regions directly, we first introduce an equivalence relation among the linear functions composing a piecewise linear function and then count those linear functions relative to that equivalence relation. Our new complexity measure can clearly distinguish between the two aforementioned models, is consistent with the classical measure, and increases exponentially with depth.

Direct-Search for a Class of Stochastic Min-Max Problems

Sotiris Anagnostidis · Aurelien Lucchi · Youssef Diouane

Recent applications in machine learning have renewed the interest of the community in min-max optimization problems. While gradient-based optimization methods are widely used to solve such problems, there are however many scenarios where these techniques are not well-suited, or even not applicable when the gradient is not accessible. We investigate the use of direct-search methods that belong to a class of derivative-free techniques that only access the objective function through an oracle. In this work, we design a novel algorithm in the context of min-max saddle point games where one sequentially updates the min and the max player. We prove convergence of this algorithm under mild assumptions, where the objective of the max-player satisfies the Polyak-\L{}ojasiewicz (PL) condition, while the min-player is characterized by a nonconvex objective. Our method only assumes dynamically adjusted accurate estimates of the oracle with a fixed probability. To the best of our knowledge, our analysis is the first one to address the convergence of a direct-search method for min-max objectives in a stochastic setting.

A Theoretical Characterization of Semi-supervised Learning with Self-training for Gaussian Mixture Models

Samet Oymak · Talha Cihad Gulcu

Self-training is a classical approach in semi-supervised learning which is successfully applied to a variety of machine learning problems. Self-training algorithms generate pseudo-labels for the unlabeled examples and progressively refine these pseudo-labels which hopefully coincides with the actual labels. This work provides theoretical insights into self-training algorithms with a focus on linear classifiers. First, we provide a sample complexity analysis for Gaussian mixture models with two components. This is established by sharp non-asymptotic characterization of the self-training iterations which captures the evolution of the model accuracy in terms of a fixed-point iteration. Our analysis reveals the provable benefits of rejecting samples with low confidence and demonstrates how self-training iterations can gracefully improve the model accuracy. Secondly, we study a generalized GMM where the component means follow a distribution. We demonstrate that ridge regularization and class margin (i.e. separation between the component means) is crucial for the success and lack of regularization may prevent self-training from identifying the core features in the data.

Causal Autoregressive Flows

Ilyes Khemakhem · Ricardo Monti · Robert Leech · Aapo Hyvarinen

Two apparently unrelated fields --- normalizing flows and causality --- have recently received considerable attention in the machine learning community. In this work, we highlight an intrinsic correspondence between a simple family of autoregressive normalizing flows and identifiable causal models. We exploit the fact that autoregressive flow architectures define an ordering over variables, analogous to a causal ordering, to show that they are well-suited to performing a range of causal inference tasks, ranging from causal discovery to making interventional and counterfactual predictions. First, we show that causal models derived from both affine and additive autoregressive flows with fixed orderings over variables are identifiable, i.e. the true direction of causal influence can be recovered. This provides a generalization of the additive noise model well-known in causal discovery. Second, we derive a bivariate measure of causal direction based on likelihood ratios, leveraging the fact that flow models can estimate normalized log-densities of data. Third, we demonstrate that flows naturally allow for direct evaluation of both interventional and counterfactual queries, the latter case being possible due to the invertible nature of flows. Finally, throughout a series of experiments on synthetic and real data, the proposed method is shown to outperform current approaches for causal discovery as well as making accurate interventional and counterfactual predictions.

Differentiating the Value Function by using Convex Duality

Sheheryar Mehmood · Peter Ochs

We consider the differentiation of the value function for parametric optimization problems. Such problems are ubiquitous in machine learning applications such as structured support vector machines, matrix factorization and min-min or minimax problems in general. Existing approaches for computing the derivative rely on strong assumptions of the parametric function. Therefore, in several scenarios there is no theoretical evidence that a given algorithmic differentiation strategy computes the true gradient information of the value function. We leverage a well known result from convex duality theory to relax the conditions and to derive convergence rates of the derivative approximation for several classes of parametric optimization problems in Machine Learning. We demonstrate the versatility of our approach in several experiments, including non-smooth parametric functions. Even in settings where other approaches are applicable, our duality based strategy shows a favorable performance.

Noisy Gradient Descent Converges to Flat Minima for Nonconvex Matrix Factorization

Tianyi Liu · Yan Li · Song Wei · Enlu Zhou · Tuo Zhao

Numerous empirical evidences have corroborated the importance of noise in nonconvex optimization problems. The theory behind such empirical observations, however, is still largely unknown. This paper studies this fundamental problem through investigating the nonconvex rectangular matrix factorization problem, which has infinitely many global minima due to rotation and scaling invariance. Hence, gradient descent (GD) can converge to any optimum, depending on the initialization. In contrast, we show that a perturbed form of GD with an arbitrary initialization converges to a global optimum that is uniquely determined by the injected noise. Our result implies that the noise imposes implicit bias towards certain optima. Numerical experiments are provided to support our theory.

Adaptive Sampling for Fast Constrained Maximization of Submodular Functions

Francesco Quinzan · Vanja Doskoc · Andreas Göbel · Tobias Friedrich

Several large-scale machine learning tasks, such as data summarization, can be approached by maximizing functions that satisfy submodularity. These optimization problems often involve complex side constraints, imposed by the underlying application. In this paper, we develop an algorithm with poly-logarithmic adaptivity for non-monotone submodular maximization under general side constraints. The adaptive complexity of a problem is the minimal number of sequential rounds required to achieve the objective.

Our algorithm is suitable to maximize a non-monotone submodular function under a p-system side constraint, and it achieves a (p + O(sqrt(p)))-approximation for this problem, after only poly-logarithmic adaptive rounds and polynomial queries to the valuation oracle function. Furthermore, our algorithm achieves a (p + O(1))-approximation when the given side constraint is a p-extendable system.

This algorithm yields an exponential speed-up, with respect to the adaptivity, over any other known constant-factor approximation algorithm for this problem. It also competes with previous known results in terms of the query complexity. We perform various experiments on various real-world applications. We find that, in comparison with commonly used heuristics, our algorithm performs better on these instances.

Sparse Gaussian Processes Revisited: Bayesian Approaches to Inducing-Variable Approximations

Simone Rossi · Markus Heinonen · Edwin Bonilla · Zheyang Shen · Maurizio Filippone

Variational inference techniques based on inducing variables provide an elegant framework for scalable posterior estimation in Gaussian process (GP) models. Besides enabling scalability, one of their main advantages over sparse approximations using direct marginal likelihood maximization is that they provide a robust alternative for point estimation of the inducing inputs, i.e. the location of the inducing variables. In this work we challenge the common wisdom that optimizing the inducing inputs in the variational framework yields optimal performance. We show that, by revisiting old model approximations such as the fully-independent training conditionals endowed with powerful sampling-based inference methods, treating both inducing locations and GP hyper-parameters in a Bayesian way can improve performance significantly. Based on stochastic gradient Hamiltonian Monte Carlo, we develop a fully Bayesian approach to scalable GP and deep GP models, and demonstrate its state-of-the-art performance through an extensive experimental campaign across several regression and classification problems.

SDF-Bayes: Cautious Optimism in Safe Dose-Finding Clinical Trials with Drug Combinations and Heterogeneous Patient Groups

Hyun-Suk Lee · Cong Shen · William Zame · Jang-Won Lee · Mihaela van der Schaar

Phase I clinical trials are designed to test the safety (non-toxicity) of drugs and find the maximum tolerated dose (MTD). This task becomes significantly more challenging when multiple-drug dose-combinations (DC) are involved, due to the inherent conflict between the exponentially increasing DC candidates and the limited patient budget. This paper proposes a novel Bayesian design, SDF-Bayes, for finding the MTD for drug combinations in the presence of safety constraints. Rather than the conventional principle of escalating or de-escalating the current dose of one drug (perhaps alternating between drugs), SDF-Bayes proceeds by cautious optimism: it chooses the next DC that, on the basis of current information, is most likely to be the MTD (optimism), subject to the constraint that it only chooses DCs that have a high probability of being safe (caution). We also propose an extension, SDF-Bayes-AR, that accounts for patient heterogeneity and enables heterogeneous patient recruitment. Extensive experiments based on both synthetic and real-world datasets demonstrate the advantages of SDF-Bayes over state of the art DC trial designs in terms of accuracy and safety.

Foundations of Bayesian Learning from Synthetic Data

Harrison Wilde · Jack Jewson · Sebastian Vollmer · Chris Holmes

There is significant growth and interest in the use of synthetic data as an enabler for machine learning in environments where the release of real data is restricted due to privacy or availability constraints. Despite a large number of methods for synthetic data generation, there are comparatively few results on the statistical properties of models learnt on synthetic data, and fewer still for situations where a researcher wishes to augment real data with another party’s synthesised data. We use a Bayesian paradigm to characterise the updating of model parameters when learning in these settings, demonstrating that caution should be taken when applying conventional learning algorithms without appropriate consideration of the synthetic data generating process and learning task at hand. Recent results from general Bayesian updating support a novel and robust approach to Bayesian synthetic-learning founded on decision theory that outperforms standard approaches across repeated experiments on supervised learning and inference problems.

Learning Bijective Feature Maps for Linear ICA

Alexander Camuto · Matthew Willetts · Chris Holmes · Brooks Paige · Stephen Roberts

Separating high-dimensional data like images into independent latent factors, i.e independent component analysis (ICA), remains an open research problem. As we show, existing probabilistic deep generative models (DGMs), which are tailor-made for image data, underperform on non-linear ICA tasks. To address this, we propose a DGM which combines bijective feature maps with a linear ICA model to learn interpretable latent structures for high-dimensional data. Given the complexities of jointly training such a hybrid model, we introduce novel theory that constrains linear ICA to lie close to the manifold of orthogonal rectangular matrices, the Stiefel manifold. By doing so we create models that converge quickly, are easy to train, and achieve better unsupervised latent factor discovery than flow-based models, linear ICA, and Variational Autoencoders on images.

A unified view of likelihood ratio and reparameterization gradients

Paavo Parmas · Masashi Sugiyama

Reparameterization (RP) and likelihood ratio (LR) gradient estimators are used to estimate gradients of expectations throughout machine learning and reinforcement learning; however, they are usually explained as simple mathematical tricks, with no insight into their nature. We use a first principles approach to explain that LR and RP are alternative methods of keeping track of the movement of probability mass, and the two are connected via the divergence theorem. Moreover, we show that the space of all possible estimators combining LR and RP can be completely parameterized by a flow field u(x) and importance sampling distribution q(x). We prove that there cannot exist a single-sample estimator of this type outside our characterized space, thus, clarifying where we should be searching for better Monte Carlo gradient estimators.

Gaming Helps! Learning from Strategic Interactions in Natural Dynamics

Yahav Bechavod · Katrina Ligett · Steven Wu · Juba Ziani

We consider an online regression setting in which individuals adapt to the regression model: arriving individuals may access the model throughout the process, and invest strategically in modifying their own features so as to improve their predicted score. Such feature manipulation, or ``gaming'', has been observed in various scenarios---from credit assessment to school admissions, posing a challenge for the learner. Surprisingly, we find that such strategic manipulation may in fact help the learner recover the meaningful variables in settings where an agent can invest in improving meaningful features---that is, the features that, when changed, affect the true label, as opposed to non-meaningful features that have no effect. We show that even simple behavior on the learner's part allows her to simultaneously i) accurately recover the meaningful features, and ii) incentivize agents to invest in these meaningful features, providing incentives for improvement.

On the Effect of Auxiliary Tasks on Representation Dynamics

Clare Lyle · Mark Rowland · Georg Ostrovski · Will Dabney

While auxiliary tasks play a key role in shaping the representations learnt by reinforcement learning agents, much is still unknown about the mechanisms through which this is achieved. This work develops our understanding of the relationship between auxiliary tasks, environment structure, and representations by analysing the dynamics of temporal difference algorithms. Through this approach, we establish a connection between the spectral decomposition of the transition operator and the representations induced by a variety of auxiliary tasks. We then leverage insights from these theoretical results to inform the selection of auxiliary tasks for deep reinforcement learning agents in sparse-reward environments.

CONTRA: Contrarian statistics for controlled variable selection

Mukund Sudarshan · Aahlad Puli · Lakshmi Subramanian · Sriram Sankararaman · Rajesh Ranganath

The holdout randomization test (HRT) discovers a set of covariates most predictive of a response. Given the covariate distribution, HRTs can explicitly control the false discovery rate (FDR). However, if this distribution is unknown and must be estimated from data, HRTs can inflate the FDR. To alleviate the inflation of FDR, we propose the contrarian randomization test (CONTRA), which is designed explicitly for scenarios where the covariate distribution must be estimated from data and may even be misspecified. Our key insight is to use an equal mixture of two “contrarian” probabilistic models in determining the importance of a covariate. One model is fit with the real data, while the other is fit using the same data, but with the covariate being tested replaced with samples from an estimate of the covariate distribution. CONTRA is flexible enough to achieve a power of 1 asymptotically, can reduce the FDR compared to state-of-the-art CVS methods when the covariate distribution is misspecified, and is computationally efficient in high dimensions and large sample sizes. We further demonstrate the effectiveness of CONTRA on numerous synthetic benchmarks, and highlight its capabilities on a genetic dataset.

Nearest Neighbour Based Estimates of Gradients: Sharp Nonasymptotic Bounds and Applications

Guillaume Ausset · Stephan Clémençon · François Portier

Motivated by a wide variety of applications, ranging from stochastic optimization to dimension reduction through variable selection, the problem of estimating gradients accurately is of crucial importance in statistics and learning theory. We consider here the classic regression setup, where a real valued square integrable r.v. Y is to be predicted upon observing a (possibly high dimensional) random vector X by means of a predictive function f(X) as accurately as possible in the mean-squared sense and study a nearest-neighbour-based pointwise estimate of the gradient of the optimal predictive function, the regression function m(x)=E[Y | X=x]. Under classic smoothness conditions combined with the assumption that the tails of Y-m(X) are sub-Gaussian, we prove nonasymptotic bounds improving upon those obtained for alternative estimation methods. Beyond the novel theoretical results established, several illustrative numerical experiments have been carried out. The latter provide strong empirical evidence that the estimation method proposed works very well for various statistical problems involving gradient estimation, namely dimensionality reduction, stochastic gradient descent optimization and quantifying disentanglement.

Self-Supervised Steering Angle Prediction for Vehicle Control Using Visual Odometry

Qadeer Khan · Patrick Wenzel · Daniel Cremers

Vision-based learning methods for self-driving cars have primarily used supervised approaches that require a large number of labels for training. However, those labels are usually difficult and expensive to obtain. In this paper, we demonstrate how a model can be trained to control a vehicle's trajectory using camera poses estimated through visual odometry methods in an entirely self-supervised fashion. We propose a scalable framework that leverages trajectory information from several different runs using a camera setup placed at the front of a car. Experimental results on the CARLA simulator demonstrate that our proposed approach performs at par with the model trained with supervision.

Robust and Private Learning of Halfspaces

Badih Ghazi · Ravi Kumar · Pasin Manurangsi · Thao Nguyen

In this work, we study the trade-off between differential privacy and adversarial robustness under $L_2$-perturbations in the context of learning halfspaces. We prove nearly tight bounds on the sample complexity of robust private learning of halfspaces for a large regime of parameters. A highlight of our results is that robust and private learning is harder than robust or private learning alone. We complement our theoretical analysis with experimental results on the MNIST and USPS datasets, for a learning algorithm that is both differentially private and adversarially robust.

Logistic Q-Learning

Joan Bas Serrano · Sebastian Curi · Andreas Krause · Gergely Neu

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.

An Analysis of the Adaptation Speed of Causal Models

Remi Le Priol · Reza Babanezhad · Yoshua Bengio · Simon Lacoste-Julien

Consider a collection of datasets generated by unknown interventions on an unknown structural causal model $G$. Recently, Bengio et al. (2020) conjectured that among all candidate models, $G$ is the fastest to adapt from one dataset to another, along with promising experiments. Indeed, intuitively $G$ has less mechanisms to adapt, but this justification is incomplete. Our contribution is a more thorough analysis of this hypothesis. We investigate the adaptation speed of cause-effect SCMs. Using convergence rates from stochastic optimization, we justify that a relevant proxy for adaptation speed is distance in parameter space after intervention. Applying this proxy to categorical and normal cause-effect models, we show two results. When the intervention is on the cause variable, the SCM with the correct causal direction is advantaged by a large factor. When the intervention is on the effect variable, we characterize the relative adaptation speed. Surprisingly, we find situations where the anticausal model is advantaged, falsifying the initial hypothesis.

Fisher Auto-Encoders

Khalil Elkhalil · Ali Hasan · Jie Ding · Sina Farsiu · Vahid Tarokh

It has been conjectured that the Fisher divergence is more robust to model uncertainty than the conventional Kullback-Leibler (KL) divergence. This motivates the design of a new class of robust generative auto-encoders (AE) referred to as Fisher auto-encoders. Our approach is to design Fisher AEs by minimizing the Fisher divergence between the intractable joint distribution of observed data and latent variables, with that of the postulated/modeled joint distribution. In contrast to KL-based variational AEs (VAEs), the Fisher AE can exactly quantify the distance between the true and the model-based posterior distributions. Qualitative and quantitative results are provided on both MNIST and celebA datasets demonstrating the competitive performance of Fisher AEs in terms of robustness compared to other AEs such as VAEs and Wasserstein AEs.

Thresholded Adaptive Validation: Tuning the Graphical Lasso for Graph Recovery

Mike Laszkiewicz · Johannes Lederer · Asja Fischer

Many Machine Learning algorithms are formulated as regularized optimization problems, but their performance hinges on a regularization parameter that needs to be calibrated to each application at hand. In this paper, we propose a general calibration scheme for regularized optimization problems and apply it to the graphical lasso, which is a method for Gaussian graphical modeling. The scheme is equipped with theoretical guarantees and motivates a thresholding pipeline that can improve graph recovery. Moreover, requiring at most one line search over the regularization path, the calibration scheme is computationally more efficient than competing schemes that are based on resampling. Finally, we show in simulations that our approach can improve on the graph recovery of other approaches considerably.

Learning Fair Scoring Functions: Bipartite Ranking under ROC-based Fairness Constraints

Robin Vogel · Aurélien Bellet · Stephan Clémençon

Many applications of AI involve scoring individuals using a learned function of their attributes. These predictive risk scores are then used to take decisions based on whether the score exceeds a certain threshold, which may vary depending on the context. The level of delegation granted to such systems in critical applications like credit lending and medical diagnosis will heavily depend on how questions of fairness can be answered. In this paper, we study fairness for the problem of learning scoring functions from binary labeled data, a classic learning task known as bipartite ranking. We argue that the functional nature of the ROC curve, the gold standard measure of ranking accuracy in this context, leads to several ways of formulating fairness constraints. We introduce general families of fairness definitions based on the AUC and on ROC curves, and show that our ROC-based constraints can be instantiated such that classifiers obtained by thresholding the scoring function satisfy classification fairness for a desired range of thresholds. We establish generalization bounds for scoring functions learned under such constraints, design practical learning algorithms and show the relevance our approach with numerical experiments on real and synthetic data.

SGD for Structured Nonconvex Functions: Learning Rates, Minibatching and Interpolation

Robert Gower · Othmane Sebbouh · Nicolas Loizou

Stochastic Gradient Descent (SGD) is being used routinely for optimizing non-convex functions. Yet, the standard convergence theory for SGD in the smooth non-convex setting gives a slow sublinear convergence to a stationary point. In this work, we provide several convergence theorems for SGD showing convergence to a global minimum for non-convex problems satisfying some extra structural assumptions. In particular, we focus on two large classes of structured non-convex functions: (i) Quasar (Strongly) Convex functions (a generalization of convex functions) and (ii) functions satisfying the Polyak-Łojasiewicz condition (a generalization of strongly-convex functions). Our analysis relies on an Expected Residual condition which we show is a strictly weaker assumption than previously used growth conditions, expected smoothness or bounded variance assumptions. We provide theoretical guarantees for the convergence of SGD for different step-size selections including constant, decreasing and the recently proposed stochastic Polyak step-size. In addition, all of our analysis holds for the arbitrary sampling paradigm, and as such, we give insights into the complexity of minibatching and determine an optimal minibatch size. Finally, we show that for models that interpolate the training data, we can dispense of our Expected Residual condition and give state-of-the-art results in this setting.

Nonlinear Functional Output Regression: A Dictionary Approach

Dimitri Bouche · Marianne Clausel · François Roueff · Florence d'Alché-Buc

To address functional-output regression, we introduce projection learning (PL), a novel dictionary-based approach that learns to predict a function that is expanded on a dictionary while minimizing an empirical risk based on a functional loss. PL makes it possible to use non orthogonal dictionaries and can then be combined with dictionary learning; it is thus much more flexible than expansion-based approaches relying on vectorial losses. This general method is instantiated with reproducing kernel Hilbert spaces of vector-valued functions as kernel-based projection learning (KPL). For the functional square loss, two closed-form estimators are proposed, one for fully observed output functions and the other for partially observed ones. Both are backed theoretically by an excess risk analysis. Then, in the more general setting of integral losses based on differentiable ground losses, KPL is implemented using first-order optimization for both fully and partially observed output functions. Eventually, several robustness aspects of the proposed algorithms are highlighted on a toy dataset; and a study on two real datasets shows that they are competitive compared to other nonlinear approaches. Notably, using the square loss and a learnt dictionary, KPL enjoys a particularily attractive trade-off between computational cost and performances.

Nested Barycentric Coordinate System as an Explicit Feature Map

Lee-Ad Gottlieb · Eran Kaufman · Aryeh Kontorovich · Gabriel Nivasch · Ofir Pele

We introduce a new embedding technique based on barycentric coordinate system. We show that our embedding can be used to transforms the problem of polytope approximation into that of finding a \emph{linear} classifier in a higher (but nevertheless quite sparse) dimensional representation. This embedding in effect maps a piecewise linear function into a single linear function, and allows us to invoke well-known algorithms for the latter problem to solve the former.

We demonstrate that our embedding has applications to the problems of approximating separating polytopes -- in fact, it can approximate any convex body and multiple convex bodies -- as well as to classification by separating polytopes and piecewise linear regression.

Latent Derivative Bayesian Last Layer Networks

Joe Watson · Jihao Andreas Lin · Pascal Klink · Joni Pajarinen · Jan Peters

Bayesian neural networks (BNN) are powerful parametric models for nonlinear regression with uncertainty quantification. However, the approximate inference techniques for weight space priors suffer from several drawbacks. The `Bayesian last layer' (BLL) is an alternative BNN approach that learns the feature space for an exact Bayesian linear model with explicit predictive distributions. However, its predictions outside of the data distribution (OOD) are typically overconfident, as the marginal likelihood objective results in a learned feature space that overfits to the data. We overcome this weakness by introducing a functional prior on the model's derivatives w.r.t. the inputs. Treating these Jacobians as latent variables, we incorporate the prior into the objective to influence the smoothness and diversity of the features, which enables greater predictive uncertainty. For the BLL, the Jacobians can be computed directly using forward mode automatic differentiation, and the distribution over Jacobians may be obtained in closed-form. We demonstrate this method enhances the BLL to Gaussian process-like performance on tasks where calibrated uncertainty is critical: OOD regression, Bayesian optimization and active learning, which include high-dimensional real-world datasets.

The Sample Complexity of Level Set Approximation

Francois Bachoc · Tommaso Cesari · Sébastien Gerchinovitz

We study the problem of approximating the level set of an unknown function by sequentially querying its values. We introduce a family of algorithms called Bisect and Approximate through which we reduce the level set approximation problem to a local function approximation problem. We then show how this approach leads to rate-optimal sample complexity guarantees for Hölder functions, and we investigate how such rates improve when additional smoothness or other structural assumptions hold true.

Improved Exploration in Factored Average-Reward MDPs

Mohammad Sadegh Talebi · Anders Jonsson · Odalric Maillard

We consider a regret minimization task under the average-reward criterion in an unknown Factored Markov Decision Process (FMDP). More specifically, we consider an FMDP where the state-action space $\mathcal X$ and the state-space $\mathcal S$ admit the respective factored forms of $\mathcal X = \otimes_{i=1}^n \mathcal X_i$ and $\mathcal S=\otimes_{i=1}^m \mathcal S_i$, and the transition and reward functions are factored over $\mathcal X$ and $\mathcal S$. Assuming a known a factorization structure, we introduce a novel regret minimization strategy inspired by the popular UCRL strategy, called DBN-UCRL, which relies on Bernstein-type confidence sets defined for individual elements of the transition function. We show that for a generic factorization structure, DBN-UCRL achieves a regret bound, whose leading term strictly improves over existing regret bounds in terms of the dependencies on the size of $\cS_i$'s and the diameter. We further show that when the factorization structure corresponds to the Cartesian product of some base MDPs, the regret of DBN-UCRL is upper bounded by the sum of regret of the base MDPs. We demonstrate, through numerical experiments on standard environments, that DBN-UCRL enjoys a substantially improved regret empirically over existing algorithms that have frequentist regret guarantees.

Kernel regression in high dimensions: Refined analysis beyond double descent

Fanghui Liu · Zhenyu Liao · Johan Suykens

In this paper, we provide a precise characterization of generalization properties of high dimensional kernel ridge regression across the under- and over-parameterized regimes, depending on whether the number of training data n exceeds the feature dimension d. By establishing a bias-variance decomposition of the expected excess risk, we show that, while the bias is (almost) independent of d and monotonically decreases with n, the variance depends on n,d and can be unimodal or monotonically decreasing under different regularization schemes. Our refined analysis goes beyond the double descent theory by showing that, depending on the data eigen-profile and the level of regularization, the kernel regression risk curve can be a double-descent-like, bell-shaped, or monotonic function of n. Experiments on synthetic and real data are conducted to support our theoretical findings.

An Analysis of LIME for Text Data

Dina Mardaoui · Damien Garreau

Text data are increasingly handled in an automated fashion by machine learning algorithms. But the models handling these data are not always well-understood due to their complexity and are more and more often referred to as ``black-boxes.'' Interpretability methods aim to explain how these models operate. Among them, LIME has become one of the most popular in recent years. However, it comes without theoretical guarantees: even for simple models, we are not sure that LIME behaves accurately. In this paper, we provide a first theoretical analysis of LIME for text data. As a consequence of our theoretical findings, we show that LIME indeed provides meaningful explanations for simple models, namely decision trees and linear models.

Couplings for Multinomial Hamiltonian Monte Carlo

Kai Xu · Tor Erlend Fjelde · Charles Sutton · Hong Ge

Hamiltonian Monte Carlo (HMC) is a popular sampling method in Bayesian inference. Recently, Heng & Jacob (2019) studied Metropolis HMC with couplings for unbiased Monte Carlo estimation, establishing a generic parallelizable scheme for HMC. However, in practice a different HMC method, multinomial HMC, is considered as the go-to method, e.g. as part of the no-U-turn sampler. In multinomial HMC, proposed states are not limited to end-points as in Metropolis HMC; instead points along the entire trajectory can be proposed. In this paper, we establish couplings for multinomial HMC, based on optimal transport for multinomial sampling in its transition. We prove an upper bound for the meeting time -- the time it takes for the coupled chains to meet -- based on the notion of local contractivity. We evaluate our methods using three targets: 1,000 dimensional Gaussians, logistic regression and log-Gaussian Cox point processes. Compared to Heng & Jacob (2019), coupled multinomial HMC generally attains a smaller meeting time, and is more robust to choices of step sizes and trajectory lengths, which allows re-use of existing adaptation methods for HMC. These improvements together paves the way for a wider and more practical use of coupled HMC methods.

Amortized Bayesian Prototype Meta-learning: A New Probabilistic Meta-learning Approach to Few-shot Image Classification

Zhuo Sun · Jijie Wu · Xiaoxu Li · Wenming Yang · Jing-Hao Xue

Probabilistic meta-learning methods recently have achieved impressive success in few-shot image classification. However, they introduce a huge number of random variables for neural network weights and thus severe computational and inferential challenges. In this paper, we propose a novel probabilistic meta-learning method called amortized Bayesian prototype meta-learning. In contrast to previous methods, we introduce only a small number of random variables for latent class prototypes rather than a huge number for network weights; we learn to learn the posterior distributions of these latent prototypes in an amortized inference way with no need for an extra amortization network, such that we can easily approximate their posteriors conditional on few labeled samples, whenever at meta-training or meta-testing stage. The proposed method can be trained end-to-end without any pre-training. Compared with other probabilistic meta-learning methods, our proposed approach is more interpretable with much less random variables, while still be able to achieve competitive performance for few-shot image classification problems on various benchmark datasets. Its excellent robustness and predictive uncertainty are also demonstrated through ablation studies.

Wyner-Ziv Estimators: Efficient Distributed Mean Estimation with Side-Information

Prathamesh Mayekar · Ananda Theertha Suresh · Himanshu Tyagi

Communication efficient distributed mean estimation is an important primitive that arises in many distributed learning and optimization scenarios such as federated learning. Without any probabilistic assumptions on the underlying data, we study the problem of distributed mean estimation where the server has access to side information. We propose \emph{Wyner-Ziv estimators}, which are efficient and near-optimal when an upper bound for the distance between the side information and the data is known. In a different direction, when there is no knowledge assumed about the distance between side information and the data, we present an alternative Wyner-Ziv estimator that uses correlated sampling. This latter setting offers {\em universal recovery guarantees}, and perhaps will be of interest in practice when the number of users is large, where keeping track of the distances between the data and the side information may not be possible.

Nonparametric Estimation of Heterogeneous Treatment Effects: From Theory to Learning Algorithms

Alicia Curth · Mihaela van der Schaar

The need to evaluate treatment effectiveness is ubiquitous in most of empirical science, and interest in flexibly investigating effect heterogeneity is growing rapidly. To do so, a multitude of model-agnostic, nonparametric meta-learners have been proposed in recent years. Such learners decompose the treatment effect estimation problem into separate sub-problems, each solvable using standard supervised learning methods. Choosing between different meta-learners in a data-driven manner is difficult, as it requires access to counterfactual information. Therefore, with the ultimate goal of building better understanding of the conditions under which some learners can be expected to perform better than others a priori, we theoretically analyze four broad meta-learning strategies which rely on plug-in estimation and pseudo-outcome regression. We highlight how this theoretical reasoning can be used to guide principled algorithm design and translate our analyses into practice by considering a variety of neural network architectures as base-learners for the discussed meta-learning strategies. In a simulation study, we showcase the relative strengths of the learners under different data-generating processes.

Stochastic Polyak Step-size for SGD: An Adaptive Learning Rate for Fast Convergence

Nicolas Loizou · Sharan Vaswani · Issam Hadj Laradji · Simon Lacoste-Julien

We propose a stochastic variant of the classical Polyak step-size (Polyak, 1987) commonly used in the subgradient method. Although computing the Polyak step-size requires knowledge of the optimal function values, this information is readily available for typical modern machine learning applications. Consequently, the proposed stochastic Polyak step-size (SPS) is an attractive choice for setting the learning rate for stochastic gradient descent (SGD). We provide theoretical convergence guarantees for SGD equipped with SPS in different settings, including strongly convex, convex and non-convex functions. Furthermore, our analysis results in novel convergence guarantees for SGD with a constant step-size. We show that SPS is particularly effective when training over-parameterized models capable of interpolating the training data. In this setting, we prove that SPS enables SGD to converge to the true solution at a fast rate without requiring the knowledge of any problem-dependent constants or additional computational overhead. We experimentally validate our theoretical results via extensive experiments on synthetic and real datasets. We demonstrate the strong performance of SGD with SPS compared to state-of-the-art optimization methods when training over-parameterized models.

Interpretable Random Forests via Rule Extraction

Clément Bénard · Gérard Biau · Sébastien da Veiga · Erwan Scornet

We introduce SIRUS (Stable and Interpretable RUle Set) for regression, a stable rule learning algorithm, which takes the form of a short and simple list of rules. State-of-the-art learning algorithms are often referred to as “black boxes” because of the high number of operations involved in their prediction process. Despite their powerful predictivity, this lack of interpretability may be highly restrictive for applications with critical decisions at stake. On the other hand, algorithms with a simple structure—typically decision trees, rule algorithms, or sparse linear models—are well known for their instability. This undesirable feature makes the conclusions of the data analysis unreliable and turns out to be a strong operational limitation. This motivates the design of SIRUS, based on random forests, which combines a simple structure, a remarkable stable behavior when data is perturbed, and an accuracy comparable to its competitors. We demonstrate the efficiency of the method both empirically (through experiments) and theoretically (with the proof of its asymptotic stability). A R/C++ software implementation sirus is available from CRAN.

Calibrated Adaptive Probabilistic ODE Solvers

Nathanael Bosch · Philipp Hennig · Filip Tronarp

Probabilistic solvers for ordinary differential equations assign a posterior measure to the solution of an initial value problem. The joint covariance of this distribution provides an estimate of the (global) approximation error. The contraction rate of this error estimate as a function of the solver’s step-size identifies it as a well-calibrated worst-case error, but its explicit numerical value for a certain step size is not automatically a good estimate of the explicit error. Addressing this issue, we introduce, discuss, and assess several probabilistically motivated ways to calibrate the uncertainty estimate. Numerical experiments demonstrate that these calibration methods interact efficiently with adaptive step-size selection, resulting in descriptive, and efficiently computable posteriors. We demonstrate the efficiency of the methodology by benchmarking against the classic, widely used Dormand-Prince 4/5 Runge-Kutta method.

Regularized ERM on random subspaces

Andrea Della Vecchia · Jaouad Mourtada · Ernesto De Vito · Lorenzo Rosasco

We study a natural extension of classical empirical risk minimization, where the hypothesis space is a random subspace of a given space. In particular, we consider possibly data dependent subspaces spanned by a random subset of the data, recovering as a special case Nyström approaches for kernel methods. Considering random subspaces naturally leads to computational savings, but the question is whether the corresponding learning accuracy is degraded. These statistical-computational tradeoffs have been recently explored for the least squares loss and self-concordant loss functions, such as the logistic loss. Here, we work to ex- tend these results to convex Lipschitz loss functions, that might not be smooth, such as the hinge loss used in support vector ma- chines. This extension requires developing new proofs, that use different technical tools. Our main results show the existence of different settings, depending on how hard the learning problem is, for which computational efficiency can be improved with no loss in performance. Theoretical results are illustrated with simple numerical experiments.

Budgeted and Non-Budgeted Causal Bandits

Vineet Nair · Vishakha Patil · Gaurav Sinha

Learning good interventions in a causal graph can be modelled as a stochastic multi-armed bandit problem with side-information. First, we study this problem when interventions are more expensive than observations and a budget is specified. If there are no backdoor paths from the intervenable nodes to the reward node then we propose an algorithm to minimize simple regret that optimally trades-off observations and interventions based on the cost of intervention. We also propose an algorithm that accounts for the cost of interventions, utilizes causal side-information, and minimizes the expected cumulative regret without exceeding the budget. Our algorithm performs better than standard algorithms that do not take side-information into account. Finally, we study the problem of learning best interventions without budget constraint in general graphs and give an algorithm that achieves constant expected cumulative regret in terms of the instance parameters when the parent distribution of the reward variable for each intervention is known. Our results are experimentally validated and compared to the best-known bounds in the current literature.

DP-MERF: Differentially Private Mean Embeddings with RandomFeatures for Practical Privacy-preserving Data Generation

Frederik Harder · Kamil Adamczewski · Mijung Park

We propose a differentially private data generation paradigm using random feature representations of kernel mean embeddings when comparing the distribution of true data with that of synthetic data. We exploit the random feature representations for two important benefits. First, we require a minimal privacy cost for training deep generative models. This is because unlike kernel-based distance metrics that require computing the kernel matrix on all pairs of true and synthetic data points, we can detach the data-dependent term from the term solely dependent on synthetic data. Hence, we need to perturb the data-dependent term once and for all and then use it repeatedly during the generator training. Second, we can obtain an analytic sensitivity of the kernel mean embedding as the random features are norm bounded by construction. This removes the necessity of hyper-parameter search for a clipping norm to handle the unknown sensitivity of a generator network. We provide several variants of our algorithm, differentially-private mean embeddings with random features (DP-MERF) to jointly generate labels and input features for datasets such as heterogeneous tabular data and image data. Our algorithm achieves drastically better privacy-utility trade-offs than existing methods when tested on several datasets.

Differentiable Divergences Between Time Series

Mathieu Blondel · Arthur Mensch · Jean-Philippe Vert

Computing the discrepancy between time series of variable sizes is notoriously challenging. While dynamic time warping (DTW) is popularly used for this purpose, it is not differentiable everywhere and is known to lead to bad local optima when used as a loss''. Soft-DTW addresses these issues, but it is not a positive definite divergence: due to the bias introduced by entropic regularization, it can be negative and it is not minimized when the time series are equal. We propose in this paper a new divergence, dubbed soft-DTW divergence, which aims to correct these issues. We study its properties; in particular, under conditions on the ground cost, we show that it is a valid divergence: it is non-negative and minimized if and only if the two time series are equal. We also propose a newsharp'' variant by further removing entropic bias.
We showcase our divergences on time series averaging and demonstrate significant accuracy improvements compared to both DTW and soft-DTW on 84 time series classification datasets.

On the Memory Mechanism of Tensor-Power Recurrent Models

Hejia Qiu · Chao Li · Ying Weng · Zhun Sun · Xingyu He · Qibin Zhao

Tensor-power (TP) recurrent model is a family of non-linear dynamical systems, of which the recurrence relation consists of a p-fold (a.k.a., degree-p) tensor product. Despite such the model frequently appears in the advanced recurrent neural networks (RNNs), to this date there is limited study on its memory property, a critical characteristic in sequence tasks. In this work, we conduct a thorough investigation of the memory mechanism of TP recurrent models. Theoretically, we prove that a large degree p is an essential condition to achieve the long memory effect, yet it would lead to unstable dynamical behaviors. Empirically, we tackle this issue by extending the degree p from discrete to a differentiable domain, such that it is efficiently learnable from a variety of datasets. Taken together, the new model is expected to benefit from the long memory effect in a stable manner. We experimentally show that the proposed model achieves competitive performance compared to various advanced RNNs in both the single-cell and seq2seq architectures.

A Linearly Convergent Algorithm for Decentralized Optimization: Sending Less Bits for Free!

Dmitry Kovalev · Anastasia Koloskova · Martin Jaggi · Peter Richtarik · Sebastian Stich

Decentralized optimization methods enable on-device training of machine learning models without a central coordinator. In many scenarios communication between devices is energy demanding and time consuming and forms the bottleneck of the entire system. We propose a new randomized first-order method which tackles the communication bottleneck by applying randomized compression operators to the communicated messages. By combining our scheme with a new variance reduction technique that progressively throughout the iterations reduces the adverse effect of the injected quantization noise, we obtain a scheme that converges linearly on strongly convex decentralized problems while using compressed communication only. We prove that our method can solve the problems without any increase in the number of communications compared to the baseline which does not perform any communication compression while still allowing for a significant compression factor which depends on the conditioning of the problem and the topology of the network. We confirm our theoretical findings in numerical experiments.

Offline detection of change-points in the mean for stationary graph signals.

Alejandro de la Concha Duarte · Nicolas Vayatis · Argyris Kalogeratos

This paper addresses the problem of segmenting a stream of graph signals: we aim to detect changes in the mean of the multivariate signal defined over the nodes of a known graph. We propose an offline algorithm that relies on the concept of graph signal stationarity and allows the convenient translation of the problem from the original vertex domain to the spectral domain (Graph Fourier Transform), where it is much easier to solve. Although the obtained spectral representation is sparse in real applications, to the best of our knowledge this property has not been much exploited in the existing related literature. Our main contribution is a change-point detection algorithm that adopts a model selection perspective, which takes into account the sparsity of the spectral representation and determines automatically the number of change-points. Our detector comes with a proof of a non-asymptotic oracle inequality, numerical experiments demonstrate the validity of our method.

ATOL: Measure Vectorization for Automatic Topologically-Oriented Learning

Martin Royer · Frederic Chazal · Clément Levrard · Yuhei Umeda · Yuichi Ike

Robust topological information commonly comes in the form of a set of persistence diagrams, finite measures that are in nature uneasy to affix to generic machine learning frameworks. We introduce a fast, learnt, unsupervised vectorization method for measures in Euclidean spaces and use it for reflecting underlying changes in topological behaviour in machine learning contexts. The algorithm is simple and efficiently discriminates important space regions where meaningful differences to the mean measure arise. It is proven to be able to separate clusters of persistence diagrams. We showcase the strength and robustness of our approach on a number of applications, from emulous and modern graph collections where the method reaches state-of-the-art performance to a geometric synthetic dynamical orbits problem. The proposed methodology comes with a single high level tuning parameter: the total measure encoding budget. We provide a completely open access software.

Efficient Statistics for Sparse Graphical Models from Truncated Samples

Arnab Bhattacharyya · Rathin Desai · Sai Ganesh Nagarajan · Ioannis Panageas

In this paper, we study high-dimensional estimation from truncated samples. We focus on two fundamental and classical problems: (i) inference of sparse Gaussian graphical models and (ii) support recovery of sparse linear models.

(i) For Gaussian graphical models, suppose d-dimensional samples x are generated from a Gaussian N(mu, Sigma) and observed only if they belong to a subset S of R^d. We show that mu and Sigma can be estimated with error epsilon in the Frobenius norm, using O~(nz(Sigma^{-1})/epsilon^2) samples from a truncated N(mu, Sigma) and having access to a membership oracle for S. The set S is assumed to have non-trivial measure under the unknown distribution but is otherwise arbitrary.

(ii) For sparse linear regression, suppose samples (x,y) are generated where y = + N(0,1) and (x, y) is seen only if y belongs to a truncation set S of the reals. We consider the case that Omega* is sparse with a support set of size k. Our main result is to establish precise conditions on the problem dimension d, the support size k, the number of observations n, and properties of the samples and the truncation that are sufficient to recover the support of Omega. Specifically, we show that under some mild assumptions, only O(k^2 log d) samples are needed to estimate Omega in the infinity-norm up to a bounded error. Similar results are also estabilished for estimating Omega* in the Euclidean norm up to arbitrary error.

For both problems, our estimator minimizes the sum of the finite population negative log-likelihood function and an ell_1-regularization term.

Last iterate convergence in no-regret learning: constrained min-max optimization for convex-concave landscapes

Qi Lei · Sai Ganesh Nagarajan · Ioannis Panageas · xiao wang

In a recent series of papers it has been established that variants of Gradient Descent/Ascent and Mirror Descent exhibit last iterate convergence in convex-concave zero-sum games. Specifically, Daskalakis et al 2018, Liang-Stokes 2019, show last iterate convergence of the so called ``Optimistic Gradient Descent/Ascent" for the case of \textit{unconstrained} min-max optimization. Moreover, in Mertikopoulos et al 2019 the authors show that Mirror Descent with an extra gradient step displays last iterate convergence for convex-concave problems (both constrained and unconstrained), though their algorithm uses \textit{vanishing stepsizes}. In this work, we show that "Optimistic Multiplicative-Weights Update (OMWU)" with \textit{constant stepsize}, exhibits last iterate convergence locally for convex-concave games, generalizing the results of Daskalakis and Panageas 2019 where last iterate convergence of OMWU was shown only for the \textit{bilinear case}. To the best of our knowledge, this is the first result about last-iterate convergence for constrained zero sum games (beyond the bilinear case) in which the dynamics use constant step-sizes.