Study on learning rates in neural networks of varying depth.
problem Dependence of maximal update learning rate on network depth.
method Analysis of random fully connected ReLU networks with mean-field weight initialization.
result Maximal update learning rate scales like L−3/2 with network depth. Deep RNNs excel at capturing long-term dependencies in sequential data.
problem Lack of a formal measure for RNNs' long-term memory capacity.
method Introduced a measure called Start-End separation rank to quantify RNNs' ability to model long-term dependencies.
result Deep RNNs support Start-End separation ranks that are combinatorially higher than shallow ones.
Self-attention models benefit equally from width and depth, but beyond a certain point, depth becomes less efficient.
problem Understanding the optimal balance between depth and width in self-attention models.
method Theoretical predictions and empirical ablations on networks of varying depths and widths.
result An optimal width of 30K is recommended for a 1-Trillion parameter network, marking a significant width for self-attention models.
Study on functions computed by deep-layered machines finds same distribution in neural networks and Boolean circuits.
problem Understanding the space of functions computed by deep-layered machines.
method Investigation of Boolean functions on random-layered machines, including neural networks and Boolean circuits.
result The space of functions computed at large depth limit is characterized and the macroscopic entropy of Boolean functions is either monotonically increasing or decreasing with depth.
Study infinite-depth limits of neural networks with fixed width.
problem Understanding the behavior of neural networks as depth increases with fixed width.
method Analyzing finite-width residual networks with random Gaussian weights, focusing on the infinite-depth limit.
result The pre-activations converge to a zero-drift diffusion process, differing from the infinite-width limit.
Study liquidity provision with exogenous competition using a reference market maker.
problem Liquidity provision in the presence of exogenous competition.
method Model a reference market maker who optimizes posted depths, competing market makers using a rule of thumb.
result Model admits approximate closed-form solution for a linear-quadratic goal functional.
New insights into how depth and width affect in-context learning in deep models.
problem Understanding how various resources impact in-context learning in deep models.
method Analyzed linear regression in a deep linear self-attention model, varying resources like depth, width, context length, and training steps.
result Increasing depth improves in-context learning even at infinite context length, contrary to previous findings.
Existing Rademacher complexity bounds for neural networks rely only on norm control of the weight matrices and depend exponentially on depth via a product of the matrix norms. Lower bounds show that this exponential dependence on depth is unavoidable when no additional properties of the training data are considered. We…
Study how depth affects inference in deep Bayesian neural networks.
problem Understanding how depth impacts inference in overparameterized linear Bayesian neural networks.
method Interpreting finite deep linear Bayesian neural networks as scale mixtures of Gaussian process predictors.
result Advances analytical understanding of how depth affects inference in a simple class of Bayesian neural networks.
We prove the precise scaling, at finite depth and width, for the mean and variance of the neural tangent kernel (NTK) in a randomly initialized ReLU network. The standard deviation is exponential in the ratio of network depth to width. Thus, even in the limit of infinite overparameterization, the NTK is not determinist…
seMCD computes depth functions with statistical guarantees using sequential Monte Carlo.
problem Computing depth functions is computationally challenging, especially in high dimensions.
method Sequential Monte Carlo methodology with theoretical and empirical guarantees.
result The seMCD method provides accurate depth approximations with fewer samples than traditional methods.
Over-parameterized CNNs show U-shaped test risk with depth increase.
problem Understanding the impact of depth on test risk in over-parameterized CNNs.
method Empirical image classification experiments and linear regression framework.
result Test risk is U-shaped with increasing depth in over-parameterized CNNs.
A new depth measure and median defined on Hadamard manifolds.
problem Statistical depth and median on Hadamard manifolds.
method Horospherical depth and Busemann median defined using renormalized distance functions.
result The Busemann median exists for every Borel probability measure on Hadamard manifolds.
Uniform consistency proven for spatial distribution and depth estimators in any dimension.
problem Uniform consistency of spatial distribution and depth estimators in arbitrary dimensions.
method Proof of uniform L1-consistency using sample size n as the only dependency. result Consistency rate is independent of dimension d and sample size n. Study shows depth improves generalization in deep learning models.
problem Understanding why and when depth improves generalization in deep learning.
method Implementation-agnostic state-transition model to analyze depth and generalization.
result Identifies geometric and semigroup mechanisms that keep entropy contribution saturated or polynomial, clarifying depth's statistical advantage.
Our study analyzes how neural network initialization affects privacy and utility in overparameterized models.
problem Privacy and utility trade-off in overparameterized neural networks.
method Analytical proof of KL divergence privacy bound, focusing on initialization, width, and depth.
result Privacy bound improvement with increasing depth under certain initializations, degradation under others.
Deeper networks are better for local labels, but shallower for global labels.
problem Understanding the effect of depth in overparameterized neural networks.
method Introduced local and global labels to investigate the advantage of depth.
result Deeper networks are better for local labels, shallower for global labels.
Let F be a closed essential surface in a hyperbolic 3-manifold M with a toroidal cusp N. The depth of F in N is the maximal distance from points of F in N to the boundary of N. It will be shown that if F is an essential pleated surface which is not coannular to the boundary torus of N then the depth…
Study on Tukey depth in machine learning using Hamilton-Jacobi equations.
problem Understanding Tukey depth in machine learning applications.
method Derive necessary conditions for Tukey depth in continuum limit, formulating them as a Hamilton-Jacobi equation.
result Prove existence and uniqueness of viscosity solutions for the derived equation, which bounds Tukey depth.
Existing depth separation results for constant-depth networks essentially show that certain radial functions in Rd, which can be easily approximated with depth 3 networks, cannot be approximated by depth 2 networks, even up to constant accuracy, unless their size is exponential in d. However, the func…
Bayesian linear networks reveal optimal depth and width trade-offs.
problem Understanding how depth, width, and dataset size affect model quality in linear networks.
method Zero noise Bayesian inference with Gaussian weight priors and mean squared error.
result Optimal predictions at infinite depth and maximized Bayesian model evidence at infinite depth.
This paper studies the expressive power of graph neural networks falling within the message-passing framework (GNNmp). Two results are presented. First, GNNmp are shown to be Turing universal under sufficient conditions on their depth, width, node attributes, and layer expressiveness. Second, it is discovered that GNNm…
We study the sample complexity of learning neural networks, by providing new bounds on their Rademacher complexity assuming norm constraints on the parameter matrix of each layer. Compared to previous work, these complexity bounds have improved dependence on the network depth, and under some additional assumptions, are…
This paper explores how neural network width and depth behave as they approach infinity.
problem Understanding the behavior of neural functions as width and depth go to infinity.
method Formal definition of commutativity framework, study of neural covariance kernel, novel proof techniques.
result Taking width and depth to infinity in a deep neural network with skip connections results in the same covariance structure, regardless of the order of taking limits.
Study of deep linear neural networks with proportional width and depth.
problem Lack of descriptive power in Gaussian limit of deep linear neural networks.
method Proportional infinite-width infinite-depth limit for deep linear neural networks.
result Characterization of limiting distribution as a nontrivial mixture of Gaussians.
We survey results on neural network expressivity described in "On the Expressive Power of Deep Neural Networks". The paper motivates and develops three natural measures of expressiveness, which all display an exponential dependence on the depth of the network. In fact, all of these measures are related to a fourth quan…
Algorithm efficiently learns deep ReLU networks with polynomial runtime in depth and parameters.
problem Learning deep ReLU networks with polynomial runtime.
method Algorithm using filtered PCA and lattice polynomial analysis.
result First nontrivial results for networks of depth more than two with polynomial runtime.
Study on size and depth of neural networks for approximating benign functions, showing barriers and explicit results.
problem Understanding how size and depth of neural networks affect their ability to approximate benign functions.
method Analyzing ReLU networks for benign functions, proving barriers and explicit results.
result Explicit benign functions that cannot be approximated by networks of certain sizes or depths, showing barriers to size and depth separation.
New bounds adaptively control spectral complexity of trained Transformers.
problem Understanding why Transformers generalize well in machine learning.
method Spectrum-adaptive post hoc generalization bounds for multi-layer Transformers.
result Bounds adaptively trade off spectral complexity against dimension and depth factors.
Recent advances in bandit tools and techniques for sequential learning are steadily enabling new applications and are promising the resolution of a range of challenging related problems. We study the game tree search problem, where the goal is to quickly identify the optimal move in a given game tree by sequentially sa…
Sharp representation theorems show depth benefits for ReLU networks.
problem Representing functions with ReLU networks of varying depths.
method Proved sharp representation results capturing depth benefits.
result Approximation rates improve with depth for less smooth functions.
This paper presents a general framework for norm-based capacity control for Lp,q weight normalized deep neural networks. We establish the upper bound on the Rademacher complexities of this family. With an Lp,q normalization where q≤p∗, and 1/p+1/p∗=1, we discuss properties of a width-independent ca…
Deep networks improve by progressively refining approximations at each layer.
problem Standard approximation theory doesn't explain the role of intermediate layers in deep neural networks.
method Developed a mixed-activation architecture with a geometric scale interpretation of depth.
result Each intermediate layer approximates the target function with a geometric rate.
Study explores how dataset breadth and depth affect Siamese Neural Network performance.
problem Impact of dataset breadth and depth on Siamese Neural Network performance.
method Experiments with three keystroke datasets varying breadth and depth factors.
result Increasing dataset breadth improves model performance, while depth's impact varies by dataset type.
Study reveals how Fisher information changes with network depth, finding it grows linearly.
problem Understanding the trainability of deep neural networks (DNNs).
method Investigates the spectral distribution of the conditional Fisher information matrix (FIM) for fully-connected networks achieving dynamical isometry.
result The conditional FIM's spectrum concentrates around the maximum and grows linearly with depth.
Study fractal and regular geometry in deep neural networks.
problem Investigate geometric properties of neural networks.
method Analyze boundary volumes of excursion sets for different activations.
result Hausdorff dimension increases with depth for non-regular activations.
Improved sample complexity for ReLU networks with norm constraints.
problem Estimating sample complexity for ReLU networks under norm constraints.
method Refined Rademacher complexity analysis for function class.
result Often no explicit depth-dependence in sample complexity bound.
ResNets approximate log-Gaussian at initialization, improving network performance.
problem Understanding the initialization behavior of deep neural networks like ResNets.
method Analyzing ReLU ResNets in the infinite-depth-and-width limit, showing log-Gaussian behavior.
result ResNets at initialization exhibit hypoactivation and interlayer correlations, which are not captured by Gaussian limits.
Transformer networks approximate Hölder and Sobolev functions with fixed-depth networks.
problem Nonparametric regression with dependent observations.
method Established novel upper bounds for Transformer networks approximating Hölder and Sobolev functions under various β-mixing data assumptions. result Explicit convergence rates for nonparametric regression problems under β-mixing data assumptions. A leading hypothesis for the surprising generalization of neural networks is that the dynamics of gradient descent bias the model towards simple solutions, by searching through the solution space in an incremental order of complexity. We formally define the notion of incremental learning dynamics and derive the conditi…
New Transformer architecture prevents rank degeneracy in deep attention models.
problem Rank degeneracy in deep attention models.
method Modified Softmax-based attention model with skip connections, centered at identity, and scaled logits.
result Existence of a stable SDE implies well-behaved covariance structure, preventing rank degeneracy.
Complex-valued neural networks can approximate any continuous function with bounded widths and depths.
problem Approximating continuous functions with complex-valued neural networks of bounded widths and depths.
method Analyzing activation functions and proving universality for complex-valued networks.
result Deep narrow complex-valued networks are universal if and only if their activation function is neither holomorphic, nor antiholomorphic, nor R-affine. Poly-GNNs achieve similar performance regardless of depth, highlighting graph noise's dominance.
problem Performance of poly-GNNs in semi-supervised node classification.
method Analysis of poly-GNNs under a contextual stochastic block model (CSBM).
result For a sufficiently large graph, depth k>1 poly-GNNs exhibit the same rate of separation as depth k=1 counterparts. WGANs improve probability distribution approximation with depth and width trade-offs.
problem Approximating complex probability distributions accurately.
method Wasserstein GANs with GroupSort discriminators, quantified generalization bound.
result High-capacity discriminators are crucial for WGANs' performance.
Residual networks' depth is mathematically equivalent to expanding an implicit ensemble size.
problem Understanding why deep residual networks are effective.
method Formal analysis of residual networks as ensembles of shallow models.
result Increasing network depth is equivalent to expanding the size of an implicit ensemble, revealing a hierarchical structure.
Much sequential data exhibits highly non-uniform information distribution. This cannot be correctly modeled by traditional Long Short-Term Memory (LSTM). To address that, recent works have extended LSTM by adding more activations between adjacent inputs. However, the approaches often use a fixed depth, which is at the …
Attention-only transformers learn from context via two stages of inference.
problem Learning from corrupted token sequences in minimal transformers.
method Two-stage empirical Bayes interpretation: kernel-weighted posterior mean and particle dynamics.
result Effective denoising without explicit noise schedules, showing posterior-mean recovery under asymptotic conditions.
We consider rotationally symmetric spaces with low regularity, which we regard as integral currents spaces or manifolds with Sobolev regularity and are assumed to have nonnegative scalar curvature. Relying on the flat distance and on Sobolev norms, we establish several nonlinear stability estimates about the ``distance…