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We present a new data structure to approximate accurately and efficiently a polynomial $f$ of degree $d$ given as a list of coefficients. Its properties allow us to improve the state-of-the-art bounds on the bit complexity for the problems of root isolation and approximate multipoint evaluation. This data structure also leads to a new geometric criterion to detect ill-conditioned polynomials, implying notably that the standard condition number of the zeros of a polynomial is at least exponential in the number of roots of modulus less than $1/2$ or greater than $2$.Given a polynomial $f$ of degree $d$ with $\|f\|_1 \leq 2^\tau$ for $\tau \geq 1$, isolating all its complex roots or evaluating it at $d$ points can be done with a quasi-linear number of arithmetic operations. However, considering the bit complexity, the state-of-the-art algorithms require at least $d^{3/2}$ bit operations even for well-conditioned polynomials and when the accuracy required is low. Given a positive integer $m$, we can compute our new data structure and evaluate $f$ at $d$ points in the unit disk with an absolute error less than $2^{-m}$ in $\widetilde O(d(\tau+m))$ bit operations, where $\widetilde O(\cdot)$ means that we omit logarithmic factors. We also show that if $\kappa$ is the absolute condition number of the zeros of $f$, then we can isolate all the roots of $f$ in $\widetilde O(d(\tau + \log \kappa))$ bit operations. Moreover, our algorithms are simple to implement. For approximating the complex roots of a polynomial, we implemented a small prototype in \verb|Python/NumPy| that is an order of magnitude faster than the state-of-the-art solver \verb/MPSolve/ for high degree polynomials with random coefficients.

A tournament graph $T = \left(V, E \right)$ is an oriented complete graph, which can be used to model a round-robin tournament between $n$ players. In this paper, we address the problem of finding a champion of the tournament, also known as Copeland winner, which is a player that wins the highest number of matches. Solving this problem has important implications on several Information Retrieval applications, including Web search, conversational IR, machine translation, question answering, recommender systems, etc. Our goal is to solve the problem by minimizing the number of times we probe the adjacency matrix, i.e., the number of matches played. We prove that any deterministic/randomized algorithm finding a champion with constant success probability requires $\Omega(\ell n)$ comparisons, where $\ell$ is the number of matches lost by the champion. We then present an optimal deterministic algorithm matching this lower bound without knowing $\ell$ and we extend our analysis to three strictly related problems. Lastly, we conduct a comprehensive experimental assessment of the proposed algorithms to speed up a state-of-the-art solution for ranking on public data. Results show that our proposals speed up the retrieval of the champion up to $13\times$ in this scenario.

A well-known line of work (Barron, 1993; Breiman, 1993; Klusowski & Barron, 2018) provides bounds on the width $n$ of a ReLU two-layer neural network needed to approximate a function $f$ over the ball $\mathcal{B}_R(\mathbb{R}^d)$ up to error $\epsilon$, when the Fourier based quantity $C_f = \frac{1}{(2\pi)^{d/2}} \int_{\mathbb{R}^d} \|\xi\|^2 |\hat{f}(\xi)| \ d\xi$ is finite. More recently Ongie et al. (2019) used the Radon transform as a tool for analysis of infinite-width ReLU two-layer networks. In particular, they introduce the concept of Radon-based $\mathcal{R}$-norms and show that a function defined on $\mathbb{R}^d$ can be represented as an infinite-width two-layer neural network if and only if its $\mathcal{R}$-norm is finite. In this work, we extend the framework of Ongie et al. (2019) and define similar Radon-based semi-norms ($\mathcal{R}, \mathcal{U}$-norms) such that a function admits an infinite-width neural network representation on a bounded open set $\mathcal{U} \subseteq \mathbb{R}^d$ when its $\mathcal{R}, \mathcal{U}$-norm is finite. Building on this, we derive sparse (finite-width) neural network approximation bounds that refine those of Breiman (1993); Klusowski & Barron (2018). Finally, we show that infinite-width neural network representations on bounded open sets are not unique and study their structure, providing a functional view of mode connectivity.

Recent years have seen tremendous growth in the amount of verified software. Proofs for complex properties can now be achieved using higher-order theories and calculi. Complex properties lead to an ever-growing number of definitions and associated lemmas, which constitute an integral part of proof construction. Following this -- whether automatic or semi-automatic -- methods for computer-aided lemma discovery have emerged. In this work, we introduce a new symbolic technique for bottom-up lemma discovery, that is, the generation of a library of lemmas from a base set of inductive data types and recursive definitions. This is known as the theory exploration problem, and so far, solutions have been proposed based either on counter-example generation or the more prevalent random testing combined with first-order solvers. Our new approach, being purely deductive, eliminates the need for random testing as a filtering phase and for SMT solvers. Therefore it is amenable compositional reasoning and for the treatment of user-defined higher-order functions. Our implementation has shown to find more lemmas than prior art, while avoiding redundancy.

The Goppa Code Distinguishing (GD) problem asks to distinguish efficiently a generator matrix of a Goppa code from a randomly drawn one. We revisit a distinguisher for alternant and Goppa codes through a new approach, namely by studying the dimension of square codes. We provide here a rigorous upper bound for the dimension of the square of the dual of an alternant or Goppa code, while the previous approach only provided algebraic explanations based on heuristics. Moreover, for Goppa codes, our proof extends to the non-binary case as well, thus providing an algebraic explanation for the distinguisher which was missing up to now. All the upper bounds are tight and match experimental evidence. Our work also introduces new algebraic results about products of trace codes in general and of dual of alternant and Goppa codes in particular, clarifying their square code structure. This might be of interest for cryptanalysis purposes.

Constrained tensor and matrix factorization models allow to extract interpretable patterns from multiway data. Therefore identifiability properties and efficient algorithms for constrained low-rank approximations are nowadays important research topics. This work deals with columns of factor matrices of a low-rank approximation being sparse in a known and possibly overcomplete basis, a model coined as Dictionary-based Low-Rank Approximation (DLRA). While earlier contributions focused on finding factor columns inside a dictionary of candidate columns, i.e. one-sparse approximations, this work is the first to tackle DLRA with sparsity larger than one. I propose to focus on the sparse-coding subproblem coined Mixed Sparse-Coding (MSC) that emerges when solving DLRA with an alternating optimization strategy. Several algorithms based on sparse-coding heuristics (greedy methods, convex relaxations) are provided to solve MSC. The performance of these heuristics is evaluated on simulated data. Then, I show how to adapt an efficient MSC solver based on the LASSO to compute Dictionary-based Matrix Factorization and Canonical Polyadic Decomposition in the context of hyperspectral image processing and chemometrics. These experiments suggest that DLRA extends the modeling capabilities of low-rank approximations, helps reducing estimation variance and enhances the identifiability and interpretability of estimated factors.

We consider the problem of learning a tree-structured Ising model from data, such that subsequent predictions computed using the model are accurate. Concretely, we aim to learn a model such that posteriors $P(X_i|X_S)$ for small sets of variables $S$ are accurate. Since its introduction more than 50 years ago, the Chow-Liu algorithm, which efficiently computes the maximum likelihood tree, has been the benchmark algorithm for learning tree-structured graphical models. A bound on the sample complexity of the Chow-Liu algorithm with respect to the prediction-centric local total variation loss was shown in [BK19]. While those results demonstrated that it is possible to learn a useful model even when recovering the true underlying graph is impossible, their bound depends on the maximum strength of interactions and thus does not achieve the information-theoretic optimum. In this paper, we introduce a new algorithm that carefully combines elements of the Chow-Liu algorithm with tree metric reconstruction methods to efficiently and optimally learn tree Ising models under a prediction-centric loss. Our algorithm is robust to model misspecification and adversarial corruptions. In contrast, we show that the celebrated Chow-Liu algorithm can be arbitrarily suboptimal.

Optimal transport distances have found many applications in machine learning for their capacity to compare non-parametric probability distributions. Yet their algorithmic complexity generally prevents their direct use on large scale datasets. Among the possible strategies to alleviate this issue, practitioners can rely on computing estimates of these distances over subsets of data, {\em i.e.} minibatches. While computationally appealing, we highlight in this paper some limits of this strategy, arguing it can lead to undesirable smoothing effects. As an alternative, we suggest that the same minibatch strategy coupled with unbalanced optimal transport can yield more robust behavior. We discuss the associated theoretical properties, such as unbiased estimators, existence of gradients and concentration bounds. Our experimental study shows that in challenging problems associated to domain adaptation, the use of unbalanced optimal transport leads to significantly better results, competing with or surpassing recent baselines.

Graph neural networks (GNNs) are a popular class of machine learning models whose major advantage is their ability to incorporate a sparse and discrete dependency structure between data points. Unfortunately, GNNs can only be used when such a graph-structure is available. In practice, however, real-world graphs are often noisy and incomplete or might not be available at all. With this work, we propose to jointly learn the graph structure and the parameters of graph convolutional networks (GCNs) by approximately solving a bilevel program that learns a discrete probability distribution on the edges of the graph. This allows one to apply GCNs not only in scenarios where the given graph is incomplete or corrupted but also in those where a graph is not available. We conduct a series of experiments that analyze the behavior of the proposed method and demonstrate that it outperforms related methods by a significant margin.

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.