The study connects the average number of solutions to mixed volumes of convex bodies.
problem Finding a relationship between the average number of solutions to systems of equations and mixed volumes of convex bodies.
method Developed Banach metrics in vector spaces, constructed Banach convex bodies in the cotangent bundle of X, and calculated the average number of solutions as the mixed symplectic volume of these bodies. result The average number of solutions is equal to the mixed symplectic volume of Banach convex bodies.
Two-Tailed Averaging improves generalization by optimizing the number of leading iterates to ignore.
problem Improving generalization in stochastic optimization with limited resources and hyperparameters.
method An anytime adaptive algorithm that balances the number of leading iterates to ignore for better generalization.
result Approximates the optimal tail at all optimization steps, improving generalization without hyperparameters.
With this study we investigate the accuracy of deep learning models for the inference of Reynolds-Averaged Navier-Stokes solutions. We focus on a modernized U-net architecture, and evaluate a large number of trained neural networks with respect to their accuracy for the calculation of pressure and velocity distribution…
Optimizes RTB campaigns by selecting user profiles and website configurations.
problem Maximizing impressions and profitability in RTB campaigns.
method Optimizes user profiles and website configurations, combines with other strategies.
result As the required number of visits increases, average profitability decreases.
We introduce a new wavelet transform suitable for analyzing functions on point clouds and graphs. Our construction is based on a generalization of the average interpolating refinement scheme of Donoho. The most important ingredient of the original scheme that needs to be altered is the choice of the interpolant. Here, …
Although stochastic gradient descent (SGD) method and its variants (e.g., stochastic momentum methods, AdaGrad) are the choice of algorithms for solving non-convex problems (especially deep learning), there still remain big gaps between the theory and the practice with many questions unresolved. For example, there is s…
ACOWA improves distributed sparse classification with extra communication round.
problem Efficiently optimizing sparse classification with limited communication.
method Introducing ACOWA, a new technique with an extra communication round.
result ACOWA achieves better approximation quality and higher accuracy.
Two methods find at least two solutions to Kazdan-Warner's problem on surfaces.
problem Finding solutions to Kazdan-Warner's problem on two-dimensional surfaces.
method Direct method on convex sets and variational method of mountain pass.
result At least two solutions to the Kazdan-Warner's problem are found.
Low precision operations can provide scalability, memory savings, portability, and energy efficiency. This paper proposes SWALP, an approach to low precision training that averages low-precision SGD iterates with a modified learning rate schedule. SWALP is easy to implement and can match the performance of full-precisi…
New method for unbiased regression reduces excess risk.
problem Least squares regression with optimal solution and Hessian matrix.
method Averaged stochastic gradient descent with time-average estimator.
result Unbiased estimator with O(1/k) expected excess risk.
Nested model averaging improves high-dimensional linear regression performance.
problem High-dimensional linear regression with predictor ordering impact.
method Combining model averaging with regularized estimators on the solution path.
result Nested model averaging with lasso and SLOPE outperforms competing methods.
In this paper we investigate a new class of growth rate maximization problems based on impulse control strategies such that the average number of trades per time unit does not exceed a fixed level. Moreover, we include proportional transaction costs to make the portfolio problem more realistic. We provide a Verificatio…
New insights into optimal portfolios and ecological equilibria reveal surprising complexity.
problem Optimal portfolio construction with ecological constraints.
method Computational analysis of multispecies Lotka-Volterra equations with unit rank interaction matrices.
result Logarithm of the average number of solutions grows as \(N^{2/3}\), with most likely solutions being much smaller.
A common approach to statistical learning with big-data is to randomly split it among m machines and learn the parameter of interest by averaging the m individual estimates. In this paper, focusing on empirical risk minimization, or equivalently M-estimation, we study the statistical error incurred by this strategy…
The study calculates the average genus of 2-bridge knots based on their crossing numbers.
problem Determining the average genus of 2-bridge knots with a given crossing number.
method Analytical approach focusing on the properties of 2-bridge knots.
result Obtained the oblique asymptote of the average genus as crossing numbers increase.
We study least squares linear regression over N uncorrelated Gaussian features that are selected in order of decreasing variance. When the number of selected features p is at most the sample size n, the estimator under consideration coincides with the principal component regression estimator; when p>n, the esti…
The study calculates the average genus of rational knots and links.
problem Finding the average genus of rational knots and links.
method Enumerating and calculating the number of rational knots and links with a given crossing number.
result A precise formula for the average minimal genus of rational knots and links.
Study shows minimizing the norm of the ERM solution stabilizes kernel ridge-less regression.
problem Stability of kernel ridge-less regression.
method Minimizing the norm of the ERM solution to minimize CV stability.
result Interpolating solution with minimum norm minimizes CV stability.
The study calculates average crosscap numbers for 2-bridge knots.
problem Determining the average crosscap number of 2-bridge knots.
method Using continued fraction expansions and recursion, the study provides exact formulas for average crosscap numbers.
result The study shows that the limit of the average crosscap number of 2-bridge knots approaches zero as the crossing number increases.
Study optimizes decisions in real-time using inexact simulation solutions.
problem Real-time decision-making in simulation optimization with inexact solutions.
method Optimize then predict (OTP) approach, analyzing bias and variance in simulation-optimization algorithms.
result Unified analysis framework for OTP, establishing convergence rates and optimal allocation of computational budget.
This paper analyzes SGD with increasingly weighted averaging for optimization and generalization.
problem Improving optimization and generalization for non-strongly convex objectives.
method Comprehensive analysis of increasingly weighted averaging schemes for convex, strongly convex, and non-convex objectives.
result The weight α affects both optimization and generalization errors, revealing a trade-off. Average signature of 2-bridge knots approximates sqrt(2c/π).
problem Estimating the average signature and 4-genus of 2-bridge knots.
method Developed a model for 2-bridge knot diagrams indexed by crossing number, and used it to derive upper bounds for the average 4-genus.
result Upper bound for the average 4-genus of a 2-bridge knot is 9.75c/log c.
We present a geometric analysis of the incompressible averaged Euler equations for an ideal inviscid fluid. We show that solutions of these equations are geodesics on the volume-preserving diffeomorphism group of a new weak right invariant pseudo metric. We prove that for precompact open subsets of Rn, thi…
A novel framework for consensus clustering is presented which has the ability to determine both the number of clusters and a final solution using multiple algorithms. A consensus similarity matrix is formed from an ensemble using multiple algorithms and several values for k. A variety of dimension reduction techniques …
We determine the expected curvature polynomial of random real projective varieties given as the zero set of independent random polynomials with Gaussian distribution, whose distribution is invariant under the action of the orthogonal group. In particular, the expected Euler characteristic of such random real projective…
New model shows average genus of 2-bridge knots grows linearly with crossing number.
problem Understanding the growth of Seifert genus for 2-bridge knots.
method Billiard table model for 2-bridge knots.
result Average genus of a 2-bridge knot with crossing number c asymptotically approaches c/4 + 1/12.
SAA method solves insurance portfolio optimization with CVaR constraints.
problem Optimal allocation under CVaR constraint in insurance.
method Sample Average Approximation (SAA) method applied to CVaR constrained portfolio optimization.
result Convergence of SAA method and solution uniqueness proved under mild assumptions.
Partial model averaging improves Federated Learning performance.
problem Periodic model averaging causes significant model discrepancy in Federated Learning.
method Proposes a partial model averaging framework that encourages local models to stay close to each other.
result Partial averaging achieves up to 2.2% higher validation accuracy than full averaging.
We propose a fully distributed actor-critic algorithm approximated by deep neural networks, named \textit{Diff-DAC}, with application to single-task and to average multitask reinforcement learning (MRL). Each agent has access to data from its local task only, but it aims to learn a policy that performs well on average …
Under-parameterized networks can either copy or average teacher weights, leading to universal optimal solutions.
problem Approximating a teacher network with an under-parameterized student network.
method Analyzing shallow neural networks with erf activation function and unitary teacher weights, proving copy-average configurations are critical points and finding the optimal solution.
result The optimal solution for under-parameterized networks has a universal structure, whether copying or averaging teacher neurons.
Tail averaging consists in averaging the last examples in a stream. Common techniques either have a memory requirement which grows with the number of samples to average, are not available at every timestep or do not accomodate growing windows. We propose two techniques with a low constant memory cost that perform tail …
In distributed optimization and distributed numerical linear algebra, we often encounter an inversion bias: if we want to compute a quantity that depends on the inverse of a sum of distributed matrices, then the sum of the inverses does not equal the inverse of the sum. An example of this occurs in distributed Newton's…
We consider a composite convex minimization problem associated with regularized empirical risk minimization, which often arises in machine learning. We propose two new stochastic gradient methods that are based on stochastic dual averaging method with variance reduction. Our methods generate a sparser solution than the…
In this thesis, we consider the suitability of using the charged cold fluid model in the description of ultra-relativistic beams. The method that we have used is the following. Firstly, the necessary notions of kinetic theory and differential geometry of second order differential equations are explained. Then an averag…
Paper explores weighted averaging schemes for SGD, achieving asymptotic normality and optimality.
problem Improving convergence of SGD in various settings.
method Develops a general weighted averaging scheme for SGD and establishes asymptotic normality.
result Establishes asymptotic normality and optimality of weighted averaged SGD solutions.
The study of 2-bridge knots reveals a linear average braid index as crossing number increases.
problem Understanding the distribution of braid indices in 2-bridge knots.
method Analyzing the asymptotic behavior of braid indices for fixed crossing numbers.
result The average braid index of 2-bridge knots of crossing number c is asymptotically $ rac{c}{3}+ rac{11}{9}$. New method for sparse kernel selection improves prediction accuracy.
problem Sparse Multiple Kernel Learning for binary classification.
method Alternating best response algorithm with semidefinite relaxations.
result Method outperforms state-of-the-art MKL approaches in prediction accuracy.
Solves a game between brokers and informed traders using stochastic differential equations.
problem Optimizing wealth in a game between brokers and informed traders with private signals.
method Closed-form solutions to a mean-field game using forward-backward SDEs.
result Optimal trading strategies for both brokers and informed traders are found.
Average teaching complexity for locating target regions among halfspace intersections is Θ(d).
problem Teaching the location of a target region among intersections of halfspaces.
method Novel insights from computational geometry to count convex polytopes and faces.
result Average-case teaching complexity is Θ(d), contrasting with Θ(n) worst-case complexity.
Lower bounds on average genus of 2-bridge knots found.
problem Finding a lower bound on the average genus of 2-bridge knots.
method Developed a random model of 2-bridge knots, counted Seifert circles, and computed a lower bound.
result Computed a lower bound for the average Seifert genus of 2-bridge knots.
First-passage times in random walks have a vast number of diverse applications in physics, chemistry, biology, and finance. In general, environmental conditions for a stochastic process are not constant on the time scale of the average first-passage time, or control might be applied to reduce noise. We investigate mome…
We investigate a hybrid quantum-classical solution method to the mean-variance portfolio optimization problems. Starting from real financial data statistics and following the principles of the Modern Portfolio Theory, we generate parametrized samples of portfolio optimization problems that can be related to quadratic b…
Unified DNN-based precoder for MIMO networks with multiple objectives.
problem Optimizing data transmission, energy harvesting, and security in MIMO networks.
method Rotation-based precoding and DNN for multi-objective optimization.
result DNN-based precoder reduces computational complexity and achieves near-optimal performance.
In this paper we study the problem of minimizing the average of a large number (n) of smooth convex loss functions. We propose a new method, S2GD (Semi-Stochastic Gradient Descent), which runs for one or several epochs in each of which a single full gradient and a random number of stochastic gradients is computed, fo…
Most systems and learning algorithms optimize average performance or average loss -- one reason being computational complexity. However, many objectives of practical interest are more complex than simply average loss. This arises, for example, when balancing performance or loss with fairness across people. We prove tha…
Improved Compressed Sensing by optimizing sparse solutions with mixed integer programming.
problem Finding sparse solutions to linear measurements with numerical tolerance.
method Introducing an ℓ2 regularized formulation, reformulating as a mixed integer second order cone program, deriving a second order cone relaxation, and developing a custom branch-and-bound algorithm. result Our approach produces solutions that are on average 6.22% more sparse compared to state-of-the-art methods.
New moving average adapts weight dynamically based on polynomial and wavefunction.
problem Lagging traditional moving averages in adjusting to changes in data.
method Develops a moving average with weight as a polynomial of a wavefunction from an eigenproblem.
result Immediate 'switch' without lag, adapting to changes in data.
Federated learning enables a large amount of edge computing devices to jointly learn a model without data sharing. As a leading algorithm in this setting, Federated Averaging (\texttt{FedAvg}) runs Stochastic Gradient Descent (SGD) in parallel on a small subset of the total devices and averages the sequences only once …