New method explains predictive uncertainty by focusing on second-order effects.
problem Explaining predictive uncertainty in machine learning models.
method CovLRP, CovGI, etc., based on second-order effects.
result Predictive uncertainty is dominated by second-order effects.
Negative step sizes improve second-order methods for neural networks.
problem Second-order methods discard negative curvature, limiting their effectiveness.
method Introduce negative step sizes in second-order methods combined with Wolfe line search.
result Negative step sizes lead to global convergence and improved performance.
AdamQLR optimizes Adam with K-FAC heuristics, achieving comparable performance to tuned benchmarks.
problem Improving the performance of Adam optimizers with stabilizing heuristics.
method Combining Adam's update directions with K-FAC's heuristics (damping and learning rate selection).
result Untuned AdamQLR can achieve comparable performance to tuned benchmarks.
Paper examines risk measure expansions under FGM dependence, improving accuracy at extreme levels.
problem Capturing higher-order tail behavior and dependence effects in risk measures.
method Second-order asymptotic expansions using extreme value theory and regular variation theory.
result Second-order approximations reduce approximation errors, especially at extreme confidence levels.
Exact second-order optimization for deep learning reduces computational cost and improves performance.
problem Inadequate use of second-order optimization methods in deep learning due to high computational cost and non-convexity.
method Developed an exact stochastic second-order Newton method that addresses the non-convexity issue and provides an expression for the stochastic Hessian.
result Exact second-order Newton direction formula and its application in deep learning datasets.
Paper proposes a second-order method for faster SVI convergence.
problem Poor convergence rate of first-order SVI algorithms.
method Derives Hessian matrix and implements two numerical schemes for efficient second-order SVI.
result Proposed approach achieves faster convergence compared to first-order SVI.
Proposes a new stock prediction method that accounts for market dynamics.
problem The dynamic nature of the stock market invalidates traditional machine learning assumptions.
method Develops a second-order learning paradigm with multi-scale patterns.
result Demonstrates effectiveness in stock prediction on real-world data.
Most of the existing approaches focus on specific visual tasks while ignoring the relations between them. Estimating task relation sheds light on the learning of high-order semantic concepts, e.g., transfer learning. How to reveal the underlying relations between different visual tasks remains largely unexplored. In th…
Paper introduces variance-based measures for second-order uncertainty quantification in classification problems.
problem Uncertainty in machine learning predictions and decision-making.
method Second-order uncertainty quantification using variance-based measures.
result Variance-based measures effectively quantify uncertainty on a class-based level and are competitive with entropy-based measures.
Natural gradient descent, which preconditions a gradient descent update with the Fisher information matrix of the underlying statistical model, is a way to capture partial second-order information. Several highly visible works have advocated an approximation known as the empirical Fisher, drawing connections between ap…
New method speeds up deep learning optimization.
problem Scalable second-order optimization for deep learning.
method Second-order optimization with algorithmic and numerical improvements.
result Significant convergence and wall-clock time improvements.
Second order Sobolev metrics on the space of regular unparametrized planar curves have several desirable completeness properties not present in lower order metrics, but numerics are still largely missing. In this paper, we present algorithms to numerically solve the initial and boundary value problems for geodesics. Th…
Multiscale stochastic volatility models have been developed as an efficient way to capture the principle effects on derivative pricing and portfolio optimization of randomly varying volatility. The recent book Fouque, Papanicolaou, Sircar and Sølna (2011, CUP) analyzes models in which the volatility of the underlying i…
New method solves stochastic optimization problems with random models.
problem Optimizing stochastic objectives with deterministic constraints.
method Trust-Region Sequential Quadratic Programming with random model.
result Global convergence guarantees for first- and second-order stationary points.
Classifies scalar second-order PDEs with low-dimensional symmetry groups.
problem Classifying differential equations with specific symmetry groups.
method Algebraic technique based on covariant form for constructing equations.
result Complete classification of quasi-linear scalar second-order PDEs with free symmetry groups of dimension ≤3.
PCA outperforms random projections in retaining second order signals from latent groups.
problem Preserving second order structure in latent groups under unsupervised linear projections.
method Theoretical framework and quasi-exhaustive enumeration of projections.
result PCA outperforms random projections in retaining second order signals across a broad range of data-generating parameters.
New technique debiases distributed optimization, improving convergence rate.
problem Bias in local estimates limits effectiveness of distributed second order optimization.
method Surrogate sketching and scaled regularization to eliminate bias.
result The debiased local estimates lead to faster convergence in distributed optimization.
The paper debiases mini-batch approximations in deep learning for more accurate optimization and uncertainty quantification.
problem Bias in mini-batch approximations distorts the shape of quadratic approximations used in deep learning.
method Developed and evaluated debiasing strategies for mini-batch approximations.
result Debiasing strategies improve the accuracy of second-order optimization and uncertainty quantification in deep learning.
Paper proposes a new method for efficient second-order neural network training.
problem Infeasibility of Hessian calculation and noisy second-order information in deep learning.
method Adopting complex-step directional derivative (CSFD) for accurate Hessian computation and designing an effective Newton Krylov procedure.
result Our method outperforms existing methods and often converges one-order faster.
Gradient-based meta-learning has proven to be highly effective at learning model initializations, representations, and update rules that allow fast adaptation from a few samples. The core idea behind these approaches is to use fast adaptation and generalization -- two second-order metrics -- as training signals on a me…
Stein variational gradient descent (SVGD) was recently proposed as a general purpose nonparametric variational inference algorithm [Liu & Wang, NIPS 2016]: it minimizes the Kullback-Leibler divergence between the target distribution and its approximation by implementing a form of functional gradient descent on a reprod…
Mini-Hes improves LFA model performance on HDI tasks with missing data.
problem Effective representation of high-dimensional, incomplete data for user behavior understanding.
method Proposes Mini-Hes, a parallelizable second-order LFA model using mini-block diagonal Hessian-free optimization.
result Mini-Hes outperforms state-of-the-art models in missing data estimation tasks on recommender system datasets.
For the system of second order quasilinear parabolic equations the problem of reducing them to the equations of diffusion type is considered. In non-degenerate case an effective algorithm for solving this problem is suggested.
With the rapid adoption of machine learning systems in sensitive applications, there is an increasing need to make black-box models explainable. Often we want to identify an influential group of training samples in a particular test prediction for a given machine learning model. Existing influence functions tackle this…
We propose a method to infer causal structures containing both discrete and continuous variables. The idea is to select causal hypotheses for which the conditional density of every variable, given its causes, becomes smooth. We define a family of smooth densities and conditional densities by second order exponential mo…
Kernel online convex optimization (KOCO) is a framework combining the expressiveness of non-parametric kernel models with the regret guarantees of online learning. First-order KOCO methods such as functional gradient descent require only O(t) time and space per iteration, and, when the only information on t…
ISAAC Newton uses input-based curvature for efficient training.
problem Efficient training in small-batch stochastic regimes.
method ISAAC Newton conditions gradients using selected second-order information based on input.
result Effective training even in small-batch stochastic regimes, competitive to first-order and second-order methods.
Value iteration is a fixed point iteration technique utilized to obtain the optimal value function and policy in a discounted reward Markov Decision Process (MDP). Here, a contraction operator is constructed and applied repeatedly to arrive at the optimal solution. Value iteration is a first order method and therefore …
A geometric reformulation of the martingale problem associated with a set of diffusion processes is proposed. This formulation, based on second order geometry and Ito integration on manifolds, allows us to give a natural and effective definition of Lie symmetries for diffusion processes.
Detects adversaries in crowdsourcing to improve accuracy.
problem Adversaries can skew results in crowdsourced classification.
method Analyzes second-order moments of annotator responses to identify and mitigate adversaries.
result The approach can identify and mitigate adversaries in crowdsourcing tasks.
Enhanced indexation with sector constraints using SSD for better portfolio performance.
problem Constructing a portfolio that outperforms a market index while respecting sector investment proportions.
method Subset second-order stochastic dominance (subset SSD) applied to asset subset constraints.
result Subset SSD approach outperforms S&P500 and standard SSD approaches.
New method for analyzing elliptic and parabolic equations.
problem Analyzing elliptic and parabolic equations.
method Level set version of partial uniform ellipticity.
result Effective approach to investigate equations.
Second-order economic theory considers new variables to improve price volatility predictions.
problem Current economic models focus on first-order variables, missing second-order variables that affect price volatility.
method Introduces second-order economic theory with new variables composed of sums of squares of agents' transactions.
result Second-order economic theory complements first-order variables and introduces new macroeconomic variables.
This paper uses second-order Poincaré inequalities to establish quantitative central limit theorems for Gaussian neural networks.
problem Establishing quantitative central limit theorems for Gaussian neural networks.
method Using second-order Poincaré inequalities to reduce the problem to computing the gradient and Hessian of the NN's output.
result Suboptimal rates of convergence for the NN's output due to the use of second-order Poincaré inequalities.
Improved algorithm finds second-order stationary points in non-convex optimization.
problem Minimizing non-convex objectives while preserving training data privacy.
method SpiderBoost framework with two gradient oracles: precise and less precise.
result Improved rates for finding second-order stationary points.
Clustering is fundamental for gaining insights from complex networks, and spectral clustering (SC) is a popular approach. Conventional SC focuses on second-order structures (e.g., edges connecting two nodes) without direct consideration of higher-order structures (e.g., triangles and cliques). This has motivated SC ext…
Second-order optimizers retain residual information after data deletion, affecting machine unlearning.
problem Residual information in second-order optimizers after data deletion.
method Comparison of first-order and second-order learners, eigendecomposition analysis.
result Second-order optimizers retain residual information, not detectable by first-order analysis.
Paper optimizes trading strategies by creating shadow prices for markets with transaction costs.
problem Optimizing trading strategies in markets with transaction costs.
method Developed shadow prices to simplify optimization into a frictionless market, considering second-order transaction costs.
result Alternative strategies outperform shadow prices for risk aversion different from one.
LIBS2ML is a library based on scalable second order learning algorithms for solving large-scale problems, i.e., big data problems in machine learning. LIBS2ML has been developed using MEX files, i.e., C++ with MATLAB/Octave interface to take the advantage of both the worlds, i.e., faster learning using C++ and easy I/O…
SOLBP extends efficient inference to uncertain Bayesian networks.
problem Inference in uncertain Bayesian networks with second-order probabilities.
method Extends Loopy Belief Propagation to second-order Bayesian networks.
result Generates inferences consistent with sum-product networks, more efficient and scalable.
Developed a theory of local convexity for second order differential equations on Lie algebroids.
problem Analyzing convexity in differential equations on Lie algebroids.
method Theory development for local convexity of SODEs on Lie algebroids.
result Extensive discussion of homogeneous quadratic SODEs on Lie algebroids.
Introduces intrinsic Riemannian cross-covariance for manifold-valued random objects.
problem Covariance estimation for random objects on Riemannian manifolds.
method Defines covariance and correlation via parallel transport.
result Proposed covariance is independent of coordinate choices.
Simplified argument for second order estimate in quaternionic Calabi-Yau problem.
problem Second order estimates for quaternionic Calabi-Yau problem on hyperkähler manifolds.
method Simplified argument to derive the estimate.
result Simplified derivation of second order estimate.
Paper studies second order tail probabilities in risk models.
problem Analyzing tail probabilities in risk models with constant interest force.
method Asymptotic expansion and weighted Kesten-type inequality for second order subexponential random variables.
result Second order asymptotic formulae for continuous-time renewal risk models are derived.
A dynamical system on the total space of the fibre bundle of second order accelerations, T2M, is defined as a third order vector field S on T2M, called semispray, which is mapped by the second order tangent structure into one of the Liouville vector field. For a regular Lagrangian of second order we prove that …
Paper studies second order symmetric parallel tensors in generalized f.pk-space forms.
problem Exploring properties of second order symmetric parallel tensors in generalized f.pk-space forms.
method Analyzes the properties of second order symmetric parallel tensors and deduces the existence or non-existence of certain tensors and hypersurfaces.
result There does not exist second order skew-symmetric parallel tensor in f.pk-space form. There is no parallel hypersurface in a generalized f.pk-space form but there is semi-parallel hypersurface.
The paper proposes a GP-based method for discovering second-order particle dynamics models.
problem Discovering a general second-order particle-based model for agent interactions.
method Gaussian Process-based approach integrating two independent GP priors on latent interaction kernels.
result The method learns effective nonlinear dynamics representations from small data sets.
We study adaptive (or online) nonlinear regression with Long-Short-Term-Memory (LSTM) based networks, i.e., LSTM-based adaptive learning. In this context, we introduce an efficient Extended Kalman filter (EKF) based second-order training algorithm. Our algorithm is truly online, i.e., it does not assume any underlying …