New RL method learns to skip states in linearly qπ-realizable MDPs, simplifying to linear MDPs.
problem Online RL in episodic MDPs with linearly qπ-realizable action-values. method Derives a novel algorithm that learns to skip states and applies a linear MDP algorithm.
result First polynomial-sample-complexity online RL algorithm for linearly qπ-realizable MDPs. TensorPlan algorithm finds δ-optimal policies with poly(H,d) queries under linearly realizable state-value function.
problem Efficient planning in MDPs with linearly realizable state-value function.
method TensorPlan algorithm using poly((dH/δ)A) simulator queries. result First algorithm with polynomial query complexity using only linear-realizability of a single competing value function.
Paper tackles sample-efficient RL for linearly realizable MDPs with limited revisiting.
problem Sample-efficient reinforcement learning for linearly realizable MDPs with limited revisiting.
method Develops a new sampling protocol that allows for backtracking and revisiting states in a controlled manner.
result Achieves polynomial sample complexity scaling with feature dimension, horizon, and inverse sub-optimality gap.
There are known infinite families of Brieskorn homology 3-spheres which can be realized as boundaries of smooth contractible 4-manifolds. In this paper we show that free periodic actions on these Brieskorn spheres do not extend smoothly over a contractible 4-manifold. We give a new infinite family of examples in which …
New method efficiently evaluates policies using trajectory data.
problem Statistically efficient policy evaluation with limited data.
method Trajectory-based approach for policy evaluation.
result Improved sample complexity for policy evaluation.
New research shows exponential lower bounds for planning in MDPs with linearly-realizable optimal action-value functions.
problem Determining the minimum number of queries needed for sound planners in MDPs with linear function approximation.
method Analyzing fixed-horizon and discounted MDPs with a generative model, showing lower bounds on the number of queries required.
result Sound planners need at least exponential number of queries in both fixed-horizon and discounted settings.
This paper examines number theoretic and topological properties of fully augmented pretzel link complements. In particular, we determine exactly when these link complements are arithmetic and exactly which are commensurable with one another. We show these link complements realize infinitely many CM-fields as invariant …
New RL difficulty shown for discounted settings.
problem Difficulty in reinforcement learning with discounted rewards.
method Adapted Wang et al. (2020) construction to 2-state MDP.
result Learning impossible even with infinite data in discounted setting.
TensorPlan shows an exponential lower bound for planning in MDPs with linearly realizable value functions.
problem Finding an exponential lower bound for planning in MDPs with linearly realizable value functions.
method TensorPlan and a few action lower bound approach.
result An exponentially large lower bound is shown for planning in MDPs with linearly realizable value functions.
Sine activation functions enable two-layer neural networks to learn modular addition more efficiently.
problem Learning modular addition with two-layer neural networks.
method Introduced and analyzed sine activation functions, providing theoretical and empirical evidence.
result Sine activation functions allow for constant-width network realizations of modular addition, whereas ReLU networks require linear width scaling.
Multi-group learners suffer a penalty in transductive learning.
problem The penalty on multi-group learners in transductive learning.
method Analyzing the relationship between the number of groups and the error rate.
result The penalty can increase linearly with the number of groups, up to the square-root of the sample size.
The paper classifies groups that can be isometry groups of infinite-genus hyperbolic surfaces.
problem Which groups can be realized as isometry groups of infinite-genus hyperbolic surfaces?
method Classification of isometry groups for infinite-genus 2-manifolds with no planar ends.
result There is an uncountable class of 2-manifolds where every countable group can be realized as an isometry group.
We present a short exposition of the following results by S. Parsa. Let L be a graph such that the join L∗{1,2,3} (i.e. the union of three cones over L along their common bases) piecewise linearly (PL) embeds into R4. Then L admits a PL embedding into R3 such that any two disjoint cycles…
New framework tackles stochastic latent subgroup heterogeneity in online decision-making.
problem Stochastic latent heterogeneity in online decision-making where individual responses vary with unobserved subgroups.
method Latent heterogeneous bandit framework using EM-greedy algorithm to learn subgroup probabilities and reward parameters.
result Achieves optimal estimation and classification guarantees, revealing a fundamental stochastic barrier in online decision-making.
The paper addresses rigid alignment of noisy patches, providing a polynomial time algorithm and convergence conditions.
problem Finding a rigid alignment of overlapping local views (patches) that minimizes alignment error in a noisy setting.
method Characterizes non-degeneracy based on kernel and positivity of a matrix, provides polynomial time algorithm for testing non-degeneracy, and uses Riemannian gradient descent for alignment.
result The algorithm converges locally linearly to a non-degenerate perfect alignment under certain conditions.
New method certifies deep graph classifiers with tighter risk bounds.
problem Certifying the reliability of deep graph classifiers.
method Linearized deep assignment flows with random initial conditions, using PAC-Bayes risk certification.
result Computes tighter out-of-sample risk certificates efficiently.
This is the second paper of a series dedicated to the study of Poisson structures of compact types (PMCTs). In this paper, we focus on regular PMCTs, exhibiting a rich transverse geometry. We show that their leaf spaces are integral affine orbifolds. We prove that the cohomology class of the leafwise symplectic form va…
New study shows exponential lower bound for RL even with constant suboptimality gap.
problem Can RL be sample-efficient with a constant suboptimality gap?
method Analyzes reinforcement learning in the online setting with a linearly realizable optimal Q-function.
result An exponential sample complexity lower bound still holds even with a constant suboptimality gap.
Paper explores how unsupervised learning can be understood through linear algebra concepts.
problem Understanding unsupervised learning through linear algebra concepts.
method Introducing the concept of linearly independent populations and using them to solve for prevalence values.
result Unsupervised learning can be realized as a generalization of supervised learning.
Bayesian approach for solving systems of linear PDEs with boundary conditions.
problem Modeling data efficiently with prior knowledge from systems of linear PDEs.
method Construct multi-output Gaussian process priors using Gröbner bases and pullback parametrizations.
result Gaussian process priors can represent solutions to systems of linear PDEs adhering to boundary conditions.
Study on singular points of translation surfaces under linearly dependent conditions.
problem Investigate singular points of translation surfaces under linearly dependent conditions.
method Use theories of generalised framed surfaces and framed surfaces.
result Introduce translation generalised framed surfaces and investigate their singular points.
We propose a new method for blind system identification. Resorting to a Gaussian regression framework, we model the impulse response of the unknown linear system as a realization of a Gaussian process. The structure of the covariance matrix (or kernel) of such a process is given by the stable spline kernel, which has b…
In this paper we study a Ricci-Hessian type manifold (M,g,φ,f,λ) which is closely related to the construction of almost Ricci soliton realized as a warped product. We classify certain classes of the Ricci-Hessian type manifolds and derive some implications for almost Ricci solitons and generalized m--qu…
We consider computational complexity of problems related to the fundamental group and the first homology group of (embeddable) 2-complexes. We show, as an extension of an earlier work, that computing first homology of 2-complexes is equivalent in computational complexity to matrix diagonalization. That is, the usua…
Improved LOO cross-validation for function approximation.
problem Estimating the Integrated Squared Error (ISE) for function approximation.
method Weighted Leave-One-Out cross-validation based on Gaussian Process.
result Significantly more precise ISE estimation compared to unweighted LOO.
Trajectory data suffices for efficient RL in linear MDPs.
problem Efficient offline RL in linear MDPs without state space scaling.
method Linear MDP approximation and trajectory data.
result Trajectory data suffices for deriving ε-optimal policies.
Universal triangulation for flat tori with 2434 triangles.
problem Embedding flat tori isometrically in 3D space.
method Adapted Burago and Zalgaller's proof for polyhedral surfaces, combined with Zalgaller's construction.
result A universal triangulation of 2434 triangles for any flat torus.
It is well known that an arbitrary closed orientable 3-manifold can be realized as the unique boundary of a compact orientable 4-manifold, that is, any closed orientable 3-manifold is cobordant to zero. In this paper, we consider the geometric cobordism problem: a hyperbolic 3-manifold is geometrically bounding…
If a closed 3-manifold M supports a closed, nonsingular, irrational 1-form which linearly deforms into contact forms, then M supports a K-contact form. On the 3-torus, a closed nonsingular 1-form deforms linearly into contact forms if and only if it is a fibration 1-form. on any other 2-torus bundle over the circle, ev…
This work improves sample efficiency in neural function approximation for reinforcement learning.
problem Improving sample efficiency in reinforcement learning with neural function approximation.
method Study of function approximation with two-layer neural networks (ReLU and polynomial activations) under generative and realizability models.
result Significant improvement in sample complexity compared to linear methods.
Flat minimal tori counterexamples refute Lu's second-gap conjecture.
problem Lu's second-gap conjecture about minimal surfaces in higher codimensions.
method Constructing closed embedded counterexamples for minimal surfaces.
result Constant values of S+λ2 realized by flat minimal tori are dense in (2,3), refuting the conjecture. In this paper, we presented a novel semi-supervised one-class classification algorithm which assumes that class is linearly separable from other elements. We proved theoretically that class is linearly separable if and only if it is maximal by probability within the sets with the same mean. Furthermore, we presented an…
New RL algorithms use lookahead info to maximize rewards.
problem RL in unknown environments without lookahead info.
method Planning using empirical reward and transition distributions.
result Achieves tight regret compared to lookahead info.
The paper explores linearly free graphs and their embeddings into 3D space.
problem Understanding the conditions under which a graph's embedding into 3D space is free.
method Developed a sufficient condition for a linear embedding to be free and applied it to specific graph cases.
result Established sufficient conditions for a graph to be linearly free and provided examples and counterexamples.
HARNet improves volatility forecasting using deep neural networks.
problem Lack of deep learning in volatility forecasting.
method HARNet based on dilated convolutional layers, explicitly initialized to match HAR model.
result HARNet significantly improves forecasting accuracy compared to HAR models.
Optimization geometrodynamics simplifies adaptive optimizer dynamics.
problem Hidden states in adaptive optimizers complicate gradient-based learning.
method Develops a variational theory to eliminate hidden states and compose across hierarchies.
result Yields interaction curvature that integrates to finite contrasts.
Large scale machine learning (ML) systems such as the Alexa automatic speech recognition (ASR) system continue to improve with increasing amounts of manually transcribed training data. Instead of scaling manual transcription to impractical levels, we utilize semi-supervised learning (SSL) to learn acoustic models (AM) …
Supervised learning frequently boils down to determining hidden and bright parameters in a parameterized hypothesis space based on finite input-output samples. The hidden parameters determine the attributions of hidden predictors or the nonlinear mechanism of an estimator, while the bright parameters characterize how h…
A quadratic line complex is a three-parameter family of lines in projective space P^3 specified by a single quadratic relation in the Plucker coordinates. Fixing a point p in P^3 and taking all lines of the complex passing through p we obtain a quadratic cone with vertex at p. This family of cones supplies P^3 with a c…
In previous work, the authors studied the linear stability of algebraic Ricci solitons on simply connected solvable Lie groups (solvsolitons), which are stationary solutions of a certain normalization of Ricci flow. Many examples were shown to be linearly stable, leading to the conjecture that all solvsolitons are line…
In his celebrated paper "Generic projections", John Mather has given a striking transversality theorem and its applications on generic projections. On the other hand, in this paper, two transversality theorems on generic linearly perturbed Cr mappings are shown (r≥1). Moreover, some applications of the two the…
Unified framework for nonconvex matrix completion with linearly parameterized factors.
problem Matrix completion with improved accuracy using linearly parameterized factors.
method Unified nonconvex optimization framework with Correlated Parametric Factorization condition.
result Uniform upper bounds for low-rank estimation at any local minimum.
We demonstrate that SDYM equations for the Lie algebra of one-dimensional vector fields represent a natural reduction in the framework of general linearly degenerate dispersionless hierarchy. We define the reduction in terms of wave functions, introduce generating relation, Lax-Sato equations and the dressing scheme fo…
Following Cao-Hamilton-Ilmanen, in this paper we study the linear stability of Perelman's ν-entropy on Einstein manifolds with positive Ricci curvature. We observe the equivalence between the linear stability restricted to the transversal traceless symmetric 2-tensors and the stability of Einstein manifolds with resp…
Randomly initialized neural networks can linearly separate arbitrary sets.
problem Mapping two arbitrary sets to linearly separable sets.
method Randomly initialized one-layer neural networks with sufficient width.
result With high probability, these networks can transform two sets into linearly separable sets.
New algorithms tackle robust RL with linear models, revealing unique challenges.
problem Distributionally robust offline RL with uncertainty in dynamics.
method Proposes minimax optimal and computationally efficient algorithms using novel function approximation mechanisms.
result Function approximation in robust offline RL is distinct and harder than in standard offline RL.
Paper confirms algebraic knots are linearly independent using twisted Blanchfield pairings.
problem Whether algebraic knots are linearly independent in the knot concordance group.
method Twisted Blanchfield pairings
result Algebraic knots are linearly independent in the knot concordance group.
The paper proves hyperbolic groups are semistable and their boundaries are linearly connected.
problem Local connectivity of hyperbolic group boundaries
method Elementary proofs based on Bestvina-Mess's original ideas
result All hyperbolic groups are semistable at infinity and their boundaries are linearly connected.