Stability properties of gradient flow dynamics for the symmetric low-rank matrix factorization problem

The symmetric low-rank matrix factorization serves as a building block in many learning tasks, including matrix recovery and training of neural networks. However, despite a flurry of recent research, the dynamics of its training via non-convex factorized gradient-descent-type methods is not fully understood especially in the over-parameterized regime where the fitted rank is higher than the true rank of the target matrix. To overcome this challenge, we characterize equilibrium points of the gradient flow dynamics and examine their local and global stability properties. To facilitate a precise global analysis, we introduce a nonlinear change of variables that brings the dynamics into a cascade connection of three subsystems whose structure is simpler than the structure of the original system. We demonstrate that the Schur complement to a principal eigenspace of the target matrix is governed by an autonomous system that is decoupled from the rest of the dynamics. In the over-parameterized regime, we show that this Schur complement vanishes at an $O(1/t)$ rate, thereby capturing the slow dynamics that arises from excess parameters. We utilize a Lyapunov-based approach to establish exponential convergence of the other two subsystems. By decoupling the fast and slow parts of the dynamics, we offer new insight into the shape of the trajectories associated with local search algorithms and provide a complete characterization of the equilibrium points and their global stability properties. Such an analysis via nonlinear control techniques may prove useful in several related over-parameterized problems.

Theoretical Insights into Overparameterized Models in Multi-Task and Replay-Based Continual Learning

Multi-task learning (MTL) is a machine learning paradigm that aims to improve the generalization performance of a model on multiple related tasks by training it simultaneously on those tasks. Unlike MTL, where the model has instant access to the training data of all tasks, continual learning (CL) involves adapting to new sequentially arriving tasks over time without forgetting the previously acquired knowledge. Despite the wide practical adoption of CL and MTL and extensive literature on both areas, there remains a gap in the theoretical understanding of these methods when used with overparameterized models such as deep neural networks. This paper studies the overparameterized linear models as a proxy for more complex models. We develop theoretical results describing the effect of various system parameters on the model's performance in an MTL setup. Specifically, we study the impact of model size, dataset size, and task similarity on the generalization error and knowledge transfer. Additionally, we present theoretical results to characterize the performance of replay-based CL models. Our results reveal the impact of buffer size and model capacity on the forgetting rate in a CL setup and help shed light on some of the state-of-the-art CL methods. Finally, through extensive empirical evaluations, we demonstrate that our theoretical findings are also applicable to deep neural networks, offering valuable guidance for designing MTL and CL models in practice.

Serpent: Scalable and Efficient Image Restoration via Multi-scale Structured State Space Models

The landscape of computational building blocks of efficient image restoration architectures is dominated by a combination of convolutional processing and various attention mechanisms. However, convolutional filters are inherently local and therefore struggle at modeling long-range dependencies in images. On the other hand, attention excels at capturing global interactions between arbitrary image regions, however at a quadratic cost in image dimension. In this work, we propose Serpent, an architecture that leverages recent advances in state space models (SSMs) in its core computational block. SSMs, originally introduced for sequence modeling, can maintain a global receptive field with a favorable linear scaling in input size. Our preliminary results demonstrate that Serpent can achieve reconstruction quality on par with state-of-the-art techniques, while requiring orders of magnitude less compute (up to $150$ fold reduction in FLOPS) and a factor of up to $5\times$ less GPU memory while maintaining a compact model size.

A Data-Free Approach to Mitigate Catastrophic Forgetting in Federated Class Incremental Learning for Vision Tasks

Deep learning models often suffer from forgetting previously learned information when trained on new data. This problem is exacerbated in federated learning (FL), where the data is distributed and can change independently for each user. Many solutions are proposed to resolve this catastrophic forgetting in a centralized setting. However, they do not apply directly to FL because of its unique complexities, such as privacy concerns and resource limitations. To overcome these challenges, this paper presents a framework for $\textbf{federated class incremental learning}$ that utilizes a generative model to synthesize samples from past distributions. This data can be later exploited alongside the training data to mitigate catastrophic forgetting. To preserve privacy, the generative model is trained on the server using data-free methods at the end of each task without requesting data from clients. Moreover, our solution does not demand the users to store old data or models, which gives them the freedom to join/leave the training at any time. Additionally, we introduce SuperImageNet, a new regrouping of the ImageNet dataset specifically tailored for federated continual learning. We demonstrate significant improvements compared to existing baselines through extensive experiments on multiple datasets.

Gradient Descent Provably Solves Nonlinear Tomographic Reconstruction

In computed tomography (CT), the forward model consists of a linear Radon transform followed by an exponential nonlinearity based on the attenuation of light according to the Beer-Lambert Law. Conventional reconstruction often involves inverting this nonlinearity as a preprocessing step and then solving a convex inverse problem. However, this nonlinear measurement preprocessing required to use the Radon transform is poorly conditioned in the vicinity of high-density materials, such as metal. This preprocessing makes CT reconstruction methods numerically sensitive and susceptible to artifacts near high-density regions. In this paper, we study a technique where the signal is directly reconstructed from raw measurements through the nonlinear forward model. Though this optimization is nonconvex, we show that gradient descent provably converges to the global optimum at a geometric rate, perfectly reconstructing the underlying signal with a near minimal number of random measurements. We also prove similar results in the under-determined setting where the number of measurements is significantly smaller than the dimension of the signal. This is achieved by enforcing prior structural information about the signal through constraints on the optimization variables. We illustrate the benefits of direct nonlinear CT reconstruction with cone-beam CT experiments on synthetic and real 3D volumes. We show that this approach reduces metal artifacts compared to a commercial reconstruction of a human skull with metal dental crowns.

Learning A Disentangling Representation For PU Learning

In this paper, we address the problem of learning a binary (positive vs. negative) classifier given Positive and Unlabeled data commonly referred to as PU learning. Although rudimentary techniques like clustering, out-of-distribution detection, or positive density estimation can be used to solve the problem in low-dimensional settings, their efficacy progressively deteriorates with higher dimensions due to the increasing complexities in the data distribution. In this paper we propose to learn a neural network-based data representation using a loss function that can be used to project the unlabeled data into two (positive and negative) clusters that can be easily identified using simple clustering techniques, effectively emulating the phenomenon observed in low-dimensional settings. We adopt a vector quantization technique for the learned representations to amplify the separation between the learned unlabeled data clusters. We conduct experiments on simulated PU data that demonstrate the improved performance of our proposed method compared to the current state-of-the-art approaches. We also provide some theoretical justification for our two cluster-based approach and our algorithmic choices.

Adapt and Diffuse: Sample-adaptive Reconstruction via Latent Diffusion Models

Inverse problems arise in a multitude of applications, where the goal is to recover a clean signal from noisy and possibly (non)linear observations. The difficulty of a reconstruction problem depends on multiple factors, such as the structure of the ground truth signal, the severity of the degradation, the implicit bias of the reconstruction model and the complex interactions between the above factors. This results in natural sample-by-sample variation in the difficulty of a reconstruction task, which is often overlooked by contemporary techniques. Recently, diffusion-based inverse problem solvers have established new state-of-the-art in various reconstruction tasks. However, they have the drawback of being computationally prohibitive. Our key observation in this paper is that most existing solvers lack the ability to adapt their compute power to the difficulty of the reconstruction task, resulting in long inference times, subpar performance and wasteful resource allocation. We propose a novel method that we call severity encoding, to estimate the degradation severity of noisy, degraded signals in the latent space of an autoencoder. We show that the estimated severity has strong correlation with the true corruption level and can give useful hints at the difficulty of reconstruction problems on a sample-by-sample basis. Furthermore, we propose a reconstruction method based on latent diffusion models that leverages the predicted degradation severities to fine-tune the reverse diffusion sampling trajectory and thus achieve sample-adaptive inference times. We utilize latent diffusion posterior sampling to maintain data consistency with observations. We perform experiments on both linear and nonlinear inverse problems and demonstrate that our technique achieves performance comparable to state-of-the-art diffusion-based techniques, with significant improvements in computational efficiency.

mL-BFGS: A Momentum-based L-BFGS for Distributed Large-Scale Neural Network Optimization

Quasi-Newton methods still face significant challenges in training large-scale neural networks due to additional compute costs in the Hessian related computations and instability issues in stochastic training. A well-known method, L-BFGS that efficiently approximates the Hessian using history parameter and gradient changes, suffers convergence instability in stochastic training. So far, attempts that adapt L-BFGS to large-scale stochastic training incur considerable extra overhead, which offsets its convergence benefits in wall-clock time. In this paper, we propose mL-BFGS, a lightweight momentum-based L-BFGS algorithm that paves the way for quasi-Newton (QN) methods in large-scale distributed deep neural network (DNN) optimization. mL-BFGS introduces a nearly cost-free momentum scheme into L-BFGS update and greatly reduces stochastic noise in the Hessian, therefore stabilizing convergence during stochastic optimization. For model training at a large scale, mL-BFGS approximates a block-wise Hessian, thus enabling distributing compute and memory costs across all computing nodes. We provide a supporting convergence analysis for mL-BFGS in stochastic settings. To investigate mL-BFGS potential in large-scale DNN training, we train benchmark neural models using mL-BFGS and compare performance with baselines (SGD, Adam, and other quasi-Newton methods). Results show that mL-BFGS achieves both noticeable iteration-wise and wall-clock speedup.

Learning Provably Robust Estimators for Inverse Problems via Jittering

Deep neural networks provide excellent performance for inverse problems such as denoising. However, neural networks can be sensitive to adversarial or worst-case perturbations. This raises the question of whether such networks can be trained efficiently to be worst-case robust. In this paper, we investigate whether jittering, a simple regularization technique that adds isotropic Gaussian noise during training, is effective for learning worst-case robust estimators for inverse problems. While well studied for prediction in classification tasks, the effectiveness of jittering for inverse problems has not been systematically investigated. In this paper, we present a novel analytical characterization of the optimal $\ell_2$-worst-case robust estimator for linear denoising and show that jittering yields optimal robust denoisers. Furthermore, we examine jittering empirically via training deep neural networks (U-nets) for natural image denoising, deconvolution, and accelerated magnetic resonance imaging (MRI). The results show that jittering significantly enhances the worst-case robustness, but can be suboptimal for inverse problems beyond denoising. Moreover, our results imply that training on real data which often contains slight noise is somewhat robustness enhancing.

Provable Multi-Task Representation Learning by Two-Layer ReLU Neural Networks

Feature learning, i.e. extracting meaningful representations of data, is quintessential to the practical success of neural networks trained with gradient descent, yet it is notoriously difficult to explain how and why it occurs. Recent theoretical studies have shown that shallow neural networks optimized on a single task with gradient-based methods can learn meaningful features, extending our understanding beyond the neural tangent kernel or random feature regime in which negligible feature learning occurs. But in practice, neural networks are increasingly often trained on {\em many} tasks simultaneously with differing loss functions, and these prior analyses do not generalize to such settings. In the multi-task learning setting, a variety of studies have shown effective feature learning by simple linear models. However, multi-task learning via {\em nonlinear} models, arguably the most common learning paradigm in practice, remains largely mysterious. In this work, we present the first results proving feature learning occurs in a multi-task setting with a nonlinear model. We show that when the tasks are binary classification problems with labels depending on only $r$ directions within the ambient $d\gg r$-dimensional input space, executing a simple gradient-based multitask learning algorithm on a two-layer ReLU neural network learns the ground-truth $r$ directions. In particular, any downstream task on the $r$ ground-truth coordinates can be solved by learning a linear classifier with sample and neuron complexity independent of the ambient dimension $d$, while a random feature model requires exponential complexity in $d$ for such a guarantee.