ASTRA improves TDA by more accurately approximating iHVP.
problem Improving insights into training data attribution.
method ASTRA uses EKFAC-preconditioner on Neumann series iterations to accurately approximate iHVP.
result Improving iHVP approximation significantly improves TDA performance.
New SGD algorithm finds critical points faster with second-order corrections.
problem Finding critical points in non-convex optimization efficiently.
method Uses Hessian-vector products to correct momentum bias in SGD.
result Finds ε-critical points in O(ε−3) time. HessFormer enables distributed Hessian computation for large models.
problem Limited Hessian computation for large deep learning models.
method Distributed stochastic Lanczos quadrature algorithm.
result Investigated Hessian spectral density of a 70 billion parameter model.
The Hessian-vector product has been utilized to find a second-order stationary solution with strong complexity guarantee (e.g., almost linear time complexity in the problem's dimensionality). In this paper, we propose to further reduce the number of Hessian-vector products for faster non-convex optimization. Previous a…
A new optimisation method efficiently scales Hessian-vector products for neural networks.
problem Challenges in applying second-order quasi-Newton methods due to large Hessian and non-convexity.
method Proposes an optimisation algorithm that asymptotically uses the exact inverse Hessian with modified eigenvalues.
result Demonstrates scalability and comparable performance to other optimisation methods in neural networks.
Stochastic Variance-Reduced Cubic regularization (SVRC) algorithms have received increasing attention due to its improved gradient/Hessian complexities (i.e., number of queries to stochastic gradient/Hessian oracles) to find local minima for nonconvex finite-sum optimization. However, it is unclear whether existing SVR…
WoodFisher improves neural network compression efficiency and accuracy.
problem Efficiently estimating inverse Hessian for neural network optimization.
method WoodFisher: a method to compute a faithful and efficient estimate of the inverse Hessian.
result WoodFisher significantly outperforms state-of-the-art methods for pruning neural networks.
We propose a reduction for non-convex optimization that can (1) turn an stationary-point finding algorithm into an local-minimum finding one, and (2) replace the Hessian-vector product computations with only gradient computations. It works both in the stochastic and the deterministic settings, without hurting the algor…
A new algorithm reduces communication rounds for distributed convex optimization.
problem Efficiently solving convex optimization problems in distributed systems.
method Proposes a stochastic Newton algorithm for homogeneous distributed stochastic convex optimization.
result Reduces the number and frequency of communication rounds compared to existing methods.
Improves understanding of neural network predictions using influence functions.
problem Challenges in understanding neural network predictions.
method Utilized NTK theory to calculate influence functions for over-parameterized neural networks.
result Proved that the approximation error of IF can be arbitrarily small in the over-parameterized regime.
Establishes statistical and computational bounds for influence diagnostics.
problem Identifying influential datapoints or subsets in machine learning models.
method Finite-sample statistical bounds and computational complexity for influence functions and approximate maximum influence perturbations.
result Established statistical and computational guarantees for influence diagnostics.
This paper proposes a stochastic variant of a classic algorithm---the cubic-regularized Newton method [Nesterov and Polyak 2006]. The proposed algorithm efficiently escapes saddle points and finds approximate local minima for general smooth, nonconvex functions in only O~(ε−3.5) stochastic gradien…
HTE improves PINNs for high-dimensional, high-order PDEs by reducing computational cost and memory usage.
problem Challenges in solving high-dimensional, high-order PDEs with PINNs due to computational cost and memory constraints.
method Introduces Hutchinson Trace Estimation (HTE) to transform Hessian matrix calculations into Hessian vector products (HVP), reducing computational cost and memory usage.
result HTE significantly reduces memory consumption and computational cost, enabling faster and more efficient solution of high-dimensional and high-order PDEs.
New method computes affine normal directions efficiently for sparse polynomials.
problem Computing affine normal directions is computationally expensive in high dimensions.
method Reduces third-order tensor contraction to matrix-free formulation using log-determinant gradient.
result Scalable implementations with near-linear scaling in dimension and sparsity.
New method for efficient sketching of gradients and Hessians.
problem Memory constraints in training machine learning models.
method A novel framework for scalable gradient and HVP sketching tailored for modern hardware.
result Theoretical guarantees and practical applications in training data attribution and Hessian spectrum analysis.
This paper proposes a family of online second order methods for possibly non-convex stochastic optimizations based on the theory of preconditioned stochastic gradient descent (PSGD), which can be regarded as an enhance stochastic Newton method with the ability to handle gradient noise and non-convexity simultaneously. …
How can we explain the predictions of a black-box model? In this paper, we use influence functions -- a classic technique from robust statistics -- to trace a model's prediction through the learning algorithm and back to its training data, thereby identifying training points most responsible for a given prediction. To …
The question of how to incorporate curvature information in stochastic approximation methods is challenging. The direct application of classical quasi- Newton updating techniques for deterministic optimization leads to noisy curvature estimates that have harmful effects on the robustness of the iteration. In this paper…
We present a novel statistical inference framework for convex empirical risk minimization, using approximate stochastic Newton steps. The proposed algorithm is based on the notion of finite differences and allows the approximation of a Hessian-vector product from first-order information. In theory, our method efficient…
Score matching is a popular method for estimating unnormalized statistical models. However, it has been so far limited to simple, shallow models or low-dimensional data, due to the difficulty of computing the Hessian of log-density functions. We show this difficulty can be mitigated by projecting the scores onto random…
New preconditioners speed up SGD on Lie groups.
problem Accelerating SGD convergence on Lie groups.
method Online updating of low-rank and matrix-free preconditioners.
result Preconditioners maintain symmetry and invariance.
New algorithm finds approximate stationary points in non-convex optimization.
problem Finding approximate stationary points in non-convex stochastic optimization.
method Design of an algorithm using O(ε−3) stochastic gradient and Hessian-vector products. result Optimal rate of O(ε−3) for finding ε-approximate stationary points, matching lower bounds. A new unbiased Hessian estimator for expectation-based objectives.
problem Estimating Hessian for objectives with non-reparameterizable nodes.
method GO Hessian estimator for expectation-based objectives.
result GO Hessian provides unbiased and low-variance estimation of Hessian.
This paper tackles efficient optimization for nonlinear embeddings in similarity learning.
problem Learning similarity with nonlinear embeddings is challenging due to the large number of pairs.
method Detailed derivations and efficient optimization methods for nonlinear embeddings are developed.
result Efficient optimization methods for nonlinear embeddings are shown to be highly effective.
The paper studies efficient Hessian fitting methods for stochastic optimization.
problem Efficient Hessian fitting for stochastic optimization.
method Preconditioned Stochastic Gradient Descent (PSGD) method and Lie groups.
result Hessian fitting problem is strongly convex in certain Lie groups.
Tensor decomposition, a collection of factorization techniques for multidimensional arrays, are among the most general and powerful tools for scientific analysis. However, because of their increasing size, today's data sets require more complex tensor decomposition involving factorization with multiple matrices and dia…
The paper analyzes convergence rates of bilevel optimization algorithms and introduces a new stochastic algorithm.
problem Nonconvex-strongly-convex bilevel optimization problems in machine learning.
method Comprehensive convergence rate analysis for deterministic bilevel optimization using AID and ITD, and a novel stochastic algorithm stocBiO.
result Theoretical convergence rates for AID and ITD methods, and stocBiO's superior performance.
Progress in deep learning is slowed by the days or weeks it takes to train large models. The natural solution of using more hardware is limited by diminishing returns, and leads to inefficient use of additional resources. In this paper, we present a large batch, stochastic optimization algorithm that is both faster tha…
We analyze the Hessian spectra of large models up to 100B parameters.
problem Accurate Hessian spectra of large foundation models are difficult to obtain.
method We use shard-local finite-difference Hessian vector products and stochastic Lanczos quadrature.
result We produce the first large-scale spectral density estimates of foundation models.
New pruning method captures global correlations for efficient neural network inference.
problem Efficiently pruning neural networks for faster inference and reduced memory usage.
method Second-order structured pruning (SOSP-H) with innovative saliency-based approaches.
result SOSP-H scales to large-scale vision tasks and improves accuracy without compromising efficiency.
New algorithms tackle complex multi-block optimization problems in machine learning.
problem Non-convex multi-block bilevel optimization with hierarchical sampling challenges.
method Blockwise stochastic variance-reduced methods with parallel speedup.
result Achieves matching complexity to single-block problems with parallel speedup.
We present the first public release of our generic neural network training algorithm, called SkyNet. This efficient and robust machine learning tool is able to train large and deep feed-forward neural networks, including autoencoders, for use in a wide range of supervised and unsupervised learning applications, such as…
Influence functions help study large language model generalization, revealing surprising decay patterns.
problem Understanding and mitigating risks in large language models (LLMs).
method Eigenvalue-corrected Kronecker-Factored Approximation (EK-FAC) to scale influence functions to LLMs.
result Influences decay to near-zero when key phrases order is flipped, revealing a surprising limitation.
Study geometric properties of loss functions to understand neural network performance.
problem Understanding the geometric properties of high-dimensional loss functions to improve neural network performance.
method Combine concepts from high-dimensional probability and differential geometry to study curvature properties in lower-dimensional loss representations.
result Mean curvature in the original loss space determines if saddle points appear as minima, maxima, or flat regions.
Machine learning (ML) problems are often posed as highly nonlinear and nonconvex unconstrained optimization problems. Methods for solving ML problems based on stochastic gradient descent are easily scaled for very large problems but may involve fine-tuning many hyper-parameters. Quasi-Newton approaches based on the lim…
New algorithms solve complex minimax problems efficiently.
problem Nonconvex-strongly concave minimax problems in machine learning.
method Gradient norm regularized trust-region (GRTR) and Levenberg-Marquardt (LMNegCur) algorithms.
result Proved iteration complexities matching best known results.
New algorithm achieves linear speedup in non-i.i.d. federated bilevel learning.
problem Linear speedup in convergence for non-i.i.d. datasets in federated bilevel optimization.
method Proposes FedMBO with a novel client sampling scheme for non-i.i.d. datasets.
result Achieves a convergence rate of O(1/√(nK) + 1/K + √n/K³/²).
New Hessian-free method improves bilevel optimization for meta-learning.
problem Efficiently solving bilevel optimization problems with limited second-order information.
method Proposes a new Hessian-free method that approximates the response Jacobian matrix via optimization path differences.
result Demonstrates superior performance on meta-learning tasks compared to baseline methods.
The paper studies a new type of submanifolds in product spaces.
problem Characterizing and understanding warped product pointwise bi-slant submanifolds.
method Introduced and studied warped product pointwise bi-slant submanifolds of locally product Riemannian manifolds.
result Characterization results and non-trivial examples of these submanifolds.
The paper extends affine connection results to singular warped and twisted products.
problem Generalizing affine connections to singular warped and twisted products.
method Study of singular multiply warped products and singular twisted products with semi-symmetric metric and non-metric connections, discussing Koszul forms and curvature.
result Theoretical results on curvature and Koszul forms for singular multiply warped and twisted products.
Defines products for fibered corners manifolds, generalizing resolutions.
problem Resolving fibered corners manifolds.
method Introduces a category of fibered corners manifolds with products and transverse fiber products, defining the 'ordered product' for wedge metrics.
result The 'ordered product' is a natural product for wedge metrics.
Productivity and credit limits affect aggregate production in non-monotonic ways.
problem Understanding how aggregate production is influenced by individual characteristics and financial constraints.
method Analytical proof of non-monotonic effects of productivity and credit limits on aggregate production in a general equilibrium model.
result Equilibrium aggregate production can be non-monotonic in both individual productivity and credit limit.
Einstein metrics on products are shown to be warped.
problem Characterizing Einstein metrics on conformal products.
method Proving Einstein metrics on conformal products are warped products under natural geometric conditions.
result Einstein metrics on conformal products are proven to be warped products.
Uniform criterion for vanishing products in bounded cohomology.
problem Vanishing of cup products and Massey products in bounded cohomology.
method Uniform vanishing criterion for products in bounded cohomology.
result Reproved and extended previous vanishing results.
The paper studies affine connections on singular warped products and their curvature.
problem Analyzing affine connections on singular warped products.
method Introducing semi-symmetric metric and non-metric Koszul forms, and expressing their curvature in terms of factor manifolds.
result Generalized results for singular multiply warped products.
We refine the intersection product in homology to an equivariant setting, which unifies several known constructions. As an application, we give a common generalisation of the Chas-Sullivan string product on a manifold and the Chataur-Menichi string product on the classifying space by defining a string product on the Bo…
New method compares geometric and standard cup products.
problem Reconciling partially defined and fully defined cochain products.
method Vector field flow through cubulation to compare products.
result Explicit cochain level comparison between intersection and cup products.
The paper examines Einstein doubly warped product manifolds with a semi-symmetric metric connection.
problem Characterizing Einstein doubly warped product manifolds with a semi-symmetric metric connection.
method Deriving curvature formulas and proving necessary and sufficient conditions for a manifold to be a warped product.
result Obtained results for Einstein doubly warped product manifolds and Einstein-like doubly warped product manifolds.