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Optimization · Non-Convex

El Mehdi Achour, Francois Malgouyres, Sebastien Gerchinovitz

We study the optimization landscape of deep linear neural networks with square loss. It is known that, under weak assumptions, there are no spurious local minima and no local maxima. However, the existence and diversity of non-strict saddle points, which can play a role in first-order algorithms' dynamics, have only been lightly studied. We go a step further with a complete analysis of the optimization landscape at order $2$. Among all critical points, we characterize global minimizers, strict saddle points, and non-strict saddle points. We enumerate all the associated critical values. The characterization is simple, involves conditions on the ranks of partial matrix products, and sheds some light on global convergence or implicit regularization that has been proved or observed when optimizing linear neural networks. In passing, we provide an explicit parameterization of the set of all global minimizers and exhibit large sets of strict and non-strict saddle points.

Applications · Health

Morgan B Talbot, Gabriel Kreiman, James DiCarlo, Guy Gaziv

The currently leading artificial neural network models of the visual ventral stream - which are derived from a combination of performance optimization and robustification methods - have demonstrated a remarkable degree of behavioral alignment with humans on visual categorization tasks. We show that image perturbations generated by these models can enhance the ability of humans to accurately report the ground truth class. Furthermore, we find that the same models can also be used out-of-the-box to predict the proportion of correct human responses to individual images, providing a simple, human-aligned estimator of the relative difficulty of each image. Motivated by these observations, we propose to augment visual learning in humans in a way that improves human categorization accuracy at test time. Our learning augmentation approach consists of (i) selecting images based on their model-estimated recognition difficulty, and (ii) applying image perturbations that aid recognition for novice learners. We find that combining these model-based strategies leads to categorization accuracy gains of 33-72% relative to control subjects without these interventions, on unmodified, randomly selected held-out test images. Beyond the accuracy gain, the training time for the augmented learning group was also shortened by 20-23%, despite both groups completing the same number of training trials. We demonstrate the efficacy of our approach in a fine-grained categorization task with natural images, as well as two tasks in clinically relevant image domains - histology and dermoscopy - where visual learning is notoriously challenging. To the best of our knowledge, our work is the first application of artificial neural networks to increase visual learning performance in humans by enhancing category-specific image features.

Theory · Everything Else

Francesco Mori, Stefano Sarao Mannelli, Francesca Mignacco

Artificial neural networks often struggle with _catastrophic forgetting_ when learning multiple tasks sequentially, as training on new tasks degrades the performance on previously learned tasks. Recent theoretical work has addressed this issue by analysing learning curves in synthetic frameworks under predefined training protocols. However, these protocols relied on heuristics and lacked a solid theoretical foundation assessing their optimality. In this paper, we fill this gap by combining exact equations for training dynamics, derived using statistical physics techniques, with optimal control methods. We apply this approach to teacher-student models for continual learning and multi-task problems, obtaining a theory for task-selection protocols maximising performance while minimising forgetting. Our theoretical analysis offers non-trivial yet interpretable strategies for mitigating catastrophic forgetting, shedding light on how optimal learning protocols modulate established effects, such as the influence of task similarity on forgetting. Finally, we validate our theoretical findings with experiments on real-world data.

Hao Sun, Yunyi Shen, Jean-Francois Ton

The Bradley-Terry (BT) model is a common and successful practice in reward modeling for Large Language Model (LLM) alignment. However, it remains unclear *why* this model --- originally developed for multi-player stochastic game matching --- can be adopted to convert pairwise response comparisons to reward values and make predictions. Especially given the fact that only a limited number of prompt-response pairs are sparsely compared with others. In this paper, we first establish the convergence rate of BT reward models based on deep neural networks using embeddings, providing a theoretical foundation for their use.Despite theoretically sound, we argue that the BT model is not a necessary choice from the perspective of downstream optimization, this is because a reward model only needs to preserve the correct ranking predictions through a monotonic transformation of the true reward. We highlight the critical concept of *order consistency* in reward modeling and demonstrate that the BT model possesses this property.Moreover, we propose a simple and straightforward upper-bound algorithm, compatible with off-the-shelf binary classifiers, as an alternative order-consistent reward modeling objective. To offer practical insights, we empirically evaluate the performance of these different reward modeling approaches across more than 12,000 experimental setups, using $6$ base LLMs, $2$ datasets, and diverse annotation designs that vary in quantity, quality, and pairing choices in preference annotations.

Deep Learning · Large Language Models

Hao Sun, Yunyi Shen, Jean-Francois Ton

The Bradley-Terry (BT) model is a common and successful practice in reward modeling for Large Language Model (LLM) alignment. However, it remains unclear *why* this model --- originally developed for multi-player stochastic game matching --- can be adopted to convert pairwise response comparisons to reward values and make predictions. Especially given the fact that only a limited number of prompt-response pairs are sparsely compared with others. In this paper, we first establish the convergence rate of BT reward models based on deep neural networks using embeddings, providing a theoretical foundation for their use.Despite theoretically sound, we argue that the BT model is not a necessary choice from the perspective of downstream optimization, this is because a reward model only needs to preserve the correct ranking predictions through a monotonic transformation of the true reward. We highlight the critical concept of *order consistency* in reward modeling and demonstrate that the BT model possesses this property.Moreover, we propose a simple and straightforward upper-bound algorithm, compatible with off-the-shelf binary classifiers, as an alternative order-consistent reward modeling objective. To offer practical insights, we empirically evaluate the performance of these different reward modeling approaches across more than 12,000 experimental setups, using $6$ base LLMs, $2$ datasets, and diverse annotation designs that vary in quantity, quality, and pairing choices in preference annotations.

Deep Learning · Graph Neural Networks

Chenyi Zi, Bowen LIU, Xiangguo SUN, Hong Cheng, Jia Li

Graph Neural Networks (GNNs) have transformed graph learning but face challenges like distribution shifts, data anomalies, and adversarial vulnerabilities. Graph prompt emerges as a novel solution, enabling data transformation to align graph data with pre-trained models without altering model parameters. This paradigm addresses negative transfer, enhances adaptability, and bridges modality gaps. Unlike traditional fine-tuning, graph prompts rewrite graph structures and features through components like prompt tokens and insertion patterns, improving flexibility and efficiency. Applications in IoT, drug discovery, fraud detection, and personalized learning demonstrate their potential to dynamically adapt graph data. While promising, challenges such as optimal design, benchmarks, and gradient issues persist. Addressing these will unlock full potential of graph prompt to advance GNNs for complex real-world tasks.

Deep Learning · Graph Neural Networks

Guibin Zhang, Xiangguo SUN, Yanwei Yue, Chonghe Jiang, Kun Wang, Tianlong Chen, Shirui Pan

Graph Neural Networks (GNNs) have demonstrated superior performance across various graph learning tasks but face significant computational challenges when applied to large-scale graphs. One effective approach to mitigate these challenges is graph sparsification, which involves removing non-essential edges to reduce computational overhead. However, previous graph sparsification methods often rely on a single global sparsity setting and uniform pruning criteria, failing to provide customized sparsification schemes for each node's complex local context.In this paper, we introduce Mixture-of-Graphs (MoG), leveraging the concept of Mixture-of-Experts (MoE), to dynamically select tailored pruning solutions for each node. Specifically, MoG incorporates multiple sparsifier experts, each characterized by unique sparsity levels and pruning criteria, and selects the appropriate experts for each node. Subsequently, MoG performs a mixture of the sparse graphs produced by different experts on the Grassmann manifold to derive an optimal sparse graph. One notable property of MoG is its entirely local nature, as it depends on the specific circumstances of each individual node. Extensive experiments on four large-scale OGB datasets and two superpixel datasets, equipped with five GNN backbones, demonstrate that MoG (I) identifies subgraphs at higher sparsity levels ($8.67\\%\sim 50.85\\%$), with performance equal to or better than the dense graph, (II) achieves $1.47-2.62\times$ speedup in GNN inference with negligible performance drop, and (III) boosts ``top-student'' GNN performance ($1.02\\%\uparrow$ on RevGNN+\textsc{ogbn-proteins} and $1.74\\%\\uparrow$ on DeeperGCN+\textsc{ogbg-ppa}). The source code is available at \url{https://github.com/yanweiyue/MoG}.

Probabilistic Methods · Everything Else

Domokos M. Kelen, Ádám Jung, Péter Kersch, Andras Benczur

Quantifying uncertainty is an essential part of predictive modeling, especially in the context of high-stakes decision-making. While classification output includes data uncertainty by design in the form of class probabilities, the regression task generally aims only to predict the expected value of the target variable. Probabilistic extensions often assume parametric distributions around the expected value, optimizing the likelihood over the resulting explicit densities. However, using parametric distributions can limit practical applicability, making it difficult for models to capture skewed, multi-modal, or otherwise complex distributions. In this paper, we propose optimizing a novel nondeterministic neural network regression architecture for loss functions derived from a sample-based approximation of the continuous ranked probability score (CRPS), enabling a truly distribution-free approach by learning to sample from the target's aleatoric distribution, rather than predicting explicit densities. Our approach allows the model to learn well-calibrated, arbitrary uni- and multivariate output distributions. We evaluate the method on a variety of synthetic and real-world tasks, including uni- and multivariate problems, function inverse approximation, and standard regression uncertainty benchmarks. Finally, we make all experiment code publicly available.

Deep Learning · Graph Neural Networks

Jie Chen

Preconditioning is at the heart of iterative solutions of large, sparse linear systems of equations in scientific disciplines. Several algebraic approaches, which access no information beyond the matrix itself, are widely studied and used, but ill-conditioned matrices remain very challenging. We take a machine learning approach and propose using graph neural networks as a general-purpose preconditioner. They show attractive performance for many problems and can be used when the mainstream preconditioners perform poorly. Empirical evaluation on over 800 matrices suggests that the construction time of these graph neural preconditioners (GNPs) is more predictable and can be much shorter than that of other widely used ones, such as ILU and AMG, while the execution time is faster than using a Krylov method as the preconditioner, such as in inner-outer GMRES. GNPs have a strong potential for solving large-scale, challenging algebraic problems arising from not only partial differential equations, but also economics, statistics, graph, and optimization, to name a few.

Deep Learning · Graph Neural Networks

Seong Ho Pahng, Sahand Hormoz

Graph Neural Networks (GNNs) traditionally employ a message-passing mechanism that resembles diffusion over undirected graphs, which often leads to homogenization of node features and reduced discriminative power in tasks such as node classification. Our key insight for addressing this limitation is to assign fuzzy edge directions---that can vary continuously from node $i$ pointing to node $j$ to vice versa---to the edges of a graph so that features can preferentially flow in one direction between nodes to enable long-range information transmission across the graph. We also introduce a novel complex-valued Laplacian for directed graphs with fuzzy edges where the real and imaginary parts represent information flow in opposite directions. Using this Laplacian, we propose a general framework, called Continuous Edge Direction (CoED) GNN, for learning on graphs with fuzzy edges and prove its expressivity limits using a generalization of the Weisfeiler-Leman (WL) graph isomorphism test for directed graphs with fuzzy edges. Our architecture aggregates neighbor features scaled by the learned edge directions and processes the aggregated messages from in-neighbors and out-neighbors separately alongside the self-features of the nodes. Since continuous edge directions are differentiable, they can be learned jointly with the GNN weights via gradient-based optimization. CoED GNN is particularly well-suited for graph ensemble data where the graph structure remains fixed but multiple realizations of node features are available, such as in gene regulatory networks, web connectivity graphs, and power grids. We demonstrate through extensive experiments on both synthetic and real graph ensemble datasets that learning continuous edge directions significantly improves performance both for undirected and directed graphs compared with existing methods.

Theory · Deep Learning

Alireza Mousavi-Hosseini, Adel Javanmard, Murat A Erdogdu

Recently, there have been numerous studies on feature learning with neural networks, specifically on learning single- and multi-index models where the target is a function of a low-dimensional projection of the input. Prior works have shown that in high dimensions, the majority of the compute and data resources are spent on recovering the low-dimensional projection; once this subspace is recovered, the remainder of the target can be learned independently of the ambient dimension. However, implications of feature learning in adversarial settings remain unexplored. In this work, we take the first steps towards understanding adversarially robust feature learning with neural networks. Specifically, we prove that the hidden directions of a multi-index model offer a Bayes optimal low-dimensional projection for robustness against $\ell_2$-bounded adversarial perturbations under the squared loss, assuming that the multi-index coordinates are statistically independent from the rest of the coordinates. Therefore, robust learning can be achieved by first performing standard feature learning, then robustly tuning a linear readout layer on top of the standard representations. In particular, we show that adversarially robust learning is just as easy as standard learning. Specifically, the additional number of samples needed to robustly learn multi-index models when compared to standard learning, does not depend on dimensionality.

Social Aspects · Accountability, Transparency and Interpretability

Thomas Norrenbrock, Timo Kaiser, Sovan Biswas, Ramesh Manuvinakurike, Bodo Rosenhahn

Understanding the classifications of deep neural networks, e.g. used in safety-critical situations, is becoming increasingly important. While recent models can locally explain a single decision, to provide a faithful global explanation about an accurate model’s general behavior is a more challenging open task. Towards that goal, we introduce the Quadratic Programming Enhanced Model (QPM), which learns globally interpretable class representations. QPM represents every class with a binary assignment of very few, typically 5, features, that are also assigned to other classes, ensuring easily comparable contrastive class representations. This compact binary assignment is found using discrete optimization based on predefined similarity measures and interpretability constraints. The resulting optimal assignment is used to fine-tune the diverse features, so that each of them becomes the shared general concept between the assigned classes. Extensive evaluations show that QPM delivers unprecedented global interpretability across small and large-scale datasets while setting the state of the art for the accuracy of interpretable models.

Haitz Sáez de Ocáriz Borde, Anastasis Kratsios, Marc T Law, Xiaowen Dong, Michael Bronstein

We propose a class of trainable deep learning-based geometries called Neural SpaceTimes (NSTs), which can universally represent nodes in weighted Directed Acyclic Graphs (DAGs) as events in a spacetime manifold. While most works in the literature focus on undirected graph representation learning or causality embedding separately, our differentiable geometry can encode both graph edge weights in its spatial dimensions and causality in the form of edge directionality in its temporal dimensions. We use a product manifold that combines a quasi-metric (for space) and a partial order (for time). NSTs are implemented as three neural networks trained in an end-to-end manner: an embedding network, which learns to optimize the location of nodes as events in the spacetime manifold, and two other networks that optimize the space and time geometries in parallel, which we call a neural (quasi-)metric and a neural partial order, respectively. The latter two networks leverage recent ideas at the intersection of fractal geometry and deep learning to shape the geometry of the representation space in a data-driven fashion, unlike other works in the literature that use fixed spacetime manifolds such as Minkowski space or De Sitter space to embed DAGs. Our main theoretical guarantee is a universal embedding theorem, showing that any $k$-point DAG can be embedded into an NST with $1+\mathcal{O}(\log(k))$ distortion while exactly preserving its causal structure. The total number of parameters defining the NST is sub-cubic in $k$ and linear in the width of the DAG. If the DAG has a planar Hasse diagram, this is improved to $\mathcal{O}(\log(k) + 2)$ spatial and 2 temporal dimensions. We validate our framework computationally with synthetic weighted DAGs and real-world network embeddings; in both cases, the NSTs achieve lower embedding distortions than their counterparts using fixed spacetime geometries.

Optimization · Learning for Optimization

Vincenzo Di Vito Francesco, Mostafa Mohammadian, Kyri Baker, Ferdinando Fioretto

Differential equations (DE) constrained optimization plays a critical role in numerous scientific and engineering fields, including energy systems, aerospace engineering, ecology, and finance, where optimal configurations or control strategies must be determined for systems governed by ordinary or stochastic differential equations. Despite its significance, the computational challenges associated with these problems have limited their practical use. To address these limitations, this paper introduces a learning-based approach to DE-constrained optimization that combines techniques from proxy optimization \citep{kotary2021end} and neural differential equations \citep{chen2019neural}. The proposed approach uses a dual-network architecture, with one approximating the control strategies, focusing on steady-state constraints, and another solving the associated DEs. This combination enables the approximation of optimal strategies while accounting for dynamic constraints in near real-time.Experiments across problems in energy optimization and finance modeling show that this method provides full compliance with dynamic constraints and it produces results up to 25 times more precise than other methods which do not explicitly model the system's dynamic equations.

Reinforcement Learning · Deep RL

Jiashun Liu, Johan S Obando Ceron, Aaron Courville, Ling Pan

The loss of plasticity in learning agents, analogous to the solidification of neural pathways in biological brains, significantly impedes learning and adaptation in reinforcement learning due to its non-stationary nature. To address this fundamental challenge, we propose a novel approach, *Neuroplastic Expansion* (NE), inspired by cortical expansion in cognitive science. NE maintains learnability and adaptability throughout the entire training process by dynamically growing the network from a smaller initial size to its full dimension. Our method is designed with three key components: (1) elastic neuron generation based on potential gradients, (2) dormant neuron pruning to optimize network expressivity, and (3) neuron consolidation via experience review to strike a balance in the plasticity-stability dilemma. Extensive experiments demonstrate that NE effectively mitigates plasticity loss and outperforms state-of-the-art methods across various tasks in MuJoCo and DeepMind Control Suite environments. NE enables more adaptive learning in complex, dynamic environments, which represents a crucial step towards transitioning deep reinforcement learning from static, one-time training paradigms to more flexible, continually adapting models.

Social Aspects · Privacy-preserving Statistics and Machine Learning

Ozan Özdenizci, Elmar Rueckert, Robert Legenstein

Lifelong learning algorithms enable models to incrementally acquire new knowledge without forgetting previously learned information. Contrarily, the field of machine unlearning focuses on explicitly forgetting certain previous knowledge from pretrained models when requested, in order to comply with data privacy regulations on the right-to-be-forgotten. Enabling efficient lifelong learning with the capability to selectively unlearn sensitive information from models presents a critical and largely unaddressed challenge with contradicting objectives. We address this problem from the perspective of simultaneously preventing catastrophic forgetting and allowing forward knowledge transfer during task-incremental learning, while ensuring exact task unlearning and minimizing memory requirements, based on a single neural network model to be adapted. Our proposed solution, privacy-aware lifelong learning (PALL), involves optimization of task-specific sparse subnetworks with parameter sharing within a single architecture. We additionally utilize an episodic memory rehearsal mechanism to facilitate exact unlearning without performance degradations. We empirically demonstrate the scalability of PALL across various architectures in image classification, and provide a state-of-the-art solution that uniquely integrates lifelong learning and privacy-aware unlearning mechanisms for responsible AI applications.

Deep Learning · Theory

Marie-Charlotte Brandenburg, Moritz Grillo, Christoph Hertrich

In this paper we contribute to the frequently studied question of how to decompose a continuous piecewise linear (CPWL) function into a difference of two convex CPWL functions. Every CPWL function has infinitely many such decompositions, but for applications in optimization and neural network theory, it is crucial to find decompositions with as few linear pieces as possible. This is a highly challenging problem, as we further demonstrate by disproving a recently proposed approach by Tran and Wang [Minimal representations of tropical rational functions. Algebraic Statistics, 15(1):27–59, 2024]. To make the problem more tractable, we propose to fix an underlying polyhedral complex determining the possible locus of nonlinearity. Under this assumption, we prove that the set of decompositions forms a polyhedron that arises as intersection of two translated cones. We prove that irreducible decompositions correspond to the bounded faces of this polyhedron and minimal solutions must be vertices. We then identify cases with a unique minimal decomposition, and illustrate how our insights have consequences in the theory of submodular functions. Finally, we improve upon previous constructions of neural networks for a given convex CPWL function and apply our framework to obtain results in the nonconvex case.

Applications · Neuroscience, Cognitive Science

Will Dorrell, Kyle Hsu, Luke Hollingsworth, Jin Hwa Lee, Jiajun Wu, Chelsea Finn, Peter Latham, Timothy Behrens, James Whittington

Why do biological and artificial neurons sometimes modularise, each encoding a single meaningful variable, and sometimes entangle their representation of many variables? In this work, we develop a theory of when biologically inspired networks---those that are nonnegative and energy efficient---modularise their representation of source variables (sources). We derive necessary and sufficient conditions on a sample of sources that determine whether the neurons in an optimal biologically-inspired linear autoencoder modularise. Our theory applies to any dataset, extending far beyond the case of statistical independence studied in previous work. Rather we show that sources modularise if their support is ``sufficiently spread''. From this theory, we extract and validate predictions in a variety of empirical studies on how data distribution affects modularisation in nonlinear feedforward and recurrent neural networks trained on supervised and unsupervised tasks. Furthermore, we apply these ideas to neuroscience data, showing that range independence can be used to understand the mixing or modularising of spatial and reward information in entorhinal recordings in seemingly conflicting experiments. Further, we use these results to suggest alternate origins of mixed-selectivity, beyond the predominant theory of flexible nonlinear classification. In sum, our theory prescribes precise conditions on when neural activities modularise, providing tools for inducing and elucidating modular representations in brains and machines.

Applications · Chemistry and Drug Discovery

Yuchen Shen, Chenhao Zhang, Sijie Fu, Chenghui Zhou, Newell Washburn, Barnabás Póczos

Recent advances in diffusion models have shown remarkable potential in the conditional generation of novel molecules. These models can be guided in two ways: (i) explicitly, through additional features representing the condition, or (ii) implicitly, using a property predictor. However, training property predictors or conditional diffusion models requires an abundance of labeled data and is inherently challenging in real-world applications. We propose a novel approach that attenuates the limitations of acquiring large labeled datasets by leveraging domain knowledge from quantum chemistry as a non-differentiable oracle to guide an unconditional diffusion model. Instead of relying on neural networks, the oracle provides accurate guidance in the form of estimated gradients, allowing the diffusion process to sample from a conditional distribution specified by quantum chemistry. We show that this results in more precise conditional generation of novel and stable molecular structures. Our experiments demonstrate that our method: (1) significantly reduces atomic forces, enhancing the validity of generated molecules when used for stability optimization; (2) is compatible with both explicit and implicit guidance in diffusion models, enabling joint optimization of molecular properties and stability; and (3) generalizes effectively to molecular optimization tasks beyond stability optimization. Our implementation is available at https://github.com/A-Chicharito-S/ChemGuide.

Theory · Online Learning and Bandits

Arun Verma, Zhongxiang Dai, Xiaoqiang Lin, Patrick Jaillet, Bryan Kian Hsiang Low

Contextual dueling bandit is used to model the bandit problems, where a learner's goal is to find the best arm for a given context using observed noisy human preference feedback over the selected arms for the past contexts. However, existing algorithms assume the reward function is linear, which can be complex and non-linear in many real-life applications like online recommendations or ranking web search results. To overcome this challenge, we use a neural network to estimate the reward function using preference feedback for the previously selected arms. We propose upper confidence bound- and Thompson sampling-based algorithms with sub-linear regret guarantees that efficiently select arms in each round. We also extend our theoretical results to contextual bandit problems with binary feedback, which is in itself a non-trivial contribution. Experimental results on the problem instances derived from synthetic datasets corroborate our theoretical results.