arXiv research

Enhances biquandle counting invariant using finite type enhancements.

We present an alternative definition for the Goussarov--Habiro filtration of the Z-module freely generated by oriented integral homology 3-spheres, by means of Lagrangian-preserving homology handlebody replacements (LP-surgeries). Garoufalidis, Goussarov and Polyak proved that the graded space (G_n)_n associated to thi…

Goussarov, Polyak, and Viro proved that finite type invariants of knots are ``finitely multi-local'', meaning that on a knot diagram, sums of quantities, defined by local information, determine the value of the knot invariant. The result implies the existence of Gauss diagram combinatorial formulas for finite type inva…

Improved convergence for Polyak steps with momentum in smooth convex optimization.

Improved Sparse Polyak for high-dimensional M-estimation with sparser solutions.

Formulae for Vassiliev invariants derived from Kauffman polynomial.

Sparse Polyak improves high-dimensional statistical estimation.

Polyak-Ruppert CLT for SA-Adam with momentum and non-convergent adaptive preconditioning

Proposes a new adaptive learning rate for SGD.

The paper improves convergence for linear systems using entropic mirror descent with Polyak stepsizes.

Polyak's momentum accelerates training of neural networks.

In this article, we investigate various properties of the pure virtual braid group PV_3. From its canonical presentation, we obtain a free product decomposition of PV_3. As a consequence, we show that PV_3 is residually torsion free nilpotent, which implies that the set of finite type invariants in the sense of Goussar…

Polyak step size GD reaches final radius of convergence after log iterations.

New adaptive scheduler improves SAM for better model training.

We construct an infinite commutative lattice of groups whose dual spaces give Kauffman finite-type invariants of long virtual knots. The lattice is based "horizontally" upon the Polyak algebra and extended "vertically" using Manturov's functorial map $f$. For each $n$, the $n$-th vertical line in the lattice contains a…

Polyak-Łojasiewicz inequality simplifies linear convergence proofs.

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.

Improved SGD methods converge faster for nonconvex optimization.

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.

Study on stochastic approximation with Polyak-Ruppert averaging for linear systems.

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…

Paper provides seven Gauss diagram formulas for degree three long virtual knots.

New step-size methods improve SHB convergence for stochastic optimization.

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…

Formula for Milnor triple linking number in link diagrams with multiple crossings.

In the present paper, we study the finite type invariants of Gauss words. In the Polyak algebra techniques, we reduce the determination of the group structure to transformation of a matrix into its Smith normal form and we give the simplified form of a universal finite type invariant by means of the isomorphism of this…

TD(0) with Polyak-Ruppert averaging achieves robust and fast convergence rates

Proposes momentum methods for Lie groups, improving on classical algorithms.

Gradient descent with biased rounding errors converges faster under certain conditions.

The paper analyzes time-dependent streaming data with biased gradient estimates and proposes improved stochastic optimization methods.

Paper proves CLTs for Q-learning with asynchronous updates.

Study Q-learning with averaging for reinforcement learning, proving efficient inference and error bounds.

New streaming methods improve convergence rates for optimization problems.

Large deviations theory applied to policy gradient methods.

New formulas classify higher-dimensional knots and links.

The paper refines optimization algorithms using Lyapunov functions and differential equations.

Gradient method converges locally linearly for overparameterized Gaussian mixtures.

Minimal sets of moves for isotopic knots and trivalent graphs identified.

New SPS variant improves non-smooth optimization without small gradients.

Stochastic GD converges linearly for CV@R learning under certain conditions.

Paper extends Stanford's equivalence to virtual knots and proves equivalence to Goussarov-Polyak-Viro's n-equivalence.

Paper improves confidence intervals for LSA with multiplier bootstrap.

Gradient descent with large momentum finds flatter minima.

Gradient descent converges linearly for overparameterized linear networks.

Study of non-convex potential functions in deep learning with Poincaré inequality.

The paper offers precise bounds for averaged LSA iterates in linear systems.

Paper analyzes normal approximation for two-timescale stochastic algorithms, revealing interaction between fast and slow timescales.

Paper improves confidence set construction for SGD using multiplier bootstrap.