New algorithm samples superlinearly growing log-gradient distributions.
problem Sampling from distributions with superlinearly growing log-gradient.
method Proposes a novel taming Langevin-based scheme called sTULA.
result Derives non-asymptotic convergence bounds in KL, TV, and W2 distances.
A new tamed stochastic gradient Hamiltonian Monte Carlo algorithm for superlinearly growing stochastic gradients.
problem Sampling and stochastic optimization problems with superlinearly growing stochastic gradients.
method Tamed Stochastic Gradient Hamiltonian Monte Carlo (tSGHMC) algorithm.
result Established a non-asymptotic error bound in Wasserstein-2 distance with a convergence rate of 1/4. Developing a scientific understanding of cities in a fast urbanizing world is essential for planning sustainable urban systems. Recently, it was shown that income and wealth creation follow increasing returns, scaling superlinearly with city size. We study scaling of per capita incomes for separate census defined incom…
RELTA-SGLD stabilizes nonconvex SGLD updates with a lighter taming scheme.
problem Stabilizing superlinear stochastic-gradient updates in nonconvex optimization.
method Threshold-based taming with relative-growth principle for stability.
result Polynomial moment stability and first-order stationary accuracy in nonconvex SGLD.
New method samples from non-log-concave distributions with weak dissipativity.
problem Sampling from distributions that are not log-concave and weakly dissipative.
method Taming scheme tailored to growth and decay properties of the target distribution.
result Explicit non-asymptotic guarantees for KL, TV, and Wasserstein distances.
Zeroth-order optimization methods lack inherent privacy guarantees.
problem Ensuring differential privacy in zeroth-order optimization methods.
method Analyzing ZO-GD with and without random initialization for convex and strongly convex objectives.
result ZO-GD is not differentially private for strongly convex objectives and can have superlinear privacy loss.
Study proves convergence of interest rate model approximations.
problem Investigating convergence of stochastic interest rate models.
method Developed analytical tools for true and truncated EM solutions, proving convergence in probability.
result True solution converges in probability to truncated EM solution as step size approaches zero.
Deep neural networks struggle with numerical instability during training.
problem Numerical instability in gradient descent training of deep neural networks.
method Analysis of floating-point arithmetic and gradient descent in ReLU neural networks.
result It is highly unlikely for ReLU networks to maintain a superlinear number of affine pieces during training.
In this work, we present a globalized stochastic semismooth Newton method for solving stochastic optimization problems involving smooth nonconvex and nonsmooth convex terms in the objective function. We assume that only noisy gradient and Hessian information of the smooth part of the objective function is available via…
The L1-regularized Gaussian maximum likelihood estimator (MLE) has been shown to have strong statistical guarantees in recovering a sparse inverse covariance matrix, or alternatively the underlying graph structure of a Gaussian Markov Random Field, from very limited samples. We propose a novel algorithm for solving the…
Measures of wealth and production have been found to scale superlinearly with the population of a city. Therefore, it makes economic sense for humans to congregate together in dense settlements. A recent model of population dynamics showed that population growth can become superexponential due to the superlinear scalin…
This paper accelerates distributed convex optimization by mitigating ill-conditioning issues.
problem Distributed convex optimization with ill-conditioned aggregate cost functions.
method Iterative pre-conditioning technique to improve convergence rate and stability.
result The proposed algorithm converges linearly with improved convergence rate and superlinearly under certain conditions.
Accelerated gradient method's stability deteriorates exponentially with steps.
problem Algorithmic stability of Nesterov's accelerated gradient method.
method Analysis of two notions of algorithmic stability for Nesterov's accelerated gradient method.
result Stability of Nesterov's accelerated method deteriorates exponentially with the number of gradient steps.
Model shows AI adoption amplifies financial market risk through prediction, herding, and cognitive dependency.
problem Systemic risk in financial markets due to AI adoption.
method Developed a unified model within an extended rational expectations framework, incorporating endogenous adoption, performative prediction, algorithmic herding, and cognitive dependency.
result Systemic risk multiplier grows superlinearly with AI penetration, implying tail-loss amplification of 18-54%.
A new method for growing neural networks by splitting neurons, improving efficiency.
problem Optimizing neural network structures, especially for lightweight architectures.
method A progressive training approach using steepest descent to adaptively grow and split neurons.
result The method provides a computationally efficient way to optimize neural network structures.
Gaussian processes (GPs) provide a powerful non-parametric framework for reasoning over functions. Despite appealing theory, its superlinear computational and memory complexities have presented a long-standing challenge. State-of-the-art sparse variational inference methods trade modeling accuracy against complexity. H…
Study on neural networks' performance under different normalizations as N grows.
problem Characterizing neural networks' performance under various normalizations.
method Developed an asymptotic expansion to analyze statistical output of shallow neural networks.
result No bias-variance trade-off exists to leading order in N, and variance decreases as normalization approaches mean field.
New method corrects bias in estimating entropic risk for better decision-making.
problem Underestimation of entropic risk when data are limited.
method Parametric bootstrap procedure to overestimate entropic risk.
result Corrected method provides better risk estimates, leading to improved decision-making.
Faster weak supervision framework using triplet methods.
problem Computational inefficiency in weak supervision models.
method Closed-form solution for latent variable models, avoiding iterative methods.
result Orders of magnitude faster than previous approaches.
Let M be a closed orientable 3-manifold with a negatively curved Riemannian metric. Let {M_i} be a collection of finite regular covers with degree d_i. (1) If the Heegaard genus of M_i grows more slowly than the square root of d_i, then M_i has positive first Betti number for all sufficiently large i. (2) The strong He…
Over the past few years, neural networks were proven vulnerable to adversarial images: targeted but imperceptible image perturbations lead to drastically different predictions. We show that adversarial vulnerability increases with the gradients of the training objective when viewed as a function of the inputs. Surprisi…
Minimal graphs grow slowly on curved spaces, proving constant solutions.
problem Characterizing minimal graphs with sublinear growth on manifolds.
method New technique to get gradient bounds by integral estimates, no further geometric assumptions.
result Entire solutions are constant when negative part grows like r/logr. Gradient flow expands curves to round shapes.
problem Expanding curves to round shapes.
method Steepest descent L2-gradient flow of entropy.
result Flow converges to a round expanding circle for various initial curves.
Time series data constitutes a distinct and growing problem in machine learning. As the corpus of time series data grows larger, deep models that simultaneously learn features and classify with these features can be intractable or suboptimal. In this paper, we present feature learning via long short term memory (LSTM) …
Gradient descent with growing learning rate enables learning non-linear features in neural networks.
problem Learning non-linear features in two-layer neural networks.
method Using gradient descent with a learning rate that grows with the sample size.
result Multiple rank-one components emerge, each corresponding to a specific polynomial feature.
Backpropagation-free trunk training improves model performance on various benchmarks.
problem Memory inefficiency and noisy gradient estimates in deep network training.
method Split Forward Gradient (Split-FG) method that splits network into trunk and head, estimating only trunk gradient.
result Split-FG achieves better performance than pure forward-gradient training and backpropagation on various benchmarks.
Cosine similarity can force points to grow in magnitude, causing convergence issues.
problem Cosine similarity loss can lead to convergence issues in deep learning.
method Analyzing under-explored settings and proposing cut-initialization.
result Cosine similarity optimization forces points to grow in magnitude, leading to convergence issues.
Training very deep networks is an important open problem in machine learning. One of many difficulties is that the norm of the back-propagated error gradient can grow or decay exponentially. Here we show that training very deep feed-forward networks (FFNs) is not as difficult as previously thought. Unlike when back-pro…
The paper proves inequalities and growth rates for Schouten solitons.
problem Understanding the properties of Schouten solitons.
method Proving inequalities and growth rates using the norm of the gradient and scalar curvature.
result The potential function of Shrinking Schouten solitons grows linearly.
Thinking LLMs struggle with stock prediction, especially as data complexity increases.
problem Evaluating the performance of 'thinking' LLMs in stock prediction, especially under varying levels of cross-sectional complexity.
method Rolling 48m/1m walk-forward evaluation, comparing direct LLMs, TLLMs, and classical learners on cross-sectional ranking loss, MSE, and backtests with transaction costs.
result TLLMs' ranking quality deteriorates as cross-sectional complexity grows, while direct LLMs remain stable.
Improved optimization technique reduces training complexity for non-convex problems.
problem Training non-convex optimization problems with exploding gradients.
method Employed variance reduction technique (SPIDER) with carefully designed learning rate.
result Improved stochastic gradient complexity to O(ε−3) for ε-stationary solutions. This paper examines a novel gradient boosting framework for regression. We regularize gradient boosted trees by introducing subsampling and employ a modified shrinkage algorithm so that at every boosting stage the estimate is given by an average of trees. The resulting algorithm, titled Boulevard, is shown to converge …
We would like to congratulate the authors of "A Bayesian Conjugate Gradient Method" on their insightful paper, and welcome this publication which we firmly believe will become a fundamental contribution to the growing field of probabilistic numerical methods and in particular the sub-field of Bayesian numerical methods…
Classical stochastic gradient methods for optimization rely on noisy gradient approximations that become progressively less accurate as iterates approach a solution. The large noise and small signal in the resulting gradients makes it difficult to use them for adaptive stepsize selection and automatic stopping. We prop…
We present and analyze several strategies for improving the performance of stochastic variance-reduced gradient (SVRG) methods. We first show that the convergence rate of these methods can be preserved under a decreasing sequence of errors in the control variate, and use this to derive variants of SVRG that use growing…
We investigate Liouville theorems and dimension estimates for the space of exponentially growing holomorphic functions on complete Kähler manifolds. While our work is motivated by the study of gradient Ricci solitons in the theory of Ricci flow, the most general results we prove here do not require any knowledge of cur…
New bounds show BBVI's gradient variance matches SGD conditions, improving parameterization efficiency.
problem Understanding and improving the convergence of black-box variational inference (BBVI).
method Showed BBVI satisfies matching gradient variance bounds corresponding to the ABC condition for smooth and quadratically-growing log-likelihoods.
result Proven BBVI's gradient variance matches SGD conditions, with superior dimensional dependence for mean-field parameterization.
Cross-regularization adapts model complexity during training.
problem Manual tuning of model complexity for overfitting prevention.
method Directly adapts regularization parameters through validation gradients during training.
result Organic emergence of architecture-specific regularization during training.
ES and FD gradients converge as optimization dimension grows.
problem Understanding the relationship between Evolution Strategies and Finite Differences gradients.
method Analyzing the convergence of gradients as the optimization dimension increases.
result ES and FD gradients converge as the dimension of the vector under optimization increases.
Deep neural networks have enabled progress in a wide variety of applications. Growing the size of the neural network typically results in improved accuracy. As model sizes grow, the memory and compute requirements for training these models also increases. We introduce a technique to train deep neural networks using hal…
DPZero fine-tunes large models privately without backpropagation.
problem Memory and privacy challenges in fine-tuning large language models.
method DPZero uses zeroth-order methods for private fine-tuning, avoiding backpropagation.
result DPZero achieves private fine-tuning of RoBERTa and OPT on various tasks.
Recent work has established an empirically successful framework for adapting learning rates for stochastic gradient descent (SGD). This effectively removes all needs for tuning, while automatically reducing learning rates over time on stationary problems, and permitting learning rates to grow appropriately in non-stati…
GradaGrad adapts learning rate non-monotonically, overcoming AdaGrad's step size decrease.
problem Fixed learning rate in AdaGrad leads to step size decrease over time.
method Introduces GradaGrad, which grows or shrinks the learning rate based on a different accumulation in the denominator.
result GradaGrad achieves similar convergence rates as AdaGrad and demonstrates non-monotone adaptation.
New approach uses 'growth' and 'harvesting' concepts to improve deep learning models.
problem Current deep learning models lack transparency and high convergence rates.
method Reconsider neural networks as single-species population dynamics with balanced growth and harvesting rates.
result SGD with balanced growth and harvesting rates outperforms adaptive methods in all three requirements.
While robust parameter estimation has been well studied in parametric density estimation, there has been little investigation into robust density estimation in the nonparametric setting. We present a robust version of the popular kernel density estimator (KDE). As with other estimators, a robust version of the KDE is u…
New method stabilizes saddle-point optimization with unbounded gradients.
problem Stochastic saddle-point optimization faces instability due to large gradients.
method Proposes a regularization technique to stabilize iterates.
result Yields meaningful performance guarantees even with unbounded gradients.
Optimal gradient quantization reduces communication costs in distributed deep learning.
problem High communication costs in distributed training of deep neural networks.
method Deduced optimal gradient quantization conditions for binary and multi-level quantization, developed novel schemes for dynamic quantization levels.
result Demonstrated superior performance of proposed quantization schemes on CIFAR and ImageNet datasets.
Direct proof shows adaptive gradient descent converges near-linearly for convex functions.
problem Proving near-linear convergence of adaptive gradient descent for convex functions.
method Direct Lyapunov-based argument for convex functions with unique minimizer.
result Direct proof of near-linear convergence for convex functions.