A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
In 1963, Polyak proposed a simple condition that is sufficient to show a global linear convergence rate for gradient descent. This condition is a special case of the Łojasiewicz inequality proposed in the same year, and it does not require strong convexity (or even convexity). In this work, we show that this much-older…
Study on stochastic approximation with Polyak-Ruppert averaging for linear systems.
problem Understanding the asymptotic and non-asymptotic properties of stochastic approximation procedures.
method Detailed analysis of linear stochastic approximation with Polyak-Ruppert averaging, focusing on asymptotic and non-asymptotic properties.
result Proves CLT and non-asymptotic concentration inequality for averaged iterates, providing refined understanding of linear stochastic approximation.
Using the recent Gauss diagram formulas for Vassiliev invariants of Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality, we prove several inequalities for positive knots relating their Vassiliev invariants, genus and degrees of the Jones polynomial. As a consequence, we prove that for any of …
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.
We prove that the norm version of the adaptive stochastic gradient method (AdaGrad-Norm) achieves a linear convergence rate for a subset of either strongly convex functions or non-convex functions that satisfy the Polyak Lojasiewicz (PL) inequality. The paper introduces the notion of Restricted Uniform Inequality of Gr…
New adaptive scheduler improves SAM for better model training.
problem Training machine learning models requires selecting a learning rate, which is often difficult and time-consuming.
method Derive Polyak schedulers tailored to SAM-style updates, proving linear convergence for strongly convex objectives and an O(1/T) rate for convex objectives.
result Polyak schedulers achieve comparable or better performance than tuned SAM baselines, reducing the need for learning-rate tuning.
We describe the space of arrow diagram formulas for virtual knot diagrams in the annulus as the kernel of a linear map, inspired from a conjecture due to M. Polyak. As a main application, we slightly improve Grishanov-Vassiliev's theorem for planar chain invariants.
In this paper, it is shown that there are no nonconstant Goussarov-Polyak-Viro finite-type invariants that are invariant under the virtualization move. As an immediate corollary, we obtain the theorem which states none of the Birman coefficients of the Jones-Kauffman polynomial are of GPV finite type.
We describe the Polyak-Viro arrow diagram formulas for the coefficients of the Conway polynomial. As a consequence, we obtain the Conway polynomial as a state sum over some subsets of the crossings of the knot diagram. It turns out to be a simplification of a special case of Jaeger's state model for the HOMFLY polynomi…
We enhance the biquandle counting invariant using elements of truncated biquandle-labeled Polyak algebras. These finite type enhancements reduce to the finite type enhancements defined by Goussarov, Polyak and Viro for the trivial biquandle of one element and determine (but are not determined by) the biquandle counting…
We present a new method to produce simple formulas for 1-cocycles of knots over the integers, inspired by Polyak-Viro's formulas for finite-type knot invariants. We conjecture that these formulas always represent finite-type cohomology classes in the sense of Vassiliev. An example of degree 3 is studied, and shown to c…
Although it is known that the dimension of the Vassiliev invariants of degree three of long virtual knots is seven, the complete list of seven distinct Gauss diagram formulas have been unknown explicitly, where only one known formula was revised without proof. In this paper, we give seven Gauss diagram formulas to pres…
The paper analyzes time-dependent streaming data with biased gradient estimates and proposes improved stochastic optimization methods.
problem Stochastic optimization in a streaming setting with time-dependent and biased gradient estimates.
method Analysis of several first-order methods including SGD, mini-batch SGD, and time-varying mini-batch SGD, along with their Polyak-Ruppert averages.
result Time-varying mini-batch SGD methods can break long- and short-range dependence structures, and biased SGD methods can achieve comparable performance to their unbiased counterparts.