arXiv subjects

Antonio Orvieto

Publications and source records attributed to Antonio Orvieto.

At least 55 records · Page 3Linked to original sources

arXiv2025

Theoretical Foundations of Deep Selective State-Space Models

Structured state-space models (SSMs) such as S4, stemming from the seminal work of Gu et al., are gaining popularity as effective approaches for modeling sequential data. Deep SSMs demonstrate outstanding performance across a diverse set of domains, at a reduced training and inference cost compared to attention-based transformers. Recent developments show that if the linear recurrence powering SSMs allows for multiplicative interactions between inputs and hidden states (e.g. GateLoop, Mamba, GLA), then the resulting architecture can surpass in both in accuracy and efficiency attention-powered foundation models trained on text, at scales of billion parameters. In this paper, we give theoretical grounding to this recent finding using tools from Rough Path Theory: we show that when random linear recurrences are equipped with simple input-controlled transitions (selectivity mechanism), then the hidden state is provably a low-dimensional projection of a powerful mathematical object called the signature of the input -- capturing non-linear interactions between tokens at distinct timescales. Our theory not only motivates the success of modern selective state-space models such as Mamba but also provides a solid framework to understand the expressive power of future SSM variants.

cs.LG↗

arXiv2024

Understanding the differences in Foundation Models: Attention, State Space Models, and Recurrent Neural Networks

Softmax attention is the principle backbone of foundation models for various artificial intelligence applications, yet its quadratic complexity in sequence length can limit its inference throughput in long-context settings. To address this challenge, alternative architectures such as linear attention, State Space Models (SSMs), and Recurrent Neural Networks (RNNs) have been considered as more efficient alternatives. While connections between these approaches exist, such models are commonly developed in isolation and there is a lack of theoretical understanding of the shared principles underpinning these architectures and their subtle differences, greatly influencing performance and scalability. In this paper, we introduce the Dynamical Systems Framework (DSF), which allows a principled investigation of all these architectures in a common representation. Our framework facilitates rigorous comparisons, providing new insights on the distinctive characteristics of each model class. For instance, we compare linear attention and selective SSMs, detailing their differences and conditions under which both are equivalent. We also provide principled comparisons between softmax attention and other model classes, discussing the theoretical conditions under which softmax attention can be approximated. Additionally, we substantiate these new insights with empirical validations and mathematical arguments. This shows the DSF's potential to guide the systematic development of future more efficient and scalable foundation models.

cs.LG↗

arXiv2024

Super Consistency of Neural Network Landscapes and Learning Rate Transfer

Recently, there has been growing evidence that if the width and depth of a neural network are scaled toward the so-called rich feature learning limit (\mup and its depth extension), then some hyperparameters -- such as the learning rate -- exhibit transfer from small to very large models. From an optimization perspective, this phenomenon is puzzling, as it implies that the loss landscape is consistently similar across very different model sizes. In this work, we study the landscape through the lens of the loss Hessian, with a focus on its largest eigenvalue (i.e. the sharpness), and find that certain spectral properties under $μ$P are largely independent of the size of the network, and remain consistent as training progresses. We name this property Super Consistency of the landscape. On the other hand, we show that in the Neural Tangent Kernel (NTK) and other scaling regimes, the sharpness exhibits very different dynamics at different scales. But what causes these differences in the sharpness dynamics? Through a connection between the Hessian's and the NTK's spectrum, we argue that the cause lies in the presence (for $μ$P) or progressive absence (for the NTK scaling) of feature learning. We corroborate our claims with a substantial suite of experiments, covering a wide range of datasets and architectures: from ResNets and Vision Transformers trained on benchmark vision datasets to Transformers-based language models trained on WikiText.

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arXiv2024

Recurrent neural networks: vanishing and exploding gradients are not the end of the story

Recurrent neural networks (RNNs) notoriously struggle to learn long-term memories, primarily due to vanishing and exploding gradients. The recent success of state-space models (SSMs), a subclass of RNNs, to overcome such difficulties challenges our theoretical understanding. In this paper, we delve into the optimization challenges of RNNs and discover that, as the memory of a network increases, changes in its parameters result in increasingly large output variations, making gradient-based learning highly sensitive, even without exploding gradients. Our analysis further reveals the importance of the element-wise recurrence design pattern combined with careful parametrizations in mitigating this effect. This feature is present in SSMs, as well as in other architectures, such as LSTMs. Overall, our insights provide a new explanation for some of the difficulties in gradient-based learning of RNNs and why some architectures perform better than others.

cs.LG↗

arXiv2024

NIMBA: Towards Robust and Principled Processing of Point Clouds With SSMs

Transformers have become dominant in large-scale deep learning tasks across various domains, including text, 2D and 3D vision. However, the quadratic complexity of their attention mechanism limits their efficiency as the sequence length increases, particularly in high-resolution 3D data such as point clouds. Recently, state space models (SSMs) like Mamba have emerged as promising alternatives, offering linear complexity, scalability, and high performance in long-sequence tasks. The key challenge in the application of SSMs in this domain lies in reconciling the non-sequential structure of point clouds with the inherently directional (or bi-directional) order-dependent processing of recurrent models like Mamba. To achieve this, previous research proposed reorganizing point clouds along multiple directions or predetermined paths in 3D space, concatenating the results to produce a single 1D sequence capturing different views. In our work, we introduce a method to convert point clouds into 1D sequences that maintain 3D spatial structure with no need for data replication, allowing Mamba sequential processing to be applied effectively in an almost permutation-invariant manner. In contrast to other works, we found that our method does not require positional embeddings and allows for shorter sequence lengths while still achieving state-of-the-art results in ModelNet40 and ScanObjectNN datasets and surpassing Transformer-based models in both accuracy and efficiency.

cs.CV↗

arXiv2024

Loss Landscape Characterization of Neural Networks without Over-Parametrization

Optimization methods play a crucial role in modern machine learning, powering the remarkable empirical achievements of deep learning models. These successes are even more remarkable given the complex non-convex nature of the loss landscape of these models. Yet, ensuring the convergence of optimization methods requires specific structural conditions on the objective function that are rarely satisfied in practice. One prominent example is the widely recognized Polyak-Lojasiewicz (PL) inequality, which has gained considerable attention in recent years. However, validating such assumptions for deep neural networks entails substantial and often impractical levels of over-parametrization. In order to address this limitation, we propose a novel class of functions that can characterize the loss landscape of modern deep models without requiring extensive over-parametrization and can also include saddle points. Crucially, we prove that gradient-based optimizers possess theoretical guarantees of convergence under this assumption. Finally, we validate the soundness of our new function class through both theoretical analysis and empirical experimentation across a diverse range of deep learning models.

cs.LG↗

arXiv2024

An Adaptive Stochastic Gradient Method with Non-negative Gauss-Newton Stepsizes

We consider the problem of minimizing the average of a large number of smooth but possibly non-convex functions. In the context of most machine learning applications, each loss function is non-negative and thus can be expressed as the composition of a square and its real-valued square root. This reformulation allows us to apply the Gauss-Newton method, or the Levenberg-Marquardt method when adding a quadratic regularization. The resulting algorithm, while being computationally as efficient as the vanilla stochastic gradient method, is highly adaptive and can automatically warmup and decay the effective stepsize while tracking the non-negative loss landscape. We provide a tight convergence analysis, leveraging new techniques, in the stochastic convex and non-convex settings. In particular, in the convex case, the method does not require access to the gradient Lipshitz constant for convergence, and is guaranteed to never diverge. The convergence rates and empirical evaluations compare favorably to the classical (stochastic) gradient method as well as to several other adaptive methods.

math.OC↗

arXiv2024

Universality of Linear Recurrences Followed by Non-linear Projections: Finite-Width Guarantees and Benefits of Complex Eigenvalues

Deep neural networks based on linear RNNs interleaved with position-wise MLPs are gaining traction as competitive approaches for sequence modeling. Examples of such architectures include state-space models (SSMs) like S4, LRU, and Mamba: recently proposed models that achieve promising performance on text, genetics, and other data that require long-range reasoning. Despite experimental evidence highlighting these architectures' effectiveness and computational efficiency, their expressive power remains relatively unexplored, especially in connection to specific choices crucial in practice - e.g., carefully designed initialization distribution and potential use of complex numbers. In this paper, we show that combining MLPs with both real or complex linear diagonal recurrences leads to arbitrarily precise approximation of regular causal sequence-to-sequence maps. At the heart of our proof, we rely on a separation of concerns: the linear RNN provides a lossless encoding of the input sequence, and the MLP performs non-linear processing on this encoding. While we show that real diagonal linear recurrences are enough to achieve universality in this architecture, we prove that employing complex eigenvalues near unit disk - i.e., empirically the most successful strategy in S4 - greatly helps the RNN in storing information. We connect this finding with the vanishing gradient issue and provide experiments supporting our claims.

cs.LG↗

arXiv2024

Recurrent Distance Filtering for Graph Representation Learning

Graph neural networks based on iterative one-hop message passing have been shown to struggle in harnessing the information from distant nodes effectively. Conversely, graph transformers allow each node to attend to all other nodes directly, but lack graph inductive bias and have to rely on ad-hoc positional encoding. In this paper, we propose a new architecture to reconcile these challenges. Our approach stems from the recent breakthroughs in long-range modeling provided by deep state-space models: for a given target node, our model aggregates other nodes by their shortest distances to the target and uses a linear RNN to encode the sequence of hop representations. The linear RNN is parameterized in a particular diagonal form for stable long-range signal propagation and is theoretically expressive enough to encode the neighborhood hierarchy. With no need for positional encoding, we empirically show that the performance of our model is comparable to or better than that of state-of-the-art graph transformers on various benchmarks, with a significantly reduced computational cost. Our code is open-source at https://github.com/skeletondyh/GRED.

cs.LG↗

arXiv2024

On the low-shot transferability of [V]-Mamba

The strength of modern large-scale neural networks lies in their ability to efficiently adapt to new tasks with few examples. Although extensive research has investigated the transferability of Vision Transformers (ViTs) to various downstream tasks under diverse constraints, this study shifts focus to explore the transfer learning potential of [V]-Mamba. We compare its performance with ViTs across different few-shot data budgets and efficient transfer methods. Our analysis yields three key insights into [V]-Mamba's few-shot transfer performance: (a) [V]-Mamba demonstrates superior or equivalent few-shot learning capabilities compared to ViTs when utilizing linear probing (LP) for transfer, (b) Conversely, [V]-Mamba exhibits weaker or similar few-shot learning performance compared to ViTs when employing visual prompting (VP) as the transfer method, and (c) We observe a weak positive correlation between the performance gap in transfer via LP and VP and the scale of the [V]-Mamba model. This preliminary analysis lays the foundation for more comprehensive studies aimed at furthering our understanding of the capabilities of [V]-Mamba variants and their distinctions from ViTs.

cs.CV↗

arXiv2024

SDEs for Minimax Optimization

Minimax optimization problems have attracted a lot of attention over the past few years, with applications ranging from economics to machine learning. While advanced optimization methods exist for such problems, characterizing their dynamics in stochastic scenarios remains notably challenging. In this paper, we pioneer the use of stochastic differential equations (SDEs) to analyze and compare Minimax optimizers. Our SDE models for Stochastic Gradient Descent-Ascent, Stochastic Extragradient, and Stochastic Hamiltonian Gradient Descent are provable approximations of their algorithmic counterparts, clearly showcasing the interplay between hyperparameters, implicit regularization, and implicit curvature-induced noise. This perspective also allows for a unified and simplified analysis strategy based on the principles of Itô calculus. Finally, our approach facilitates the derivation of convergence conditions and closed-form solutions for the dynamics in simplified settings, unveiling further insights into the behavior of different optimizers.

cs.LG↗

arXiv2024

Dynamics of SGD with Stochastic Polyak Stepsizes: Truly Adaptive Variants and Convergence to Exact Solution

Recently, Loizou et al. (2021), proposed and analyzed stochastic gradient descent (SGD) with stochastic Polyak stepsize (SPS). The proposed SPS comes with strong convergence guarantees and competitive performance; however, it has two main drawbacks when it is used in non-over-parameterized regimes: (i) It requires a priori knowledge of the optimal mini-batch losses, which are not available when the interpolation condition is not satisfied (e.g., regularized objectives), and (ii) it guarantees convergence only to a neighborhood of the solution. In this work, we study the dynamics and the convergence properties of SGD equipped with new variants of the stochastic Polyak stepsize and provide solutions to both drawbacks of the original SPS. We first show that a simple modification of the original SPS that uses lower bounds instead of the optimal function values can directly solve issue (i). On the other hand, solving issue (ii) turns out to be more challenging and leads us to valuable insights into the method's behavior. We show that if interpolation is not satisfied, the correlation between SPS and stochastic gradients introduces a bias, which effectively distorts the expectation of the gradient signal near minimizers, leading to non-convergence - even if the stepsize is scaled down during training. To fix this issue, we propose DecSPS, a novel modification of SPS, which guarantees convergence to the exact minimizer - without a priori knowledge of the problem parameters. For strongly-convex optimization problems, DecSPS is the first stochastic adaptive optimization method that converges to the exact solution without restrictive assumptions like bounded iterates/gradients.

math.OC↗

arXiv2023

An SDE for Modeling SAM: Theory and Insights

We study the SAM (Sharpness-Aware Minimization) optimizer which has recently attracted a lot of interest due to its increased performance over more classical variants of stochastic gradient descent. Our main contribution is the derivation of continuous-time models (in the form of SDEs) for SAM and two of its variants, both for the full-batch and mini-batch settings. We demonstrate that these SDEs are rigorous approximations of the real discrete-time algorithms (in a weak sense, scaling linearly with the learning rate). Using these models, we then offer an explanation of why SAM prefers flat minima over sharp ones~--~by showing that it minimizes an implicitly regularized loss with a Hessian-dependent noise structure. Finally, we prove that SAM is attracted to saddle points under some realistic conditions. Our theoretical results are supported by detailed experiments.

cs.LG↗

arXiv2023

Anticorrelated Noise Injection for Improved Generalization

Injecting artificial noise into gradient descent (GD) is commonly employed to improve the performance of machine learning models. Usually, uncorrelated noise is used in such perturbed gradient descent (PGD) methods. It is, however, not known if this is optimal or whether other types of noise could provide better generalization performance. In this paper, we zoom in on the problem of correlating the perturbations of consecutive PGD steps. We consider a variety of objective functions for which we find that GD with anticorrelated perturbations ("Anti-PGD") generalizes significantly better than GD and standard (uncorrelated) PGD. To support these experimental findings, we also derive a theoretical analysis that demonstrates that Anti-PGD moves to wider minima, while GD and PGD remain stuck in suboptimal regions or even diverge. This new connection between anticorrelated noise and generalization opens the field to novel ways to exploit noise for training machine learning models.

stat.ML↗

arXiv2023

On the effectiveness of Randomized Signatures as Reservoir for Learning Rough Dynamics

Many finance, physics, and engineering phenomena are modeled by continuous-time dynamical systems driven by highly irregular (stochastic) inputs. A powerful tool to perform time series analysis in this context is rooted in rough path theory and leverages the so-called Signature Transform. This algorithm enjoys strong theoretical guarantees but is hard to scale to high-dimensional data. In this paper, we study a recently derived random projection variant called Randomized Signature, obtained using the Johnson-Lindenstrauss Lemma. We provide an in-depth experimental evaluation of the effectiveness of the Randomized Signature approach, in an attempt to showcase the advantages of this reservoir to the community. Specifically, we find that this method is preferable to the truncated Signature approach and alternative deep learning techniques in terms of model complexity, training time, accuracy, robustness, and data hungriness.

cs.LG↗

arXiv2023

Achieving a Better Stability-Plasticity Trade-off via Auxiliary Networks in Continual Learning

In contrast to the natural capabilities of humans to learn new tasks in a sequential fashion, neural networks are known to suffer from catastrophic forgetting, where the model's performances on old tasks drop dramatically after being optimized for a new task. Since then, the continual learning (CL) community has proposed several solutions aiming to equip the neural network with the ability to learn the current task (plasticity) while still achieving high accuracy on the previous tasks (stability). Despite remarkable improvements, the plasticity-stability trade-off is still far from being solved and its underlying mechanism is poorly understood. In this work, we propose Auxiliary Network Continual Learning (ANCL), a novel method that applies an additional auxiliary network which promotes plasticity to the continually learned model which mainly focuses on stability. More concretely, the proposed framework materializes in a regularizer that naturally interpolates between plasticity and stability, surpassing strong baselines on task incremental and class incremental scenarios. Through extensive analyses on ANCL solutions, we identify some essential principles beneath the stability-plasticity trade-off.

cs.LG↗

arXiv2023

Resurrecting Recurrent Neural Networks for Long Sequences

Recurrent Neural Networks (RNNs) offer fast inference on long sequences but are hard to optimize and slow to train. Deep state-space models (SSMs) have recently been shown to perform remarkably well on long sequence modeling tasks, and have the added benefits of fast parallelizable training and RNN-like fast inference. However, while SSMs are superficially similar to RNNs, there are important differences that make it unclear where their performance boost over RNNs comes from. In this paper, we show that careful design of deep RNNs using standard signal propagation arguments can recover the impressive performance of deep SSMs on long-range reasoning tasks, while also matching their training speed. To achieve this, we analyze and ablate a series of changes to standard RNNs including linearizing and diagonalizing the recurrence, using better parameterizations and initializations, and ensuring proper normalization of the forward pass. Our results provide new insights on the origins of the impressive performance of deep SSMs, while also introducing an RNN block called the Linear Recurrent Unit that matches both their performance on the Long Range Arena benchmark and their computational efficiency.

cs.LG↗

arXiv2023

Explicit Regularization in Overparametrized Models via Noise Injection

Injecting noise within gradient descent has several desirable features, such as smoothing and regularizing properties. In this paper, we investigate the effects of injecting noise before computing a gradient step. We demonstrate that small perturbations can induce explicit regularization for simple models based on the L1-norm, group L1-norms, or nuclear norms. However, when applied to overparametrized neural networks with large widths, we show that the same perturbations can cause variance explosion. To overcome this, we propose using independent layer-wise perturbations, which provably allow for explicit regularization without variance explosion. Our empirical results show that these small perturbations lead to improved generalization performance compared to vanilla gradient descent.

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