We study the task of agnostically learning general (as opposed to homogeneous) ReLUs under the Gaussian distribution with respect to the squared loss. In the passive learning setting, recent work gave a computationally efficient algorithm that uses poly(d,1/ϵ)labeled examples and outputs a hypothesis with error O(opt)+ϵ, where optis the squared loss of the best fit ReLU. Here we focus on the interactive setting, where the learner has some form of query access to the labels of unlabeled examples. Our main result is the first computationally efficient learner that uses dpolylog(1/ϵ)+ O(min{1/p,1/ϵ})black-box label queries, where pis the bias of the target function, and achieves error O(opt)+ϵ. We complement our algorithmic result by showing that its query complexity bound is qualitatively near-optimal, even ignoring computational constraints. Finally, we establish that query access is essentially necessary to improve on the label complexity of passive learning. Specifically, for pool-based active learning, any active learner requires Ω(d/ϵ) labels, unless it draws a super-polynomial number of unlabeled examples.

Numerous studies have demonstrated that the Transformer architecture possesses the capability for in-context learning (ICL). In scenarios involving function approximation, context can serve as a control parameter for the model, endowing it with the universal approximation property (UAP). In practice, context is represented by tokens from a finite set, referred to as a vocabulary, which is the case considered in this paper, i.e., vocabulary in-context learning (VICL). We demonstrate that VICL in single-layer Transformers, without positional encoding, does not possess the UAP; however, it is possible to achieve the UAP when positional encoding is included. Several sufficient conditions for the positional encoding are provided. Our findings reveal the benefits of positional encoding from an approximation theory perspective in the context of ICL.

We study a shallow variant of XNet, a neural architecture whose activation functions are derived from the Cauchy integral formula. While prior work focused on deep variants, we show that even a single-layer XNet exhibits near-exponential approximation rates--exceeding the polynomial bounds of MLPs and spline-based networks such as Kolmogorov-Arnold Networks (KANs). Empirically, XNet reduces approximation error by over 600 on discontinuous functions, achieves up to 20,000 lower residuals in physics-informed PDEs, and improves policy accuracy and sample efficiency in PPO-based reinforcement learning--while maintaining comparable or better computational efficiency than KAN baselines. These results demonstrate that expressive approximation can stem from principled activation design rather than depth alone, offering a compact, theoretically grounded alternative for function approximation, scientific computing, and control.

Equivariant neural networks outperform traditional deep neural networks on a number of tasks. The theoretical understanding of their generalization properties remains, however, limited. In this paper, we analyze the generalization capabilities of Group Convolutional Neural Networks (GCNNs) with ReLU activation function through the lens of Vapnik-Chervonenkis (VC) dimension theory. By deriving upper and lower bounds, we investigate how the network architecture affects the VC dimension.

Computation in recurrent networks of neurons has been hypothesized to occur at the level of low-dimensional latent dynamics, both in artificial systems and in the brain. This hypothesis seems at odds with evidence from large-scale neuronal recordings in mice showing that neuronal population activity is high-dimensional. To demonstrate that low-dimensional latent dynamics and high-dimensional activity can be two sides of the same coin, we present an analytically solvable recurrent neural network (RNN) model whose dynamics can be exactly reduced to a lowdimensional dynamical system, but generates an activity manifold that has a high linear embedding dimension. This raises the question: Do low-dimensional latents explain the high-dimensional activity observed in mouse visual cortex? Spectral theory tells us that the covariance eigenspectrum alone does not allow us to recover the dimensionality of the latents, which can be low or high, when neurons are nonlinear. To address this indeterminacy, we develop Neural Cross-Encoder (NCE), an interpretable, nonlinear latent variable modeling method for neuronal recordings, and find that high-dimensional neuronal responses to drifting gratings and spontaneous activity in visual cortex can be reduced to low-dimensional latents, while the responses to natural images cannot. We conclude that the high-dimensional activity measured in certain conditions, such as in the absence of a stimulus, is explained by low-dimensional latents that are nonlinearly processed by individual neurons.

In this paper, we examine the impact and significance of bias terms in Implicit Neural Representations (INRs). While bias terms are known to enhance nonlinear capacity by shifting activations in typical neural networks, we discover their functionality differs markedly in neural representation networks. Our analysis reveals that INR performance neither scales with increased number of bias terms nor shows substantial improvement through bias term gradient propagation. We demonstrate that bias terms in INRs primarily serve to eliminate spatial aliasing caused by symmetry from both coordinates and activation functions, with inputlayer bias terms yielding the most significant benefits. These findings challenge the conventional practice of implementing full-bias INR architecture. We propose using freezing bias terms exclusively in input layers, which consistently outperforms fully biased networks in signal fitting tasks. Furthermore, we introduce Feature-Biased INRs (Feat-Bias), which initialize input-layer bias with high-level features extracted from pre-trained models. This feature-biasing approach effectively addresses the limited performance in INR post-processing tasks due to neural parameter uninterpretability, achieving superior accuracy while reducing parameter count and improving reconstruction quality. Our code is available at this link.

Deep reinforcement learning (RL) agents frequently suffer from neuronal activity loss, which impairs their ability to adapt to new data and learn continually. A common method to quantify and address this issue is the τ-dormant neuron ratio, which uses activation statistics to measure the expressive ability of neurons. While effective for simple MLP-based agents, this approach loses statistical power in more complex architectures. To address this, we argue that in advanced RL agents, maintaining a neuron's learning capacity, its ability to adapt via gradient updates, is more critical than preserving its expressive ability. Based on this insight, we shift the statistical objective from activations to gradients, and introduce GraMa (Gradient Magnitude Neural Activity Metric), a lightweight, architecture-agnostic metric for quantifying neuron-level learning capacity. We show that GraMaeffectively reveals persistent neuron inactivity across diverse architectures, including residual networks, diffusion models, and agents with varied activation functions. Moreover, resetting neurons guided by GraMa (ReGraMa) consistently improves learning performance across multiple deep RL algorithms and benchmarks, such as MuJoCo and the DeepMind Control Suite. We make our code available2.

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, ai and aj, diverge to infinity, and their input weight vectors, wi and wj, become equal to each other. At convergence, the two neurons implement a gated linear unit: aiσ(wi x) + ajσ(wj x) cσ(w x) + (v x)σ (w 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 these quasi-flat regions 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.

The brain's diverse spatiotemporal activity patterns are fundamental to cognition and consciousness, yet how these macroscopic dynamics emerge from microscopic neural circuitry remains a critical challenge. We take a step in this direction by developing a spatially extended neural network model integrated with a spectral theory of its connectivity matrix. Our theory quantitatively demonstrates how local structural parameters, such as E/I neuron projection ranges, connection strengths, and density determine distinct features of the eigenvalue spectrum, specifically outlier eigenvalues and a bulk disk. These spectral signatures, in turn, precisely predict the network's emergent global dynamical regime, encompassing asynchronous states, synchronous states, oscillations, localized activity bumps, traveling waves, and chaos. Motivated by observations of shifting cortical dynamics in mice across arousal states, our framework not only provides a possible explanation for repertoire of behaviors but also offers a principled starting point for inferring underlying effective connectivity changes from macroscopic brain activity. By mechanistically linking neural structure to dynamics, this work advances a principled framework for dissecting how large-scale activity patterns--central to cognition and open questions in consciousness research--arise from, and constrain, local circuitry.

Activation functions are fundamental elements of deep learning architectures as they significantly influence training dynamics. ReLU, while widely used, is prone to the dying neuron problem, which has been mitigated by variants such as LeakyReLU, PReLU, and ELU that better handle negative neuron outputs. Recently, self-gated activations like GELU and Swish have emerged as state-of-the-art alternatives, leveraging their smoothness to ensure stable gradient flow and prevent neuron inactivity.