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Optimization · Neural Network Optimization

Yong Liu, Zirui Zhu, Chaoyu Gong, Minhao Cheng, Cho-Jui Hsieh, Yang You

While fine-tuning large language models (LLMs) for specific tasks often yields impressive results, it comes at the cost of memory inefficiency due to back-propagation in gradient-based training. Memory-efficient Zeroth-order (MeZO) optimizers, recently proposed to address this issue, only require forward passes during training, making them more memory-friendly. However, compared with exact gradients, ZO-based gradients usually exhibit an estimation error, which can significantly hurt the optimization process, leading to slower convergence and suboptimal solutions. In addition, we find that the estimation error will hurt more when adding to large weights instead of small weights. Based on this observation, this paper introduces Sparse MeZO, a novel memory-efficient zeroth-order optimization approach that applies ZO only to a carefully chosen subset of parameters. We propose a simple yet effective parameter selection scheme that yields significant performance gains with Sparse-MeZO. Additionally, we develop a memory-optimized implementation for sparse masking, ensuring the algorithm requires only inference-level memory consumption, allowing Sparse-MeZO to fine-tune LLaMA-30b on a single A100 GPU. Experimental results illustrate that Sparse-MeZO consistently improves both performance and convergence speed over MeZO without any overhead. For example, it achieves a 9% absolute accuracy improvement and 3.5x speedup over MeZO on the RTE task.

Social Aspects · Privacy-preserving Statistics and Machine Learning

Rob Romijnders, Antti Koskela

The hidden state threat model of differential privacy (DP) assumes that the adversary has access only to the final trained machine learning (ML) model, without seeing intermediate states during training. However, the current privacy analyses under this model are restricted to convex optimization problems, reducing their applicability to multi-layer neural networks, which are essential in modern deep learning applications. Notably, the most successful applications of the hidden state privacy analyses in classification tasks have only been for logistic regression models. We demonstrate that it is possible to privately train convex problems with privacy-utility trade-offs comparable to those of 2-layer ReLU networks trained with DP stochastic gradient descent (DP-SGD). This is achieved through a stochastic approximation of a dual formulation of the ReLU minimization problem, resulting in a strongly convex problem. This enables the use of existing hidden state privacy analyses and provides accurate privacy bounds also for the noisy cyclic mini-batch gradient descent (NoisyCGD) method with fixed disjoint mini-batches. Empirical results on benchmark classification tasks demonstrate that NoisyCGD can achieve privacy-utility trade-offs on par with DP-SGD applied to 2-layer ReLU networks.

Social Aspects · Accountability, Transparency and Interpretability

Sai Sumedh R. Hindupur, Ekdeep S Lubana, Thomas Fel, Demba Ba

Sparse Autoencoders (SAEs) are widely used to interpret neural networks by identifying meaningful concepts from their representations. However, do SAEs truly uncover all concepts a model relies on, or are they inherently biased toward certain kinds of concepts? We introduce a unified framework that recasts SAEs as solutions to a bilevel optimization problem, revealing a fundamental challenge: each SAE imposes structural assumptions about how concepts are encoded in model representations, which in turn shapes what it can and cannot detect. This means different SAEs are not interchangeable -- switching architectures can expose entirely new concepts or obscure existing ones. To systematically probe this effect, we evaluate SAEs across a spectrum of settings: from controlled toy models that isolate key variables, to semi-synthetic experiments on real model activations and finally to large-scale, naturalistic datasets. Across this progression, we examine two fundamental properties that real-world concepts often exhibit: heterogeneity in intrinsic dimensionality (some concepts are inherently low-dimensional, others are not) and nonlinear separability. We show that SAEs fail to recover concepts when these properties are ignored, and we design a new SAE that explicitly incorporates both, enabling the discovery of previously hidden concepts and reinforcing our theoretical insights. Our findings challenge the idea of a universal SAE and underscores the need for architecture-specific choices in model interpretability. Overall, we argue an SAE does not just reveal concepts -- it determines what can be seen at all.

Deep Learning · Everything Else

Malyaban Bal, Brian Matejek, Susmit Jha, Adam Cobb

Video Temporal Grounding (VTG) aims to retrieve precise temporal segments in a video conditioned on natural language queries. Unlike conventional neural frameworks that rely heavily on computationally expensive dense matrix multiplications, Spiking Neural Networks (SNNs)—previously underexplored in this domain—offer a unique opportunity to tackle VTG tasks through bio-plausible spike-based communication and an event-driven accumulation-based computational paradigm. We introduce SpikingVTG, a multi-modal spiking detection transformer, designed to harness the computational simplicity and sparsity of SNNs for VTG tasks. Leveraging the temporal dynamics of SNNs, our model introduces a Saliency Feedback Gating (SFG) mechanism that assigns dynamic saliency scores to video clips and applies multiplicative gating to highlight relevant clips while suppressing less informative ones. SFG enhances performance and reduces computational overhead by minimizing neural activity. We analyze the layer-wise convergence dynamics of SFG-enabled model and apply implicit differentiation at equilibrium to enable efficient, BPTT-free training. To improve generalization and maximize performance, we enable knowledge transfer by optimizing a Cos-L2 representation matching loss that aligns the layer-wise representation and attention maps of a non-spiking teacher with those of our student SpikingVTG. Additionally, we present Normalization-Free (NF)-SpikingVTG, which eliminates non-local operations like softmax and layer normalization, and an extremely quantized 1-bit (NF)-SpikingVTG variant for potential deployment on edge devices. Our models achieve competitive results on QVHighlights, Charades-STA, TACoS, and YouTube Highlights, establishing a strong baseline for multi-modal spiking VTG solutions.

General Machine Learning · Everything Else

Reza Ramezanpour, Victor Manuel Tenorio Gomez, Antonio G. Marques, Ashutosh Sabharwal, Santiago Segarra

Graphons, as limit objects of dense graph sequences, play a central role in the statistical analysis of network data. However, existing graphon estimation methods often struggle with scalability to large networks and resolution-independent approximation, due to their reliance on estimating latent variables or costly metrics such as the Gromov-Wasserstein distance. In this work, we propose a novel, scalable graphon estimator that directly recovers the graphon via moment matching, leveraging implicit neural representations (INRs). Our approach avoids latent variable modeling by training an INR--mapping coordinates to graphon values--to match empirical subgraph counts (i.e., moments) from observed graphs. This direct estimation mechanism yields a polynomial-time solution and crucially sidesteps the combinatorial complexity of Gromov-Wasserstein optimization. Building on foundational results, we establish a theoretical guarantee: when the observed subgraph motifs sufficiently represent those of the true graphon (a condition met with sufficiently large or numerous graph samples), the estimated graphon achieves a provable upper bound in cut distance from the ground truth. Additionally, we introduce MomentMixup, a data augmentation technique that performs mixup in the moment space to enhance graphon-based learning. Our graphon estimation method achieves strong empirical performance--demonstrating high accuracy on small graphs and superior computational efficiency on large graphs--outperforming state-of-the-art scalable estimators in 75\% of benchmark settings and matching them in the remaining cases. Furthermore, MomentMixup demonstrated improved graph classification accuracy on the majority of our benchmarks.

Theory · Learning Theory

Hugo Ninou, Jonathan Kadmon, N Alex Cayco Gajic

Gradient-based algorithms are a cornerstone of artificial neural network training, yet it remains unclear whether biological neural networks use similar gradient-based strategies during learning. Experiments often discover a diversity of synaptic plasticity rules, but whether these amount to an approximation to gradient descent is unclear. Here we investigate a previously overlooked possibility: that learning dynamics may include fundamentally non-gradient "curl"-like components while still being able to effectively optimize a loss function. Curl terms naturally emerge in networks with excitatory-inhibitory connectivity or Hebbian/anti-Hebbian plasticity, resulting in learning dynamics that cannot be framed as gradient descent on any objective. To investigate the impact of these curl terms, we analyze feedforward networks within an analytically tractable student-teacher framework, systematically introducing non-gradient dynamics through rule-flipped neurons. Small curl terms preserve the stability of the original solution manifold, resulting in learning dynamics similar to gradient descent. Beyond a critical value, strong curl terms destabilize the solution manifold. Depending on the network architecture, this loss of stability can lead to chaotic learning dynamics that destroy performance. In other cases, the curl terms can counterintuitively speed up learning compared to gradient descent by allowing the weight dynamics to escape saddles by temporarily ascending the loss. Our results identify specific architectures capable of supporting robust learning via diverse learning rules, providing an important counterpoint to normative theories of gradient-based learning in neural networks.

Deep Learning · Theory

Viet-Hoang Tran, Van Hoan Trinh, Khanh-Vinh Bui, Tan Nguyen

Linear Mode Connectivity (LMC) is a notable phenomenon in the loss landscapes of neural networks, wherein independently trained models have been observed to be connected—up to permutation symmetries—by linear paths in parameter space along which the loss remains consistently low. This observation challenges classical views of non-convex optimization and has implications for model ensembling, generalization, and our understanding of neural loss geometry. Inspired by recent studies on LMC in standard neural networks, we systematically investigate this phenomenon within Mixture-of-Experts (MoE) architectures—a class of models known for their scalability and computational efficiency, which combine traditional neural networks—referred to as experts—through a learnable gating mechanism. We begin by conducting a comprehensive analysis of both dense and sparse gating regimes, demonstrating that the symmetries inherent to MoE architectures are fully characterized by permutations acting on both the expert components and the gating function. Building on these foundational findings, we propose a matching algorithm that enables alignment between independently trained MoEs, thereby facilitating the discovery of LMC. Finally, we empirically validate the presence of LMC using our proposed algorithm across diverse MoE configurations—including dense, sparse, and shared-expert variants—under a wide range of model settings and datasets of varying scales and modalities. Our results confirm the existence of LMC in MoE architectures and offer fundamental insights into the functional landscape and optimization dynamics of deep learning models.

Optimization · Discrete Optimization

Yuanyao Chen, Rongsheng Chen, Fu Luo, Zhenkun Wang

Neural Combinatorial Optimization (NCO) has emerged as a promising learning-based paradigm for addressing Vehicle Routing Problems (VRPs) by minimizing the need for extensive manual engineering. While existing NCO methods, trained on small-scale instances (e.g., 100 nodes), have demonstrated considerable success on problems of similar scale, their performance significantly degrades when applied to large-scale scenarios. This degradation arises from the distributional shift between training and testing data, rendering policies learned on small instances ineffective for larger problems. To overcome this limitation, we introduce a novel learning framework driven by Large Language Models (LLMs). This framework learns a projection between the training and testing distributions, which is then deployed to enhance the scalability of the NCO model. Notably, unlike prevailing techniques that necessitate joint training with the neural network, our approach operates exclusively during the inference phase, obviating the need for model retraining. Extensive experiments demonstrate that our method enables a backbone model (trained on 100-node instances) to achieve superior performance on large-scale Traveling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) of up to 100K nodes from diverse distributions. The source code can be found in https://github.com/CIAM-Group/TTPL.

Optimization · Neural Network Optimization

Zixuan XIa, Aram Davtyan, Paolo Favaro

We propose KOALA++, a scalable Kalman-based optimization algorithm that explicitly models structured gradient uncertainty in neural network training. Unlike second-order methods, which rely on expensive second order gradient calculation, our method directly estimates the parameter covariance matrix by recursively updating compact gradient covariance products. This design improves upon the original KOALA framework that assumed diagonal covariance by implicitly capturing richer uncertainty structure without storing the full covariance matrix and avoiding large matrix inversions. Across diverse tasks, including image classification and language modeling, KOALA++ achieves accuracy on par or better than state-of-the-art second-order optimizers while maintaining the efficiency of first-order methods.

Deep Learning · Theory

Liu Ziyin, Yizhou Xu, Isaac Chuang

With the rapid discovery of emergent phenomena in deep learning and large language models, understanding their cause has become an urgent need. Here, we propose a rigorous entropic-force theory for understanding the learning dynamics of neural networks trained with stochastic gradient descent (SGD) and its variants. Building on the theory of parameter symmetries and an entropic loss landscape, we show that representation learning is crucially governed by emergent entropic forces arising from stochasticity and discrete-time updates. These forces systematically break continuous parameter symmetries and preserve discrete ones, leading to a series of gradient balance phenomena that resemble the equipartition property of thermal systems. These phenomena, in turn, (a) explain the universal alignment of neural representations between AI models and lead to a proof of the Platonic Representation Hypothesis, and (b) reconcile the seemingly contradictory observations of sharpness- and flatness-seeking behavior of deep learning optimization. Our theory and experiments demonstrate that a combination of entropic forces and symmetry breaking is key to understanding emergent phenomena in deep learning.

Social Aspects · Trustworthy Machine Learning

Nikolaos Spanos, Maria Lymperaiou, Giorgos Filandrianos, Konstantinos Thomas, Athanasios Voulodimos, Giorgos Stamou

Recent black-box counterfactual generation frameworks fail to take into account the semantic content of the proposed edits, while relying heavily on training to guide the generation process. We propose a novel, plug-and-play black-box counterfactual generation framework, which suggests step-by-step edits based on theoretical guarantees of optimal edits to produce human-level counterfactual explanations with zero training. Our framework utilizes a pre-trained image editing diffusion model, and operates without access to the internals of the classifier, leading to an explainable counterfactual generation process. Throughout our experimentation, we showcase the explanatory gap between human reasoning and neural model behavior by utilizing both Convolutional Neural Network (CNN), Vision Transformer (ViT) and Large Vision Language Model (LVLM) classifiers, substantiated through a comprehensive human evaluation.

Deep Learning · Graph Neural Networks

Francesco Cozzi, Marco Pangallo, Alan Perotti, André Panisson, Corrado Monti

Agent-Based Models (ABMs) are powerful tools for studying emergent properties in complex systems. In ABMs, agent behaviors are governed by local interactions and stochastic rules. However, these rules are ad hoc and, in general, non-differentiable, limiting the use of gradient-based methods for optimization, and thus integration with real-world data. We propose a novel framework to learn a differentiable surrogate of any ABM by observing its generated data. Our method combines diffusion models to capture behavioral stochasticity and graph neural networks to model agent interactions. Distinct from prior surrogate approaches, our method introduces a fundamental shift: rather than approximating system-level outputs, it models individual agent behavior directly, preserving the decentralized, bottom-up dynamics that define ABMs. We validate our approach on two ABMs (Schelling's segregation model and a Predator-Prey ecosystem) showing that it replicates individual-level patterns and accurately forecasts emergent dynamics beyond training. Our results demonstrate the potential of combining diffusion models and graph learning for data-driven ABM simulation.

Deep Learning · Theory

Alexander Theus, Alessandro Cabodi, Sotiris Anagnostidis, Antonio Orvieto, Sidak Pal Singh, Valentina Boeva

Understanding the geometry of neural network loss landscapes is a central question in deep learning, with implications for generalization and optimization. A striking phenomenon is $\textit{linear mode connectivity}$ (LMC), where independently trained models can be connected by low- or zero-barrier paths, despite appearing to lie in separate loss basins. However, this is often obscured by symmetries in parameter space—such as neuron permutations—which make functionally equivalent models appear dissimilar. Prior work has predominantly focused on neuron reordering through permutations, but such approaches are limited in scope and fail to capture the richer symmetries exhibited by modern architectures such as Transformers. In this work, we introduce a unified framework that captures four symmetry classes—permutations, semi-permutations, orthogonal transformations, and general invertible maps—broadening the set of valid reparameterizations and subsuming many previous approaches as special cases. Crucially, this generalization enables, for the first time, the discovery of low- and zero-barrier linear interpolation paths between independently trained Vision Transformers and GPT-2 models. Furthermore, our framework extends beyond pairwise alignment, to multi-model and width-heterogeneous settings, enabling alignment across architectures of different sizes. These results reveal deeper structure in the loss landscape and underscore the importance of symmetry-aware analysis for understanding model space geometry.

Deep Learning · Everything Else

Flavio Martinelli, Alexander van Meegen, Berfin Simsek, Wulfram Gerstner, Johanni Brea

The loss landscapes of neural networks contain minima and saddle points that may be connected in flat regions or appear in isolation. We identify and characterize a special structure in the loss landscape: channels along which the loss decreases extremely slowly, while the output weights of at least two neurons, $a_i$ and $a_j$, diverge to $\pm$infinity, and their input weight vectors, $\mathbf{w_i}$ and $\mathbf{w_j}$, become equal to each other. At convergence, the two neurons implement a gated linear unit: $a_i\sigma(\mathbf{w_i} \cdot \mathbf{x}) + a_j\sigma(\mathbf{w_j} \cdot \mathbf{x}) \rightarrow c \sigma(\mathbf{w} \cdot \mathbf{x}) + (\mathbf{v} \cdot \mathbf{x}) \sigma'(\mathbf{w} \cdot \mathbf{x})$. Geometrically, these channels to infinity are asymptotically parallel to symmetry-induced lines of critical points. Gradient flow solvers, and related optimization methods like SGD or ADAM, reach the channels with high probability in diverse regression settings, but without careful inspection they look like flat local minima with finite parameter values. Our characterization provides a comprehensive picture of this quasi-flat region in terms of gradient dynamics, geometry, and functional interpretation. The emergence of gated linear units at the end of the channels highlights a surprising aspect of the computational capabilities of fully connected layers.

Optimization · Neural Network Optimization

Tianyue Zhang, Lucas Maes, Alan Milligan, Alexia Jolicoeur-Martineau, Ioannis Mitliagkas, Damien Scieur, Simon Lacoste-Julien, Charles Guille-Escuret

Despite its widespread adoption, Adam's advantage over Stochastic Gradient Descent (SGD) lacks a comprehensive theoretical explanation. This paper investigates Adam's sensitivity to rotations of the parameter space. We observe that Adam's performance in training transformers degrades under random rotations of the parameter space, indicating a crucial sensitivity to the choice of basis in practice. This reveals that conventional rotation-invariant assumptions are insufficient to capture Adam's advantages theoretically. To better understand the rotation-dependent properties that benefit Adam, we also identify structured rotations that preserve or even enhance its empirical performance. We then examine the rotation-dependent assumptions in the literature and find that they fall short in explaining Adam's behavior across various rotation types. In contrast, we verify the orthogonality of the update as a promising indicator of Adam’s basis sensitivity, suggesting it may be the key quantity for developing rotation-dependent theoretical frameworks that better explain its empirical success.

Applications · Neuroscience, Cognitive Science

Mete Erdogan, Cengiz Pehlevan, Alper Erdogan

We introduce *Error Broadcast and Decorrelation* (EBD), a novel learning framework for neural networks that addresses credit assignment by directly broadcasting output errors to individual layers, circumventing weight transport of backpropagation. EBD is rigorously grounded in the stochastic orthogonality property of Minimum Mean Square Error estimators. This fundamental principle states that the error of an optimal estimator is orthogonal to functions of the input. Guided by this insight, EBD defines layerwise loss functions that directly penalize correlations between layer activations and output errors, thereby establishing a principled foundation for error broadcasting. This theoretically sound mechanism naturally leads to the experimentally observed three-factor learning rule and integrates with biologically plausible frameworks to enhance performance and plausibility. Numerical experiments demonstrate EBD’s competitive or better performance against other error-broadcast methods on benchmark datasets. Our findings establish EBD as an efficient, biologically plausible, and principled alternative for neural network training.

Theory · Learning Theory

WEI-KAI CHANG, Rajiv Khanna

Understanding the dynamics of optimization algorithms in deep learning has become increasingly critical, especially as models grow in scale and complexity. Despite the empirical success of stochastic gradient descent (SGD) and its variants in finding solutions that generalize well, the precise mechanisms underlying this generalization remain poorly understood. A particularly intriguing aspect of this phenomenon is the bias of optimization algorithms towards certain types of minima—often flatter or simpler—especially in overparameterized regimes. While prior works have associated flatness of the loss landscape with better generalization, tools to mechanistically connect data, optimization algorithms, and the nature of the resulting minima are still limited. For instance, methods like Sharpness-Aware Minimization (SAM) have shown practical gains by explicitly promoting flatness, but lack a unified theoretical framework explaining their influence across different data structures and model architectures. In this work, we introduce a comprehensive linear stability analysis framework to dissect the behavior of optimization algorithms—SGD, random perturbations, and SAM—in neural networks, focusing particularly on two-layer ReLU models. Our approach is built upon a novel coherence measure that captures the interaction between data geometry and gradient similarity, providing new insights into why and how certain solutions are favored.

Optimization · Large Scale, Parallel and Distributed

Alexander Long, Chamin Hewa Koneputugodage, Thalaiyasingam Ajanthan, Yan Zuo, Gil Avraham, Violetta Shevchenko, Hadi Mohaghegh Dolatabadi, Sameera Ramasinghe

We consider a decentralized setup in which the participants collaboratively train and serve a large neural network, and where each participant only processes a subset of the model. In this setup, we explore the possibility of unmaterializable weights, where a full weight set is never available to any one participant. We introduce Unextractable Protocol Models (UPMs): a training and inference framework that leverages the sharded model setup to ensure model shards (i.e.,, subsets) held by participants are incompatible at different time steps. UPMs periodically inject time-varying, random, invertible transforms at participant boundaries; preserving the overall network function yet rendering cross-time assemblies incoherent. On Qwen-2.5-0.5B and Llama-3.2-1B, 10 000 transforms leave FP32 perplexity unchanged ($\Delta$PPL$< 0.01$; Jensen–Shannon drift $<4 \times 10^{-5}$), and we show how to control growth for lower precision datatypes. Applying a transform every 30s adds 3% latency, 0.1% bandwidth, and 10% GPU-memory overhead at inference, while training overhead falls to 1.6% time and < 1% memory. We consider several attacks, showing that the requirements of direct attacks are impractical and easy to defend against, and that gradient-based fine-tuning of stitched partitions consumes $\geq 60\%$ of the tokens required to train from scratch. By enabling models to be collaboratively trained yet not extracted, UPMs make it practical to embed programmatic incentive mechanisms in community-driven decentralized training.

Optimization

Roman Jacome, Romario Gualdrón-Hurtado, León Suárez-Rodríguez, Henry Arguello

Imaging inverse problems aim to recover high-dimensional signals from undersampled, noisy measurements, a fundamentally ill-posed task with infinite solutions in the null-space of the sensing operator. To resolve this ambiguity, prior information is typically incorporated through handcrafted regularizers or learned models that constrain the solution space. However, these priors typically ignore the task-specific structure of that null-space. In this work, we propose Non-Linear Projections of the Null-Space (NPN), a novel class of regularization that, instead of enforcing structural constraints in the image domain, promotes solutions that lie in a low-dimensional projection of the sensing matrix's null-space with a neural network. Our approach has two key advantages: (1) Interpretability: by focusing on the structure of the null-space, we design sensing-matrix-specific priors that capture information orthogonal to the signal components that are fundamentally blind to the sensing process. (2) Flexibility: NPN is adaptable to various inverse problems, compatible with existing reconstruction frameworks, and complementary to conventional image-domain priors. We provide theoretical guarantees on convergence and reconstruction accuracy when used within plug-and-play methods. Empirical results across diverse sensing matrices demonstrate that NPN priors consistently enhance reconstruction fidelity in various imaging inverse problems, such as compressive sensing, deblurring, super-resolution, computed tomography, and magnetic resonance imaging, with plug-and-play methods, unrolling networks, deep image prior, and diffusion models.

Computer Vision · Vision Models & Multimodal

Jingjun Yang, Liangwei Fan, Jinpu Zhang, Xiangkai Lian, Hui Shen, Dewen Hu

The integration of image and event streams offers a promising approach for achieving robust visual object tracking in complex environments. However, current fusion methods achieve high performance at the cost of significant computational overhead and struggle to efficiently extract the sparse, asynchronous information from event streams, failing to leverage the energy-efficient advantages of event-driven spiking paradigms. To address this challenge, we propose the first fully Spiking Frame-Event Tracking framework called SpikeFET. This network achieves synergistic integration of convolutional local feature extraction and Transformer-based global modeling within the spiking paradigm, effectively fusing frame and event data. To overcome the degradation of translation invariance caused by convolutional padding, we introduce a Random Patchwork Module (RPM) that eliminates positional bias through randomized spatial reorganization and learnable type encoding while preserving residual structures. Furthermore, we propose a Spatial-Temporal Regularization (STR) strategy that overcomes similarity metric degradation from asymmetric features by enforcing spatio-temporal consistency among temporal template features in latent space. Extensive experiments across multiple benchmarks demonstrate that the proposed framework achieves superior tracking accuracy over existing methods while significantly reducing power consumption, attaining an optimal balance between performance and efficiency.