We consider the problem of solving a large-scale Quadratically Constrained Quadratic Program. Such problems occur naturally in many scientific and web applications. Although there are efficient methods which tackle this problem, they are mostly not scalable. In this paper, we develop a method that transforms the quadra…
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48 results for “Quadratic scaling”
Gaussian equivalence fails for simple polynomial embeddings in quadratic scaling RF models.
problem Failure of Gaussian equivalence in polynomial feature embeddings under quadratic scaling.
method Introduced Conditional Gaussian Equivalent (CGE) model to capture non-Gaussian behavior.
result Correct asymptotics derived for training and test errors in CGE model.
Study how generalization scales with model size and data in quadratic neural networks.
problem Understanding how generalization scales with model size and data in quadratic neural networks.
method Analyzed -regularized empirical test error minimization in a quadratic two-layer network with finite-sample setting and structured data.
result Revealed a phase diagram with distinct scaling regimes as the number of parameters varies, showing data-dependent power laws controlled by spectral structure of the target.
Paper presents an ADMM-based approach to efficiently integrate quadratic programming layers into neural networks.
problem Integrating quadratic programs into neural networks for optimization.
method An ADMM-based network layer architecture for solving quadratic programs efficiently.
result The ADMM layer is approximately an order of magnitude faster than existing methods for medium scaled problems.
New model captures asymmetric rough volatility with Zumbach effect.
problem Capturing asymmetric rough volatility and Zumbach effect.
method Proposes a bivariate QHawkes process to model asymmetric buying and selling actions.
result Derives a super-rough-Heston model preserving the Zumbach effect.
We study super--replication of European contingent claims in an illiquid market with insider information. Illiquidity is captured by quadratic transaction costs and insider information is modeled by an investor who can peek into the future. Our main result describes the scaling limit of the super--replication prices wh…
Study on SGD dynamics and scaling laws for training quadratic neural networks in high dimensions.
problem Optimizing and understanding the training dynamics of quadratic neural networks in high-dimensional settings.
method Sharp analysis of SGD dynamics, combining matrix Riccati differential equations and matrix monotonicity arguments.
result Derivation of scaling laws for prediction risk, highlighting power-law dependencies on optimization time, sample size, and model width.
We propose HAMSI (Hessian Approximated Multiple Subsets Iteration), which is a provably convergent, second order incremental algorithm for solving large-scale partially separable optimization problems. The algorithm is based on a local quadratic approximation, and hence, allows incorporating curvature information to sp…
Study on neural network dynamics in high dimensions with quadratic activation.
problem Understanding training dynamics in overparameterized neural networks.
method Derivation of gradient flow equations and analysis under l2-regularization.
result Characterization of estimator performance and spectral properties in the high-dimensional limit.
Global minima found for multidimensional scaling with penalties.
problem Finding global minima in multidimensional scaling.
method Combining stress loss function with a quadratic penalty term to find minimizers.
result Trajectory of minimizers leads to global minima.
We consider the problem of high-dimensional classification between the two groups with unequal covariance matrices. Rather than estimating the full quadratic discriminant rule, we propose to perform simultaneous variable selection and linear dimension reduction on original data, with the subsequent application of quadr…
Sharp Capacity Thresholds in Linear Associative Memory: From Winner-Take-All to Listwise Retrievalstat.ML
Linear memory stores associations up to a logarithmic scale, but listwise retrieval can handle a quadratic scale.
problem How many key-value associations can a linear memory store?
method Analyzed linear memory models for top-1 and listwise retrieval, proving phase transitions and developing asymptotic theories.
result Linear memory has a logarithmic capacity for top-1 retrieval and a quadratic capacity for listwise retrieval.
The study sets limits on how well systems can be controlled adaptively.
problem Learning to control unknown linear Gaussian systems with quadratic costs.
method Combining ideas from experiment design, estimation theory, and perturbation bounds of information matrices.
result Regret lower bounds of the order of in the time horizon accurately capture control-theoretic parameters.
Streaming Approach to Quadratic Covariation Estimation Using Financial Ultra-High-Frequency Dataq-fin.CP
A streaming algorithm estimates quadratic covariation from financial data efficiently.
problem Estimating quadratic covariation from ultra-high-frequency financial data with limited memory.
method Formulated multi-scale, realized kernel, pre-averaging, and modulated realized covariance estimators with fixed bandwidth.
result Fixed bandwidth estimators require higher bandwidth for positive semidefiniteness.
Study shows certainty equivalent policy minimizes regret in continuous-time systems.
problem Minimizing regret in continuous-time stochastic linear-quadratic systems.
method Theoretical analysis of randomized certainty equivalent policy.
result Establishes square-root of time regret bounds and linear scaling with parameters.
New algorithms achieve logarithmic regret in learning linear quadratic control systems.
problem Learning in Linear Quadratic Control systems with unknown parameters.
method Efficient algorithms for two scenarios: unknown or with certain conditions.
result Regret scales logarithmically with the number of steps, not square root.
This paper gives examples of explicit arbitrage-free term structure models with Lévy jumps via state price density approach. By generalizing quadratic Gaussian models, it is found that the probability density function of a Lévy process is a "natural" scale for the process to be the state variable of a market.
Proposes SPFB method for optimizing partition functions in stochastic learning.
problem Optimizing partition functions in stochastic learning settings.
method Stochastic Gradient Bound (SPFB) method based on upper-bounding the partition function with a quadratic surrogate.
result Sub-linear convergence rate of SPFB method and efficient training of deep learning models.
We reconsider the problem of optimal trading in the presence of linear and quadratic costs, for arbitrary linear costs but in the limit where quadratic costs are small. Using matched asymptotic expansion techniques, we find that the trading speed vanishes inside a band that is narrower than in the absence of quadratic …
Paper connects MoE and self-attention, proposing active-attention.
problem Improving efficiency and performance of self-attention mechanisms.
method Established connection between MoE and self-attention, analyzed quadratic gating functions, proposed active-attention mechanism.
result Active-attention outperforms standard self-attention in various tasks.
Enhances power of covariance matrix tests for high-dimensional data.
problem Testing large covariance matrices in high-dimensional data.
method Proposes a new Fisher's combined probability test for quadratic form and maximum form statistics.
result Boosts power against more general alternatives.
Motivated by electricity consumption metering, we extend existing nonnegative matrix factorization (NMF) algorithms to use linear measurements as observations, instead of matrix entries. The objective is to estimate multiple time series at a fine temporal scale from temporal aggregates measured on each individual serie…
New model-free algorithm achieves similar LQR regret guarantees.
problem Model-free control of linear dynamical systems under quadratic costs.
method Online policy gradient scheme with policy space cost analysis.
result Achieves regret scaling with √T, matching model-based methods.
New algorithm achieves logarithmic regret for adversarial online control.
problem Online linear-quadratic control in systems with adversarial disturbances.
method Characterization of optimal offline control law, reduced to online learning with approximate advantage functions.
result First algorithm with logarithmic regret for arbitrary adversarial disturbance sequences.
BBVI with STL converges geometrically under perfect specification, with quadratic variance bound.
problem Convergence rate of BBVI with STL estimator.
method Proved geometric convergence rate with quadratic variance bound for BBVI with STL estimator.
result BBVI with STL converges geometrically under perfect variational family specification.
We study the performance of the certainty equivalent controller on Linear Quadratic (LQ) control problems with unknown transition dynamics. We show that for both the fully and partially observed settings, the sub-optimality gap between the cost incurred by playing the certainty equivalent controller on the true system …
Improved kernel Stein discrepancy for large-scale data.
problem Efficiently testing probability distributions with kernel methods.
method Nyström approximation to reduce runtime complexity.
result Nyström-based KSD is -consistent and applicable for large datasets.
We study the problem of learning similarity functions over very large corpora using neural network embedding models. These models are typically trained using SGD with sampling of random observed and unobserved pairs, with a number of samples that grows quadratically with the corpus size, making it expensive to scale to…
We propose a novel end-to-end non-minimax algorithm for training optimal transport mappings for the quadratic cost (Wasserstein-2 distance). The algorithm uses input convex neural networks and a cycle-consistency regularization to approximate Wasserstein-2 distance. In contrast to popular entropic and quadratic regular…
Based on the new type of random walk process called the Potentials of Unbalanced Complex Kinetics (PUCK) model, we theoretically show that the price diffusion in large scales is amplified 2/(2 + b) times, where b is the coefficient of quadratic term of the potential. In short time scales the price diffusion depends on …
This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space satisfying integrability conditions on their first variation. Firstly, the study of pointwise power decay rates almost everywhere of the quadratic tilt-excess is completed by establishing the precise decay rate for two-di…
Globally solving the Gromov-Wasserstein problem for point clouds in low dimensional Euclidean spacesmath.OC
Paper solves Gromov-Wasserstein for point clouds efficiently.
problem Quantifying similarity between two formations or shapes.
method Reformulates QAP as low-rank concave quadratic optimization problem.
result Global solution for large-scale problems with thousands of points.
Quadratic discriminant analysis (QDA) is a standard tool for classification due to its simplicity and flexibility. Because the number of its parameters scales quadratically with the number of the variables, QDA is not practical, however, when the dimensionality is relatively large. To address this, we propose a novel p…
Proposes sparse QSVM for better generalization and interpretability.
problem Overfitting and difficulty in interpreting full quadratic classifiers.
method Enforces -norm constraint to promote sparsity and develops a penalty decomposition algorithm.
result The proposed model enhances generalization and produces sparse solutions.
Study utility indifference pricing with delayed investment information in a Bachelier model.
problem Investment decisions based on delayed information in a Bachelier model.
method Developed discrete-time duality and used techniques from [7] to compute scaling limits.
result Utility indifference prices scaling limit for vanishing delay with quadratic penalty.
New method distinguishes stochastic from deterministic signals using excursion counts.
problem Distinguishing between stochastic and deterministic signals in discrete time series.
method Excursion and crossing theorems for continuous semimartingales, comparing empirical excursion counts to theoretical expectation.
result A robust data-driven diffusion test that classifies signals based on log-log slope deviation.
Bayesian neural networks are shown to be minimax and admissible under certain conditions.
problem Optimality of Bayesian neural networks in deep learning models.
method Analysis of decision rules induced by BNNs in the normal location model under quadratic loss.
result A hyperprior on the effective output variance yields a minimax and admissible decision rule.
Unified framework for fast large-scale portfolio optimization.
problem Efficient portfolio optimization for large-scale financial data.
method Incorporates shrinkage and regularization techniques, addressing multiple objectives.
result AP-Trees and PCA-based factor models consistently outperform other approaches in out-of-sample portfolio performance.
We attempt to unveil the fine structure of volatility feedback effects in the context of general quadratic autoregressive (QARCH) models, which assume that today's volatility can be expressed as a general quadratic form of the past daily returns. The standard ARCH or GARCH framework is recovered when the quadratic kern…
New method for online inference of constrained optimization problems.
problem Online inference of constrained stochastic optimization problems.
method Random scaling of Sketched Stochastic Sequential Quadratic Programming (SSQP).
result Asymptotically valid confidence intervals and matrix-free computation.
A homogeneous nilpotent Lie group has a scaling automorphism determined by a grading of its Lie algebra. Many proofs of upper bounds for the Dehn function of such a group depend on being able to fill curves with discs compatible with this grading; the area of such discs changes predictably under the scaling automorphis…
Learning DAG or Bayesian network models is an important problem in multi-variate causal inference. However, a number of challenges arises in learning large-scale DAG models including model identifiability and computational complexity since the space of directed graphs is huge. In this paper, we address these issues in …
Semidefinite programs (SDP) are important in learning and combinatorial optimization with numerous applications. In pursuit of low-rank solutions and low complexity algorithms, we consider the Burer--Monteiro factorization approach for solving SDPs. We show that all approximate local optima are global optima for the pe…
Improved loss scaling for stochastic momentum algorithms in high dimensions.
problem Improving loss scaling for stochastic momentum algorithms in high dimensions.
method Dimension-adapted Nesterov acceleration (DANA) scales momentum hyperparameters based on model size and data complexity.
result DANA improves loss scaling exponents across various data and target complexities.
New Q-Newton's method avoids saddle points and converges quadratically.
problem Optimizing functions with saddle points and ensuring convergence guarantees.
method Modified New Q-Newton's method with Backtracking line search.
result Theorem for Morse functions: quadratic convergence to local minima.
Training of one-vs.-rest SVMs can be parallelized over the number of classes in a straight forward way. Given enough computational resources, one-vs.-rest SVMs can thus be trained on data involving a large number of classes. The same cannot be stated, however, for the so-called all-in-one SVMs, which require solving a …
A new robust and flexible classification method for non-Gaussian data.
problem Robustness to scale changes and non-Gaussian distributions in classical discriminant analysis.
method FEMDA uses arbitrary Elliptically Symmetrical distributions and scale parameters for each data point.
result FEMDA is robust to scale changes and outperforms other methods.
Mamba struggles with long context lengths, but spectrum scaling improves performance.
problem Mamba's performance degrades with increasing context length.
method Spectrum scaling applied to pre-trained Mamba models to improve long-context generalization.
result Spectrum scaling significantly improves performance in long-context settings.