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Vlad Niculae

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17 papers

2 author rows

JMLR Journal 2025 Journal Article

Hopfield-Fenchel-Young Networks: A Unified Framework for Associative Memory Retrieval

Saul Santos
Vlad Niculae
Daniel McNamee
Andre F.T. Martins

Associative memory models, such as Hopfield networks and their modern variants, have garnered renewed interest due to advancements in memory capacity and connections with self-attention in transformers. In this work, we introduce a unified framework-Hopfield-Fenchel-Young networks-which generalizes these models to a broader family of energy functions. Our energies are formulated as the difference between two Fenchel-Young losses: one, parameterized by a generalized entropy, defines the Hopfield scoring mechanism, while the other applies a post-transformation to the Hopfield output. By utilizing Tsallis and norm entropies, we derive end-to-end differentiable update rules that enable sparse transformations, uncovering new connections between loss margins, sparsity, and exact retrieval of single memory patterns. We further extend this framework to structured Hopfield networks using the SparseMAP transformation, allowing the retrieval of pattern associations rather than a single pattern. Our framework unifies and extends traditional and modern Hopfield networks and provides an energy minimization perspective for widely used post-transformations like $\ell_2$-normalization and layer normalization-all through suitable choices of Fenchel-Young losses and by using convex analysis as a building block. Finally, we validate our Hopfield-Fenchel-Young networks on diverse memory recall tasks, including free and sequential recall. Experiments on simulated data, image retrieval, multiple instance learning, and text rationalization demonstrate the effectiveness of our approach. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2025. ( edit, beta )

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TMLR Journal 2025 Journal Article

Keep your distance: learning dispersed embeddings on $\mathbb{S}_{m}$

Evgeniia Tokarchuk
Hua Chang Bakker
Vlad Niculae

Learning well-separated features in high-dimensional spaces, such as text or image embeddings, is crucial for many machine learning applications. Achieving such separation can be effectively accomplished through the dispersion of embeddings, where unrelated vectors are pushed apart as much as possible. By constraining features to be on a hypersphere, we can connect dispersion to well-studied problems in mathematics and physics, where optimal solutions are known for limited low-dimensional cases. However, in representation learning we typically deal with a large number of features in high-dimensional space, and moreover, dispersion is usually traded off with some other task-oriented training objective, making existing theoretical and numerical solutions inapplicable. Therefore, it is common to rely on gradient-based methods to encourage dispersion, usually by minimizing some function of the pairwise distances. In this work, we first give an overview of existing methods from disconnected literature, making new connections and highlighting similarities. Next, we introduce some new angles. We propose to reinterpret pairwise dispersion using a maximum mean discrepancy (MMD) motivation. We then propose an online variant of the celebrated Lloyd’s algorithm, of K-Means fame, as an effective alternative regularizer for dispersion on generic domains. Finally, we revise and empirically assess sliced regularizers that directly exploit properties of the hypersphere, proposing a new, simple but effective one. Our experiments show the importance of dispersion in image classification and natural language processing tasks, and how algorithms exhibit different trade-offs in different regimes

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TMLR Journal 2025 Journal Article

On the Low-Rank Parametrization of Reward Models for Controlled Language Generation

Sergey Troshin
Vlad Niculae
Antske Fokkens

Language models trained on large amounts of data are known to produce inappropriate content in some cases and require careful tuning to be used in the real world. We revisit an effective and modular approach for controllability of the language models, when an external expert model guides the decoding. Particularly, we zoom in into the parametrization choice of an external expert, highlighting the difference between low-rank and higher-rank parametrizations. Higher-rank experts are designed to support high flexibility when representing the rewards, leading to higher computational costs during decoding. However, we demonstrate that they might not use their full flexibility. By analyzing the recently proposed reward-augmented decoding approach (RAD), which uses a higher-rank expert model, we introduce a simpler but more efficient low-rank parametrization of the expert model enabling fast and effective guided decoding. We empirically show that the low-rank RAD performs on par with the more flexible RAD on a detoxification and a sentiment control task, while requiring only a single reward model call per generated token.

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ICML Conference 2024 Conference Paper

Sparse and Structured Hopfield Networks

Saul José Rodrigues dos Santos
Vlad Niculae
Daniel C. McNamee
André F. T. Martins

Modern Hopfield networks have enjoyed recent interest due to their connection to attention in transformers. Our paper provides a unified framework for sparse Hopfield networks by establishing a link with Fenchel-Young losses. The result is a new family of Hopfield-Fenchel-Young energies whose update rules are end-to-end differentiable sparse transformations. We reveal a connection between loss margins, sparsity, and exact memory retrieval. We further extend this framework to structured Hopfield networks via the SparseMAP transformation, which can retrieve pattern associations instead of a single pattern. Experiments on multiple instance learning and text rationalization demonstrate the usefulness of our approach.

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ICLR Conference 2023 Conference Paper

DAG Learning on the Permutahedron

Valentina Zantedeschi
Luca Franceschi 0001
Jean Kaddour
Matt J. Kusner
Vlad Niculae

We propose a continuous optimization framework for discovering a latent directed acyclic graph (DAG) from observational data. Our approach optimizes over the polytope of permutation vectors, the so-called Permutahedron, to learn a topological ordering. Edges can be optimized jointly, or learned conditional on the ordering via a non-differentiable subroutine. Compared to existing continuous optimization approaches our formulation has a number of advantages including: 1. validity: optimizes over exact DAGs as opposed to other relaxations optimizing approximate DAGs; 2. modularity: accommodates any edge-optimization procedure, edge structural parameterization, and optimization loss; 3. end-to-end: either alternately iterates between node-ordering and edge-optimization, or optimizes them jointly; We demonstrate, on real-world data problems in protein-signaling and transcriptional network discovery, that our approach lies on the Pareto frontier of two key metrics, the SID and SHD.

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TMLR Journal 2023 Journal Article

Wrapped $\beta$-Gaussians with compact support for exact probabilistic modeling on manifolds

Sergey Troshin
Vlad Niculae

We introduce wrapped $\beta$-Gaussians, a family of wrapped distributions on Riemannian manifolds, supporting efficient reparametrized sampling, as well as exact density estimation, effortlessly supporting high dimensions and anisotropic scale parameters. We extend Fenchel-Young losses for geometry-aware learning with wrapped $\beta$-Gaussians, and demonstrate the efficacy of our proposed family in a suite of experiments on hypersphere and rotation manifolds: data fitting, hierarchy encoding, generative modeling with variational autoencoders, and multilingual word embedding alignment.

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ICML Conference 2022 Conference Paper

Modeling Structure with Undirected Neural Networks

Tsvetomila Mihaylova
Vlad Niculae
André F. T. Martins

Neural networks are powerful function estimators, leading to their status as a paradigm of choice for modeling structured data. However, unlike other structured representations that emphasize the modularity of the problem {–} e. g. , factor graphs {–} neural networks are usually monolithic mappings from inputs to outputs, with a fixed computation order. This limitation prevents them from capturing different directions of computation and interaction between the modeled variables. In this paper, we combine the representational strengths of factor graphs and of neural networks, proposing undirected neural networks (UNNs): a flexible framework for specifying computations that can be performed in any order. For particular choices, our proposed models subsume and extend many existing architectures: feed-forward, recurrent, self-attention networks, auto-encoders, and networks with implicit layers. We demonstrate the effectiveness of undirected neural architectures, both unstructured and structured, on a range of tasks: tree-constrained dependency parsing, convolutional image classification, and sequence completion with attention. By varying the computation order, we show how a single UNN can be used both as a classifier and a prototype generator, and how it can fill in missing parts of an input sequence, making them a promising field for further research.

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ICLR Conference 2022 Conference Paper

Sparse Communication via Mixed Distributions

António Farinhas
Wilker Aziz
Vlad Niculae
André F. T. Martins

Neural networks and other machine learning models compute continuous representations, while humans communicate mostly through discrete symbols. Reconciling these two forms of communication is desirable for generating human-readable interpretations or learning discrete latent variable models, while maintaining end-to-end differentiability. Some existing approaches (such as the Gumbel-Softmax transformation) build continuous relaxations that are discrete approximations in the zero-temperature limit, while others (such as sparsemax transformations and the Hard Concrete distribution) produce discrete/continuous hybrids. In this paper, we build rigorous theoretical foundations for these hybrids, which we call "mixed random variables.'' Our starting point is a new "direct sum'' base measure defined on the face lattice of the probability simplex. From this measure, we introduce new entropy and Kullback-Leibler divergence functions that subsume the discrete and differential cases and have interpretations in terms of code optimality. Our framework suggests two strategies for representing and sampling mixed random variables, an extrinsic ("sample-and-project'’) and an intrinsic one (based on face stratification). We experiment with both approaches on an emergent communication benchmark and on modeling MNIST and Fashion-MNIST data with variational auto-encoders with mixed latent variables.

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JMLR Journal 2022 Journal Article

Sparse Continuous Distributions and Fenchel-Young Losses

André F. T. Martins
Marcos Treviso
António Farinhas
Pedro M. Q. Aguiar
Mário A. T. Figueiredo
Mathieu Blondel
Vlad Niculae

Exponential families are widely used in machine learning, including many distributions in continuous and discrete domains (e.g., Gaussian, Dirichlet, Poisson, and categorical distributions via the softmax transformation). Distributions in each of these families have fixed support. In contrast, for finite domains, recent work on sparse alternatives to softmax (e.g., sparsemax, $\alpha$-entmax, and fusedmax), has led to distributions with varying support. This paper develops sparse alternatives to continuous distributions, based on several technical contributions: First, we define $\Omega$-regularized prediction maps and Fenchel-Young losses for arbitrary domains (possibly countably infinite or continuous). For linearly parametrized families, we show that minimization of Fenchel-Young losses is equivalent to moment matching of the statistics, generalizing a fundamental property of exponential families. When $\Omega$ is a Tsallis negentropy with parameter $\alpha$, we obtain “deformed exponential families,” which include $\alpha$-entmax and sparsemax ($\alpha=2$) as particular cases. For quadratic energy functions, the resulting densities are $\beta$-Gaussians, an instance of elliptical distributions that contain as particular cases the Gaussian, biweight, triweight, and Epanechnikov densities, and for which we derive closed-form expressions for the variance, Tsallis entropy, and Fenchel-Young loss. When $\Omega$ is a total variation or Sobolev regularizer, we obtain a continuous version of the fusedmax. Finally, we introduce continuous-domain attention mechanisms, deriving efficient gradient backpropagation algorithms for $\alpha \in \{1,\frac{4}{3}, \frac{3}{2}, 2\}$. Using these algorithms, we demonstrate our sparse continuous distributions for attention-based audio classification and visual question answering, showing that they allow attending to time intervals and compact regions. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2022. ( edit, beta )

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ICML Conference 2021 Conference Paper

Learning Binary Decision Trees by Argmin Differentiation

Valentina Zantedeschi
Matt J. Kusner
Vlad Niculae

We address the problem of learning binary decision trees that partition data for some downstream task. We propose to learn discrete parameters (i. e. , for tree traversals and node pruning) and continuous parameters (i. e. , for tree split functions and prediction functions) simultaneously using argmin differentiation. We do so by sparsely relaxing a mixed-integer program for the discrete parameters, to allow gradients to pass through the program to continuous parameters. We derive customized algorithms to efficiently compute the forward and backward passes. This means that our tree learning procedure can be used as an (implicit) layer in arbitrary deep networks, and can be optimized with arbitrary loss functions. We demonstrate that our approach produces binary trees that are competitive with existing single tree and ensemble approaches, in both supervised and unsupervised settings. Further, apart from greedy approaches (which do not have competitive accuracies), our method is faster to train than all other tree-learning baselines we compare with.

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NeurIPS Conference 2020 Conference Paper

Efficient Marginalization of Discrete and Structured Latent Variables via Sparsity

Gonçalo Correia
Vlad Niculae
Wilker Aziz
André Martins

Training neural network models with discrete (categorical or structured) latent variables can be computationally challenging, due to the need for marginalization over large or combinatorial sets. To circumvent this issue, one typically resorts to sampling-based approximations of the true marginal, requiring noisy gradient estimators (e. g. , score function estimator) or continuous relaxations with lower-variance reparameterized gradients (e. g. , Gumbel-Softmax). In this paper, we propose a new training strategy which replaces these estimators by an exact yet efficient marginalization. To achieve this, we parameterize discrete distributions over latent assignments using differentiable sparse mappings: sparsemax and its structured counterparts. In effect, the support of these distributions is greatly reduced, which enables efficient marginalization. We report successful results in three tasks covering a range of latent variable modeling applications: a semisupervised deep generative model, a latent communication game, and a generative model with a bit-vector latent representation. In all cases, we obtain good performance while still achieving the practicality of sampling-based approximations.

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JMLR Journal 2020 Journal Article

Learning with Fenchel-Young losses

Mathieu Blondel
André F.T. Martins
Vlad Niculae

Over the past decades, numerous loss functions have been been proposed for a variety of supervised learning tasks, including regression, classification, ranking, and more generally structured prediction. Understanding the core principles and theoretical properties underpinning these losses is key to choose the right loss for the right problem, as well as to create new losses which combine their strengths. In this paper, we introduce Fenchel-Young losses, a generic way to construct a convex loss function for a regularized prediction function. We provide an in-depth study of their properties in a very broad setting, covering all the aforementioned supervised learning tasks, and revealing new connections between sparsity, generalized entropies, and separation margins. We show that Fenchel-Young losses unify many well-known loss functions and allow to create useful new ones easily. Finally, we derive efficient predictive and training algorithms, making Fenchel-Young losses appealing both in theory and practice. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2020. ( edit, beta )

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ICML Conference 2020 Conference Paper

LP-SparseMAP: Differentiable Relaxed Optimization for Sparse Structured Prediction

Vlad Niculae
André F. T. Martins

Structured predictors require solving a combinatorial optimization problem over a large number of structures, such as dependency trees or alignments. When embedded as structured hidden layers in a neural net, argmin differentiation and efficient gradient computation are further required. Recently, SparseMAP has been proposed as a differentiable, sparse alternative to maximum a posteriori (MAP) and marginal inference. SparseMAP returns an interpretable combination of a small number of structures; its sparsity being the key to efficient optimization. However, SparseMAP requires access to an exact MAP oracle in the structured model, excluding, e. g. , loopy graphical models or logic constraints, which generally require approximate inference. In this paper, we introduce LP-SparseMAP, an extension of SparseMAP addressing this limitation via a local polytope relaxation. LP-SparseMAP uses the flexible and powerful language of factor graphs to define expressive hidden structures, supporting coarse decompositions, hard logic constraints, and higher-order correlations. We derive the forward and backward algorithms needed for using LP-SparseMAP as a structured hidden or output layer. Experiments in three structured tasks show benefits versus SparseMAP and Structured SVM.

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NeurIPS Conference 2020 Conference Paper

Sparse and Continuous Attention Mechanisms

André Martins
António Farinhas
Marcos Treviso
Vlad Niculae
Pedro Aguiar
Mario Figueiredo

Exponential families are widely used in machine learning; they include many distributions in continuous and discrete domains (e. g. , Gaussian, Dirichlet, Poisson, and categorical distributions via the softmax transformation). Distributions in each of these families have fixed support. In contrast, for finite domains, there has been recent work on sparse alternatives to softmax (e. g. , sparsemax and alpha-entmax), which have varying support, being able to assign zero probability to irrelevant categories. These discrete sparse mappings have been used for improving interpretability of neural attention mechanisms. This paper expands that work in two directions: first, we extend alpha-entmax to continuous domains, revealing a link with Tsallis statistics and deformed exponential families. Second, we introduce continuous-domain attention mechanisms, deriving efficient gradient backpropagation algorithms for alpha in {1, 2}. Experiments on attention-based text classification, machine translation, and visual question answering illustrate the use of continuous attention in 1D and 2D, showing that it allows attending to time intervals and compact regions.

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ICML Conference 2018 Conference Paper

SparseMAP: Differentiable Sparse Structured Inference

Vlad Niculae
André F. T. Martins
Mathieu Blondel
Claire Cardie

Structured prediction requires searching over a combinatorial number of structures. To tackle it, we introduce SparseMAP, a new method for sparse structured inference, together with corresponding loss functions. SparseMAP inference is able to automatically select only a few global structures: it is situated between MAP inference, which picks a single structure, and marginal inference, which assigns probability mass to all structures, including implausible ones. Importantly, SparseMAP can be computed using only calls to a MAP oracle, hence it is applicable even to problems where marginal inference is intractable, such as linear assignment. Moreover, thanks to the solution sparsity, gradient backpropagation is efficient regardless of the structure. SparseMAP thus enables us to augment deep neural networks with generic and sparse structured hidden layers. Experiments in dependency parsing and natural language inference reveal competitive accuracy, improved interpretability, and the ability to capture natural language ambiguities, which is attractive for pipeline systems.

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NeurIPS Conference 2017 Conference Paper

A Regularized Framework for Sparse and Structured Neural Attention

Vlad Niculae
Mathieu Blondel

Modern neural networks are often augmented with an attention mechanism, which tells the network where to focus within the input. We propose in this paper a new framework for sparse and structured attention, building upon a smoothed max operator. We show that the gradient of this operator defines a mapping from real values to probabilities, suitable as an attention mechanism. Our framework includes softmax and a slight generalization of the recently-proposed sparsemax as special cases. However, we also show how our framework can incorporate modern structured penalties, resulting in more interpretable attention mechanisms, that focus on entire segments or groups of an input. We derive efficient algorithms to compute the forward and backward passes of our attention mechanisms, enabling their use in a neural network trained with backpropagation. To showcase their potential as a drop-in replacement for existing ones, we evaluate our attention mechanisms on three large-scale tasks: textual entailment, machine translation, and sentence summarization. Our attention mechanisms improve interpretability without sacrificing performance; notably, on textual entailment and summarization, we outperform the standard attention mechanisms based on softmax and sparsemax.

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NeurIPS Conference 2017 Conference Paper

Multi-output Polynomial Networks and Factorization Machines

Mathieu Blondel
Vlad Niculae
Takuma Otsuka
Naonori Ueda

Factorization machines and polynomial networks are supervised polynomial models based on an efficient low-rank decomposition. We extend these models to the multi-output setting, i. e. , for learning vector-valued functions, with application to multi-class or multi-task problems. We cast this as the problem of learning a 3-way tensor whose slices share a common basis and propose a convex formulation of that problem. We then develop an efficient conditional gradient algorithm and prove its global convergence, despite the fact that it involves a non-convex basis selection step. On classification tasks, we show that our algorithm achieves excellent accuracy with much sparser models than existing methods. On recommendation system tasks, we show how to combine our algorithm with a reduction from ordinal regression to multi-output classification and show that the resulting algorithm outperforms simple baselines in terms of ranking accuracy.

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