Introduces gradient decay in Softmax for better generalization.
problem Improving generalization performance in neural networks.
method Gradient decay hyperparameter in Softmax for varying gradient rates based on probability.
result Gradient decay rate affects generalization performance and can be tuned for better optimization.
We show that the probability that a finitely supported random walk on a non-elementary subgroup of the the mapping class group gives a non-pseudo-Anosov element decays exponentially in the length of the random walk. More generally, we show that if R is a set of mapping class group elements with an upper bound on their …
Random knots generated from Chebyshev diagrams, showing probability decay.
problem Generating and analyzing random knots using Chebyshev diagrams.
method Assigning crossings randomly, defining reduction moves, calculating probabilities.
result Probability of any knot decays to zero as the number of crossings increases.
We study the tick dynamical behavior of the bond futures in Korean Futures Exchange(KOFEX) market. Since the survival probability in the continuous-time random walk theory is applied to the bond futures transaction, the form of the decay function in our bond futures model is discussed from two kinds of Korean Treasury …
In this paper we study the asymptotic decay of finite time ruin probabilities for an insurance company that faces heavy-tailed claims, uses predictable investment strategies and makes investments in risky assets whose prices evolve according to quite general semimartingales. We show that the ruin problem corresponds to…
The paper explores how the probability of default estimation changes with temporal correlation decay.
problem Difficulty in estimating the probability of default due to correlations between borrowers.
method Hierarchical Bayesian estimation using beta binomial distribution with temporal correlation.
result A phase transition occurs in the PD estimator, with convergence depending on the power decay index of temporal correlation.
Random walks on hyperbolic spaces show linear progress with exponential decay.
problem Understanding progress and decay in random walks on hyperbolic spaces.
method Analyzing random walks on separable, geodesic hyperbolic metric spaces with specific step distributions.
result Exponential decay in progress is extended to non-acylindrical actions.
The article studies knot distributions in petal diagrams and proves probabilities of specific knot types decay as the number of petals increases.
problem Understanding the probability of specific knot types in petal diagrams as the number of petals grows.
method Established properties of the randomized knot model, proving probabilities decay to zero, improved bounds on crossing number and petal number relationships.
result The n-petal model represents at least exponentially many distinct knots, with probabilities of specific knot types decaying as the number of petals increases.
We consider a random walk on the mapping class group of a surface of finite type. We assume that the random walk is determined by a probability measure whose support is finite and generates a non-elementary subgroup H. We further assume that H is not consisting only of lifts with respect to any one covering. Then w…
The goal of this paper is to prove a result conjectured in Föllmer and Schachermayer [FS07], even in slightly more general form. Suppose that S is a continuous semimartingale and satisfies a large deviations estimate; this is a particular growth condition on the mean-variance tradeoff process of S. We show that S then …
We found that factors decay over time, with momentum fitting best.
problem Understanding how factors decay over time and their impact on performance.
method Derived a hyperbolic decay model for factors, tested against linear and exponential alternatives.
result Momentum exhibits hyperbolic decay, outperforming linear and exponential models.
We consider the least-square linear regression problem with regularization by the ℓ1-norm, a problem usually referred to as the Lasso. In this paper, we first present a detailed asymptotic analysis of model consistency of the Lasso in low-dimensional settings. For various decays of the regularization parameter, w…
We consider the least-square linear regression problem with regularization by the l1-norm, a problem usually referred to as the Lasso. In this paper, we present a detailed asymptotic analysis of model consistency of the Lasso. For various decays of the regularization parameter, we compute asymptotic equivalents of the …
We present the results of computer experiments suggesting that the probability that a random multiword in a free group is virtually geometric decays to zero exponentially quickly in the length of the multiword. We then prove this fact.
Geometric step decay schedules improve stochastic algorithms' convergence on sharp nonconvex problems.
problem Convergence of stochastic algorithms on sharp nonconvex problems.
method Geometric step decay schedule applied to stochastic algorithms.
result Geometric step decay schedules lead to local linear convergence rates for sharp nonconvex problems.
Estimate arrival times in random recursive trees using iterated Jordan centralities.
problem Estimate arrival times in random recursive trees.
method Pointwise approach using iterated Jordan centralities.
result Tail bounds for relative estimation error.
Model for optimal execution with passive market impact.
problem Optimizing execution strategies in markets with passive price impact.
method Developed a mesoscopic model incorporating empirical price impact features.
result Obtained a passive impact rate that decays exponentially with quote distance.
New policy tackles evolving externalities in contextual bandits.
problem Difficulty in recovering from wrong decisions over time.
method Rejection-based policy to achieve low regret.
result Low regret achieved regardless of reward matrix structure.
We show that, for any (symmetric) finite generating set of the Torelli group of a closed surface, the probability that a random word is not pseudo-Anosov decays exponentially in terms of the length of the word.
Unified framework for analyzing gradient flows of measures with exponential decay of entropy.
problem Analyzing exponential decay of entropy functionals in gradient flows of measures.
method Characterization of global exponential decay behaviors using Hellinger-Kantorovich geometry, shape-mass decomposition, and Polyak-Łojasiewicz-type inequalities.
result Unified theoretical framework for gradient flows with complete analysis of exponential decay behaviors.
Study on crossing numbers of random two-bridge knots.
problem Understanding the distribution of crossing numbers in random two-bridge knots.
method Used billiard table diagrams to model random knots and derived a closed formula for their crossing numbers.
result Closed formula for the distribution of crossing numbers and exponential decay of knot appearance probability.
Link invariants fail to detect most links with high probability.
problem Detecting specific link types using invariants.
method Mathematical proof and big-data analysis.
result Link invariants have a zero probability of detecting alternating links.
New dimension concept for groups based on percolation probability.
problem Defining a new dimension for groups using percolation probability.
method Introducing percolation dimension pdim(G) for groups G using symmetric probability measures. result The percolation dimension pdim(G) has natural properties like monotonicity and coincides with growth rate exponents for various groups. In the Black-Scholes context we consider the probability distribution function (PDF) of financial returns implied by volatility smile and we study the relation between the decay of its tails and the fitting parameters of the smile. We show that, considering a scaling law derived from data, it is possible to get a new f…
Paper proposes efficient inference for hidden Markov models with memory decay.
problem Challenges in scalability due to dependencies in hidden Markov model observation data.
method Utilizes memory decay to carry out forward and backward probabilities with subsequences, enabling efficient inference over long sequences.
result Developed an efficient algorithm to numerically estimate the gap of top Lyapunov exponents, which determines the length of subsequences.
The paper analyzes multivariate Hawkes processes and their induced population processes.
problem Analyzing the time-dependent joint probability distribution of multivariate Hawkes processes.
method Exact and asymptotic analysis of general multivariate Hawkes processes and their induced population processes.
result Full characterization of the time-dependent joint transform of the multivariate population process and its intensity process.
The quotient of normal variables has power-law decay, applied to asset price tails.
problem Understanding the distribution of relative price changes in finance.
method Analyzing the quotient of normal distributions and deriving power-law decay densities.
result Relative price changes follow a power-law distribution with density f(x)≃f0x−2. Classic studies of the probability density of price fluctuations g for stocks and foreign exchanges of several highly developed economies have been interpreted using a {\it power-law} probability density function P(g)∼g−(α+1) with exponent values α>2, which are outside the Lévy-stable regime 0<α<2. …
In this paper, we discuss the Cramér-Lundberg model with investments, where the price of the invested risk asset follows a geometric Brownian motion with drift a and volatility σ>0. By assuming there is a cap on the claim sizes, we prove that the probability of ruin has at least an algebraic decay rate if $2a/σ^2 …
Study on random knots and their projections with varying decay rates of coefficients.
problem Understanding the probability of knot types with a given number of crossings.
method Defined random knots as random periodic functions with Gaussian coefficients, analyzed decay rates of coefficients, and examined projections.
result For certain decay rates of coefficients, the probability of forming a knot type with a given number of crossings decays at least as fast as \(1/N\).
Optimal tests for goodness of fit and two-sample problems using MMD and KSD.
problem Asymptotically optimal tests for goodness of fit and two-sample problems.
method Maximum Mean Discrepancy (MMD) and Kernel Stein Discrepancy (KSD) based tests.
result Optimal tests achieve the maximum exponential decay rate under specific conditions.
Lower bound on portfolio underperformance risk over time.
problem Minimizing risk of a portfolio underperforming a benchmark over long periods.
method Modelled prices of securities as geometric Brownian motions with nonlinear coefficients and economic factor modeled by Ito equation. Obtained a tight lower bound on underperformance probability.
result Lower bound on decay rate of underperformance probability is tight and can be achieved with epsilon-optimal portfolios under certain conditions.
New research shows fixed-budget best-arm identification cannot match static oracle performance.
problem Fixed-budget best-arm identification's performance limitations.
method Analysis of various adaptive and static algorithms for best-arm identification.
result For any algorithm, there exists at least one instance where the error decay rate is at most \((1 + \frac{\log(K)}{8})^{-1}\) times that of the static oracle.
We explain theoretically a curious empirical phenomenon: "Approximating a matrix by deterministically selecting a subset of its columns with the corresponding largest leverage scores results in a good low-rank matrix surrogate". To obtain provable guarantees, previous work requires randomized sampling of the columns wi…
Adaptive algorithm identifies best arm with abstention, showing phase transition from polynomial to exponential error probability.
problem Bayesian best-arm identification with abstention to reduce undetected error.
method Adaptive algorithm PGWS that optimally uses abstention budget.
result Introducing any positive abstention budget induces an exponential decay in undetected error probability.
Decentralized clustering and linking by networked agents learns and tracks models without prior knowledge.
problem Decentralized clustering and estimation over multi-task networks with unknown models and agents.
method Proposes a decentralized clustering algorithm integrating learning and clustering tasks.
result Error probabilities decay exponentially to zero with the step-size parameter.
Analyzes multifractality caused by fat-tailed distributions in time series.
problem Quantifying multifractality induced by fat-tailed distributions in time series data.
method Examines different types of fat-tailed distributions using Tsallis statistics and nonextensive analysis.
result Developed semi-analytical formulas to distinguish true multifractality from spurious multifractality.
Adversarial training effectiveness varies widely due to inconsistent training settings.
problem Variability in adversarial training effectiveness due to inconsistent training settings.
method Comprehensive evaluation of 10+ adversarial training methods and their hyperparameters.
result Basic training settings like weight decay can significantly impact adversarial robustness.
The study uses the Merton model to estimate PD and finds a phase transition affecting convergence speed.
problem Estimating the probability of default (PD) using limited historical data.
method Adopted the Merton model and analyzed phase transitions in default correlation.
result PD estimation converges slowly when temporal correlation decays by power law less than one.
Upper bounds on Wasserstein distance for empirical measures in unbounded functional spaces.
problem Analyzing convergence and concentration of empirical measures in unbounded functional spaces.
method Generalized upper bounds using Wasserstein distance, covering large dimensional Euclidean spaces and Gaussian processes.
result Rate-optimal upper bounds for functional data distributions with specific decay rates.
Paper tackles causal inference with partially labeled data, introducing robust methods.
problem Challenges in causal inference due to partially labeled datasets and potential bias.
method Decaying missing-at-random framework and BRSS estimator for doubly robust causal inference.
result Established asymptotic normality of BRSS estimator under decaying labeling propensity scores.
Financial markets provide an ideal frame for the study of crossing or first-passage time events of non-Gaussian correlated dynamics mainly because large data sets are available. Tick-by-tick data of six futures markets are herein considered resulting in fat tailed first-passage time probabilities. The scaling of the re…
HED Score improves temporal evaluation of detection accuracy.
problem Temporal agnosticism in existing evaluation frameworks for non-stationary processes.
method Measure-theoretic HED Score integrating exponentially decaying kernel over posterior probability stream.
result HED Score achieves 388.8% improvement over ROC/AUC on NSL-KDD benchmark.
Study on Wasserstein gradient flow for MMD between Coulomb measures.
problem Analyzing the long-time behavior of MMD between probability and target measures using Coulomb kernels.
method Existence of global weak solutions, ultracontractive estimate, regularity analysis, exponential decay proof, defective Polyak-Lojasiewicz inequality.
result Exponential decay of squared MMD toward a uniformly positive target measure on flat torus.
New bandit algorithm for non-i.i.d. noise, improving standard rates.
problem Linear stochastic bandit with non-i.i.d. observation noise.
method Developed new confidence sequences and an algorithm based on optimism in uncertainty.
result Regret bounds for the new algorithm, showing recovery of standard rates up to a factor of the mixing time.
Price fluctuations of commodities like cotton and wheat are thought to display probability distributions of returns that follow a Lévy stable distribution. Recent analysis of stocks and foreign exchange markets show that the probability distributions are not Lévy stable, a plausible result since commodity markets have …
Optimal trading strategy derived for nonlinear price impact models.
problem Optimal trading with nonlinear price impact induced by alpha signals.
method Variational approach, nonlinear Fredholm equation, iterative scheme.
result Existence and uniqueness of optimal trading strategy under monotonicity condition.
We investigate the probability distribution of the volatility return intervals τ for the Chinese stock market. We rescale both the probability distribution Pq(τ) and the volatility return intervals τ as Pq(τ)=1/τˉf(τ/τˉ) to obtain a uniform scaling curve for different threshold value q. The scali…