Transformers improve logical reasoning on longer proofs but struggle with length.
problem Understanding systematic generalization in neural proof generation.
method Soft theorem-proving using Transformer models, evaluating logical consistency and inference accuracy.
result Transformers improve generalization with longer proofs but have difficulty with length.
In this work we study input gradient regularization of deep neural networks, and demonstrate that such regularization leads to generalization proofs and improved adversarial robustness. The proof of generalization does not overcome the curse of dimensionality, but it is independent of the number of layers in the networ…
Benchmark for math reasoning models from human proofs.
problem Measuring and accelerating machine learning models in high-level mathematical reasoning.
method Built a non-synthetic dataset from theorem prover proofs, defined a task for model to fill in missing propositions, used hierarchical transformer to improve performance.
result Neural models can capture non-trivial mathematical reasoning, hierarchical transformer outperforms baseline.
Lean Copilot uses LLMs to assist theorem proving in Lean, improving efficiency and automation.
problem Challenges in using existing neural theorem provers to prove novel theorems autonomously.
method Introduces Lean Copilot, a framework for integrating LLMs into Lean's theorem proving process.
result Lean Copilot automates 74.2% of proof steps on average, significantly improving over existing methods.
Proof of learning rate transfer in MLPs with μP parameterization.
problem Understanding and optimizing learning rates in neural networks with different parameterizations.
method Theoretical analysis and empirical validation of learning rate transfer in MLPs with μP, SP, and NTP parameterizations. result The optimal learning rate converges to a non-zero constant as width goes to infinity under μP, explaining learning rate transfer. New bound on neural nets complexity for approximating functions.
problem Approximating continuous functions with shallow neural networks.
method Inspired by Stone-Weierstrass theorem, constructive proof.
result General upper bound on neuron count for accuracy.
A common way to train neural networks is the Backpropagation. This algorithm includes a gradient descent method, which needs an adaptive step size. In the area of neural networks, the ADAM-Optimizer is one of the most popular adaptive step size methods. It was invented in \cite{Kingma.2015} by Kingma and Ba. The 5865…
Due to space limitations, our submission "Source Separation and Clustering of Phase-Locked Subspaces", accepted for publication on the IEEE Transactions on Neural Networks in 2011, presented some results without proof. Those proofs are provided in this paper.
AI generates theorems and proofs for training theorem provers.
problem Limited human-written theorems and proofs for supervised learning.
method Proposes a neural generator to automatically synthesize theorems and proofs.
result Synthetic data improves automated theorem proving in Metamath.
We present a simple proof for the universality of invariant and equivariant tensorized graph neural networks. Our approach considers a restricted intermediate hypothetical model named Graph Homomorphism Model to reach the universality conclusions including an open case for higher-order output. We find that our proposed…
This work provides an additional step in the theoretical understanding of neural networks. We consider neural networks with one hidden layer and show that when learning symmetric functions, one can choose initial conditions so that standard SGD training efficiently produces generalization guarantees. We empirically ver…
We present a dataset and experiments on applying recurrent neural networks (RNNs) for guiding clause selection in the connection tableau proof calculus. The RNN encodes a sequence of literals from the current branch of the partial proof tree to a hidden vector state; using it, the system selects a clause for extending …
We provide a proof of backpropagation algorithm in matrix notation.
problem The lack of a full induction proof of backpropagation algorithm in matrix notation.
method We provide a full induction proof of the BP algorithm in matrix notation, situating it in the framework of matrix differential calculus.
result We prove the validity of the backpropagation algorithm in inductive form.
INEUS solves high-dimensional PIDEs efficiently with neural networks.
problem Solving high-dimensional partial integro-differential equations (PIDEs) efficiently.
method INEUS uses iterative neural networks to replace nonlocal integrals with sampling and reformulates PIDE solving as recursive regression.
result INEUS delivers accurate and scalable solutions for high-dimensional linear and nonlinear PIDEs.
For nonconvex optimization in machine learning, this article proves that every local minimum achieves the globally optimal value of the perturbable gradient basis model at any differentiable point. As a result, nonconvex machine learning is theoretically as supported as convex machine learning with a handcrafted basis …
Paper analyzes neural network complexity for planning problems.
problem Understanding neural network complexity for planning policies.
method Circuit complexity analysis for relational neural networks.
result Three classes of planning problems identified based on network complexity.
Proposes a method to generate prediction intervals using weighted asymmetric loss functions.
problem Generating reliable prediction intervals for neural network models.
method Uses a weighted asymmetric loss function to estimate prediction intervals.
result The method produces reliable prediction intervals in complex machine learning scenarios.
Gradient descent proves global convergence for deep networks with a single wide layer.
problem Proving global convergence of gradient descent for deep ReLU networks.
method Simplified proof using a single wide layer, leveraging ReLU's Lipschitz property.
result Gradient descent converges globally for networks with a single wide layer.
New proof shows neural networks can represent all multivariate functions.
problem Representing all multivariate functions with neural networks.
method Proved that three-layer neural networks can represent both continuous and discontinuous functions.
result Three-layer neural networks can represent all multivariate functions, including discontinuous ones.
UNIPoint universally approximates point process intensities.
problem How to precisely describe the flexibility of point process models.
method Proof using Stone-Weierstrass Theorem, transfer functions, and recurrent neural networks.
result UNIPoint performs better than other models on synthetic and real-world datasets.
The paper proves deep neural networks with analytic activation can approximate any function.
problem Approximating functions with neural networks using analytic activation functions.
method Elementary proofs for real and complex networks, Stone-Weierstrass theorem, Mergelyan's theorem.
result Closure of neural network classes equals space of polynomials for analytic activation.
New method selects facts in proofs using stateful recurrent neural networks.
problem Selecting facts for proving new goals over large formal libraries.
method Stateful architecture based on recurrent neural networks with data augmentation.
result Significantly better performance and solving many new problems compared to previous methods.
Blind Descent avoids gradient issues, using a different learning approach.
problem Gradient issues like exploding and vanishing gradients.
method Does not use gradients to guide learning; instead, it is a more fundamental learning process.
result Gradient descent is a specific case of Blind Descent.
Graph neural networks over-smooth when layers increase, reducing discriminative power.
problem Over-smoothing in graph neural networks reduces model performance as the number of layers increases.
method Analyzed over-smoothing in general graph neural network architecture using Dirichlet energy.
result The Dirichlet energy of embeddings converges to zero, leading to loss of discriminative power.
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.
The adaptive moment estimation algorithm Adam (Kingma and Ba) is a popular optimizer in the training of deep neural networks. However, Reddi et al. have recently shown that the convergence proof of Adam is problematic and proposed a variant of Adam called AMSGrad as a fix. In this paper, we show that the convergence pr…
This paper proves a generalization bound for complex-valued neural networks scaling with spectral complexity.
problem Ensuring the performance of complex-valued neural networks on unseen data.
method Theoretical derivation using Maurey Sparsification Lemma and Dudley Entropy Integral, empirical validation on various datasets.
result The spectral complexity of weight matrices is a significant factor in the generalization ability of complex-valued neural networks.
New proof shows random neural networks contain sparse subnetworks.
problem Finding sparse subnetworks in random neural networks.
method Proof of the Strong Lottery Ticket Hypothesis using the Random Fixed-Size Subset Sum Problem.
result Guaranteed sparsity of subnetworks in random neural networks.
Graph neural networks help infer causal effects from partially observable data.
problem Inferring causal effects from partially observable data.
method Theoretical analysis of GNN and SCM connections.
result Established a new model class for GNN-based causal inference.
Inspired by recent work on neural network image generation which rely on backpropagation towards the network inputs, we present a proof-of-concept system for speech texture synthesis and voice conversion based on two mechanisms: approximate inversion of the representation learned by a speech recognition neural network,…
Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.
problem Quantifying the efficiency of neural operators for solving nonlinear parabolic PDEs.
method Deriving approximation rates by transferring PDEs to integral equations and leveraging Picard's iteration.
result Neural operators can efficiently approximate solution operators of nonlinear PDEs without exponential complexity growth.
Neural symbolic processing aims to combine the generalization of logical learning approaches and the performance of neural networks. The Neural Theorem Proving (NTP) model by Rocktaschel et al (2017) learns embeddings for concepts and performs logical unification. While NTP is promising and effective in predicting fact…
In this paper we prove that, in the deep limit, the stochastic gradient descent on a ResNet type deep neural network, where each layer shares the same weight matrix, converges to the stochastic gradient descent for a Neural ODE and that the corresponding value/loss functions converge. Our result gives, in the context o…
This paper establishes rates of universal approximation for the shallow neural tangent kernel (NTK): network weights are only allowed microscopic changes from random initialization, which entails that activations are mostly unchanged, and the network is nearly equivalent to its linearization. Concretely, the paper has …
Gradient descent amplifies random features in neural networks to useful ones.
problem Generalization in neural networks trained on corrupted data.
method Characterization of feature-learning process in two-layer ReLU networks trained by gradient descent.
result Gradient descent amplifies random features to useful ones, achieving near optimal generalization error.
GD with early stopping trains shallow neural nets for nonparametric regression robustly.
problem Learning Lipschitz regression functions with noisy labels.
method Overparameterized shallow neural networks trained by GD with early stopping.
result Optimal rates of convergence for nonparametric regression.
Adam converges to stationary points under relaxed conditions.
problem Understanding and proving convergence of Adam under realistic assumptions.
method New proof of boundedness of gradients and variance-reduced Adam.
result Adam converges to ε-stationary points with O(ε⁻⁴) gradient complexity under realistic conditions.
Debona improves neural network verification by faster and tighter bounds.
problem Proving adversarial robustness of neural networks is computationally hard.
method Decouples upper and lower bounds computation, re-implements Neurify.
result 94% reduction in runtime for proving robustness of max-pooling layers.
In this brief we investigate the generalization properties of a recently-proposed class of non-parametric activation functions, the kernel activation functions (KAFs). KAFs introduce additional parameters in the learning process in order to adapt nonlinearities individually on a per-neuron basis, exploiting a cheap ker…
Existence of minimizers proven for residual ANNs with ReLU activation.
problem Existence of minimizers in neural network optimization landscapes.
method Proof using closure of search space containing ANNs and additional discontinuous responses.
result Existence of minimizers proven for residual ANNs with ReLU activation.
Deep neural networks have shown remarkable performance across a wide range of vision-based tasks, particularly due to the availability of large-scale datasets for training and better architectures. However, data seen in the real world are often affected by distortions that not accounted for by the training datasets. In…
Dropout neural networks can approximate any function with high probability.
problem Approximating functions with dropout neural networks.
method Two universal approximation theorems for dropout neural networks in random and deterministic modes.
result Dropout neural networks can approximate any function in probability and in Lq. Sparse neural encoding can store more memories as targets become sparser.
problem Storing sparse input-target associations in neural networks.
method Mathematical proofs using properties of random polytopes and sub-gaussian random vector variables.
result The capacity of neural maps increases with sparsity in target layers.
Real-world problems of operations research are typically high-dimensional and combinatorial. Linear programs are generally used to formulate and efficiently solve these large decision problems. However, in multi-period decision problems, we must often compute expected downstream values corresponding to current decision…
Neural networks can approximate any continuous function with a single hidden layer.
problem Approximating any continuous function using neural networks.
method Direct algebraic proof and explicit quantification of hidden units required.
result Explicit bounds on the number of hidden units needed for approximation.
Nonlinear ICA is a fundamental problem for unsupervised representation learning, emphasizing the capacity to recover the underlying latent variables generating the data (i.e., identifiability). Recently, the very first identifiability proofs for nonlinear ICA have been proposed, leveraging the temporal structure of the…
We establish large deviation principles for convolutional neural networks.
problem Understanding the behavior of convolutional neural networks in the infinite-channel limit.
method We establish large deviation principles for convolutional neural networks under Gaussian prior and posterior distributions.
result We provide a large deviation principle for the sequence of conditional covariance matrices and the posterior distribution.
General lower bounds on neural network approximation in L^p norm.
problem Fundamental limits of neural network expressivity.
method General lower bound proof on approximation in L^p norm, applied to feed-forward neural networks.
result Neural networks can't approximate certain functions as well as previously thought.