Rahul Parhi

Rahul Parhi, Robert D. Nowak

Deep learning has been wildly successful in practice and most state-of-the-art machine learning methods are based on neural networks. Lacking, however, is a rigorous mathematical theory that adequately explains the amazing performance of deep neural networks. In this article, we present a relatively new mathematical framework that provides the beginning of a deeper understanding of deep learning. This framework precisely characterizes the functional properties of neural networks that are trained to fit to data. The key mathematical tools which support this framework include transform-domain sparse regularization, the Radon transform of computed tomography, and approximation theory, which are all techniques deeply rooted in signal processing. This framework explains the effect of weight decay regularization in neural network training, the use of skip connections and low-rank weight matrices in network architectures, the role of sparsity in neural networks, and explains why neural networks can perform well in high-dimensional problems.

Ronald DeVore, Robert D. Nowak, Rahul Parhi et al.

We investigate the approximation of functions $f$ on a bounded domain $Ω\subset \mathbb{R}^d$ by the outputs of single-hidden-layer ReLU neural networks of width $n$. This form of nonlinear $n$-term dictionary approximation has been intensely studied since it is the simplest case of neural network approximation (NNA). There are several celebrated approximation results for this form of NNA that introduce novel model classes of functions on $Ω$ whose approximation rates do not grow unbounded with the input dimension. These novel classes include Barron classes, and classes based on sparsity or variation such as the Radon-domain BV classes. The present paper is concerned with the definition of these novel model classes on domains $Ω$. The current definition of these model classes does not depend on the domain $Ω$. A new and more proper definition of model classes on domains is given by introducing the concept of weighted variation spaces. These new model classes are intrinsic to the domain itself. The importance of these new model classes is that they are strictly larger than the classical (domain-independent) classes. Yet, it is shown that they maintain the same NNA rates.

Rahul Parhi, Michael Unser

We investigate the function-space optimality (specifically, the Banach-space optimality) of a large class of shallow neural architectures with multivariate nonlinearities/activation functions. To that end, we construct a new family of Banach spaces defined via a regularization operator, the $k$-plane transform, and a sparsity-promoting norm. We prove a representer theorem that states that the solution sets to learning problems posed over these Banach spaces are completely characterized by neural architectures with multivariate nonlinearities. These optimal architectures have skip connections and are tightly connected to orthogonal weight normalization and multi-index models, both of which have received recent interest in the neural network community. Our framework is compatible with a number of classical nonlinearities including the rectified linear unit (ReLU) activation function, the norm activation function, and the radial basis functions found in the theory of thin-plate/polyharmonic splines. We also show that the underlying spaces are special instances of reproducing kernel Banach spaces and variation spaces. Our results shed light on the regularity of functions learned by neural networks trained on data, particularly with multivariate nonlinearities, and provide new theoretical motivation for several architectural choices found in practice.

Anil Kamber, Rahul Parhi

Weight decay is ubiquitous in training deep neural network architectures. Its empirical success is often attributed to capacity control; nonetheless, our theoretical understanding of its effect on the loss landscape and the set of minimizers remains limited. In this paper, we show that $\ell^2$-regularized deep matrix factorization/deep linear network training problems with squared-error loss admit a unique end-to-end minimizer for all target matrices subject to factorization, except for a set of Lebesgue measure zero formed by the depth and the regularization parameter. This observation reveals fundamental properties of the loss landscape of regularized deep matrix factorization problems: the Hessian spectrum is constant across all minimizers of the regularized deep scalar factorization problem with squared-error loss. Moreover, we show that, in regularized deep matrix factorization problems with squared-error loss, if the target matrix does not belong to the Lebesgue measure-zero set, then the Frobenius norm of each layer is constant across all minimizers. This, in turn, yields a global lower bound on the trace of the Hessian evaluated at any minimizer of the regularized deep matrix factorization problem. Furthermore, we establish a critical threshold for the regularization parameter above which the unique end-to-end minimizer collapses to zero.

Rahul Parhi, Pakshal Bohra, Ayoub El Biari et al.

We consider a large class of shallow neural networks with randomly initialized parameters and rectified linear unit activation functions. We prove that these random neural networks are well-defined non-Gaussian processes. As a by-product, we demonstrate that these networks are solutions to stochastic differential equations driven by impulsive white noise (combinations of random Dirac measures). These processes are parameterized by the law of the weights and biases as well as the density of activation thresholds in each bounded region of the input domain. We prove that these processes are isotropic and wide-sense self-similar with Hurst exponent 3/2. We also derive a remarkably simple closed-form expression for their autocovariance function. Our results are fundamentally different from prior work in that we consider a non-asymptotic viewpoint: The number of neurons in each bounded region of the input domain (i.e., the width) is itself a random variable with a Poisson law with mean proportional to the density parameter. Finally, we show that, under suitable hypotheses, as the expected width tends to infinity, these processes can converge in law not only to Gaussian processes, but also to non-Gaussian processes depending on the law of the weights. Our asymptotic results provide a new take on several classical results (wide networks converge to Gaussian processes) as well as some new ones (wide networks can converge to non-Gaussian processes).

Luke McDermott, Robert W. Heath, Rahul Parhi

The per-token cost of transformer inference scales with context length, preventing its application to lifelong in-context learning. Linear attention is an efficient alternative that maintains a constant memory footprint, even on infinite context lengths. While this is a potential candidate for lifelong learning, it falls short in memory capacity. In this paper, we propose LoLA, a training-free augmentation to linear attention that boosts associative recall. LoLA distributes past key-value pairs from context into three memory systems: (i) recent pairs in a local sliding window cache; (ii) difficult-to-memorize pairs in a sparse, global cache; and (iii) generic pairs in the recurrent hidden state of linear attention. We show through ablations that our self-recall error metric is crucial to efficiently manage long-term associative memories. On pass-key retrieval tasks, LoLA improves the base model's performance from 0.6% to 97.4% accuracy. This is achieved with a 4.6x smaller cache than Llama-3.1 8B on 4K context length. LoLA also outperforms other 1B and 8B parameter subquadratic models on zero-shot commonsense reasoning tasks.

Tongtong Liang, Dan Qiao, Yu-Xiang Wang et al.

We study the implicit bias of flatness / low (loss) curvature and its effects on generalization in two-layer overparameterized ReLU networks with multivariate inputs -- a problem well motivated by the minima stability and edge-of-stability phenomena in gradient-descent training. Existing work either requires interpolation or focuses only on univariate inputs. This paper presents new and somewhat surprising theoretical results for multivariate inputs. On two natural settings (1) generalization gap for flat solutions, and (2) mean-squared error (MSE) in nonparametric function estimation by stable minima, we prove upper and lower bounds, which establish that while flatness does imply generalization, the resulting rates of convergence necessarily deteriorate exponentially as the input dimension grows. This gives an exponential separation between the flat solutions vis-à-vis low-norm solutions (i.e., weight decay), which knowingly do not suffer from the curse of dimensionality. In particular, our minimax lower bound construction, based on a novel packing argument with boundary-localized ReLU neurons, reveals how flat solutions can exploit a kind of ''neural shattering'' where neurons rarely activate, but with high weight magnitudes. This leads to poor performance in high dimensions. We corroborate these theoretical findings with extensive numerical simulations. To the best of our knowledge, our analysis provides the first systematic explanation for why flat minima may fail to generalize in high dimensions.

Tongtong Liang, Esha Singh, Rahul Parhi et al.

We study how architectural inductive bias reshapes the implicit regularization induced by the edge-of-stability phenomenon in gradient descent. Prior work has established that for fully connected networks, the strength of this regularization is governed solely by the global input geometry; consequently, it is insufficient to prevent overfitting on difficult distributions such as the high-dimensional sphere. In this paper, we show that locality and weight sharing fundamentally change this picture. Specifically, we prove that provided the receptive field size $m$ remains small relative to the ambient dimension $d$, these networks generalize on spherical data with a rate of $n^{-\frac{1}{6} +O(m/d)}$, a regime where fully connected networks provably fail. This theoretical result confirms that weight sharing couples the learned filters to the low-dimensional patch manifold, thereby bypassing the high dimensionality of the ambient space. We further corroborate our theory by analyzing the patch geometry of natural images, showing that standard convolutional designs induce patch distributions that are highly amenable to this stability mechanism, thus providing a systematic explanation for the superior generalization of convolutional networks over fully connected baselines.

Tongtong Liang, Alexander Cloninger, Rahul Parhi et al.

Understanding generalization in overparameterized neural networks hinges on the interplay between the data geometry, neural architecture, and training dynamics. In this paper, we theoretically explore how data geometry controls this implicit bias. This paper presents theoretical results for overparameterized two-layer ReLU networks trained below the edge of stability. First, for data distributions supported on a mixture of low-dimensional balls, we derive generalization bounds that provably adapt to the intrinsic dimension. Second, for a family of isotropic distributions that vary in how strongly probability mass concentrates toward the unit sphere, we derive a spectrum of bounds showing that rates deteriorate as the mass concentrates toward the sphere. These results instantiate a unifying principle: When the data is harder to "shatter" with respect to the activation thresholds of the ReLU neurons, gradient descent tends to learn representations that capture shared patterns and thus finds solutions that generalize well. On the other hand, for data that is easily shattered (e.g., data supported on the sphere) gradient descent favors memorization. Our theoretical results consolidate disparate empirical findings that have appeared in the literature.

Anil Kamber, Rahul Parhi

Understanding the geometry of the loss landscape near a minimum is key to explaining the implicit bias of gradient-based methods in non-convex optimization problems such as deep neural network training and deep matrix factorization. A central quantity to characterize this geometry is the maximum eigenvalue of the Hessian of the loss, which measures the sharpness of the landscape. Currently, its precise role has been obfuscated because no exact expressions for this sharpness measure were known in general settings. In this paper, we present the first exact expression for the maximum eigenvalue of the Hessian of the squared-error loss at any minimizer in general overparameterized deep matrix factorization (i.e., deep linear neural network training) problems, resolving an open question posed by Mulayoff & Michaeli (2020). To complement our theory, we empirically investigate an escape phenomenon observed during gradient-based training near a minimum that crucially relies on our exact expression of the sharpness.

Luke McDermott, Rahul Parhi

Recent works have shown that Dataset Distillation, the process for summarizing the training data, can be leveraged to accelerate the training of deep learning models. However, its impact on training dynamics, particularly in neural network pruning, remains largely unexplored. In our work, we use distilled data in the inner loop of iterative magnitude pruning to produce sparse, trainable subnetworks at initialization -- more commonly known as lottery tickets. While using 150x less training points, our algorithm matches the performance of traditional lottery ticket rewinding on ResNet-18 & CIFAR-10. Previous work highlights that lottery tickets can be found when the dense initialization is stable to SGD noise (i.e. training across different ordering of the data converges to the same minima). We extend this discovery, demonstrating that stable subnetworks can exist even within an unstable dense initialization. In our linear mode connectivity studies, we find that pruning with distilled data discards parameters that contribute to the sharpness of the loss landscape. Lastly, we show that by first generating a stable sparsity mask at initialization, we can find lottery tickets at significantly higher sparsities than traditional iterative magnitude pruning.

Akash Kumar, Rahul Parhi, Mikhail Belkin

Recent works have characterized the function-space inductive bias of infinite-width bounded-norm single-hidden-layer neural networks as a kind of bounded-variation-type space. This novel neural network Banach space encompasses many classical multivariate function spaces, including certain Sobolev spaces and the spectral Barron spaces. Notably, this Banach space also includes functions that exhibit less classical regularity, such as those that only vary in a few directions. On bounded domains, it is well-established that the Gaussian reproducing kernel Hilbert space (RKHS) strictly embeds into this Banach space, demonstrating a clear gap between the Gaussian RKHS and the neural network Banach space. It turns out that when investigating these spaces on unbounded domains, e.g., all of $\mathbb{R}^d$, the story is fundamentally different. We establish the following fundamental result: Certain functions that lie in the Gaussian RKHS have infinite norm in the neural network Banach space. This provides a nontrivial gap between kernel methods and neural networks by exhibiting functions that kernel methods easily represent, whereas neural networks cannot.

Joseph Shenouda, Rahul Parhi, Kangwook Lee et al.

This paper introduces a novel theoretical framework for the analysis of vector-valued neural networks through the development of vector-valued variation spaces, a new class of reproducing kernel Banach spaces. These spaces emerge from studying the regularization effect of weight decay in training networks with activations like the rectified linear unit (ReLU). This framework offers a deeper understanding of multi-output networks and their function-space characteristics. A key contribution of this work is the development of a representer theorem for the vector-valued variation spaces. This representer theorem establishes that shallow vector-valued neural networks are the solutions to data-fitting problems over these infinite-dimensional spaces, where the network widths are bounded by the square of the number of training data. This observation reveals that the norm associated with these vector-valued variation spaces encourages the learning of features that are useful for multiple tasks, shedding new light on multi-task learning with neural networks. Finally, this paper develops a connection between weight-decay regularization and the multi-task lasso problem. This connection leads to novel bounds for layer widths in deep networks that depend on the intrinsic dimensions of the training data representations. This insight not only deepens the understanding of the deep network architectural requirements, but also yields a simple convex optimization method for deep neural network compression. The performance of this compression procedure is evaluated on various architectures.

Rahul Parhi, Robert D. Nowak

We study the problem of estimating an unknown function from noisy data using shallow ReLU neural networks. The estimators we study minimize the sum of squared data-fitting errors plus a regularization term proportional to the squared Euclidean norm of the network weights. This minimization corresponds to the common approach of training a neural network with weight decay. We quantify the performance (mean-squared error) of these neural network estimators when the data-generating function belongs to the second-order Radon-domain bounded variation space. This space of functions was recently proposed as the natural function space associated with shallow ReLU neural networks. We derive a minimax lower bound for the estimation problem for this function space and show that the neural network estimators are minimax optimal up to logarithmic factors. This minimax rate is immune to the curse of dimensionality. We quantify an explicit gap between neural networks and linear methods (which include kernel methods) by deriving a linear minimax lower bound for the estimation problem, showing that linear methods necessarily suffer the curse of dimensionality in this function space. As a result, this paper sheds light on the phenomenon that neural networks seem to break the curse of dimensionality.

Rahul Parhi, Robert D. Nowak

We develop a variational framework to understand the properties of functions learned by fitting deep neural networks with rectified linear unit activations to data. We propose a new function space, which is reminiscent of classical bounded variation-type spaces, that captures the compositional structure associated with deep neural networks. We derive a representer theorem showing that deep ReLU networks are solutions to regularized data fitting problems over functions from this space. The function space consists of compositions of functions from the Banach spaces of second-order bounded variation in the Radon domain. These are Banach spaces with sparsity-promoting norms, giving insight into the role of sparsity in deep neural networks. The neural network solutions have skip connections and rank bounded weight matrices, providing new theoretical support for these common architectural choices. The variational problem we study can be recast as a finite-dimensional neural network training problem with regularization schemes related to the notions of weight decay and path-norm regularization. Finally, our analysis builds on techniques from variational spline theory, providing new connections between deep neural networks and splines.

Rahul Parhi, Robert D. Nowak

We develop a variational framework to understand the properties of the functions learned by neural networks fit to data. We propose and study a family of continuous-domain linear inverse problems with total variation-like regularization in the Radon domain subject to data fitting constraints. We derive a representer theorem showing that finite-width, single-hidden layer neural networks are solutions to these inverse problems. We draw on many techniques from variational spline theory and so we propose the notion of polynomial ridge splines, which correspond to single-hidden layer neural networks with truncated power functions as the activation function. The representer theorem is reminiscent of the classical reproducing kernel Hilbert space representer theorem, but we show that the neural network problem is posed over a non-Hilbertian Banach space. While the learning problems are posed in the continuous-domain, similar to kernel methods, the problems can be recast as finite-dimensional neural network training problems. These neural network training problems have regularizers which are related to the well-known weight decay and path-norm regularizers. Thus, our result gives insight into functional characteristics of trained neural networks and also into the design neural network regularizers. We also show that these regularizers promote neural network solutions with desirable generalization properties.

Rahul Parhi, Robert D. Nowak

A wide variety of activation functions have been proposed for neural networks. The Rectified Linear Unit (ReLU) is especially popular today. There are many practical reasons that motivate the use of the ReLU. This paper provides new theoretical characterizations that support the use of the ReLU, its variants such as the leaky ReLU, as well as other activation functions in the case of univariate, single-hidden layer feedforward neural networks. Our results also explain the importance of commonly used strategies in the design and training of neural networks such as "weight decay" and "path-norm" regularization, and provide a new justification for the use of "skip connections" in network architectures. These new insights are obtained through the lens of spline theory. In particular, we show how neural network training problems are related to infinite-dimensional optimizations posed over Banach spaces of functions whose solutions are well-known to be fractional and polynomial splines, where the particular Banach space (which controls the order of the spline) depends on the choice of activation function.

Rahul Parhi, Michael Schliep, Nicholas Hopper

This paper proposes MP3, the second privacy-preserving presence protocol that leaks no information about the graph structure of the social network. Several cryptographic techniques are applied to improve the existing DP5 protocol---the first privacy-preserving presence protocol---while maintaining the same level of privacy. The key contribution of this paper is the use of a dynamic broadcast encryption scheme to reduce the size of the presence database. This enables cheaper registration and lookup required for the protocol. As compared to DP5, MP3 requires on the order of ten times less bandwidth of the servers during registration, and requires on the order of two times less bandwidth for lookup, for a small number of users ($N=10000$). Furthermore, these savings asymptotically increase with the number of users. The client-side latency is also improved significantly in MP3, as compared with DP5. We provide an evaluation of the performance and scalability of both protocols.