Paper proposes a policy-search algorithm to learn entropy-maximizing exploration policies in reward-free environments.
problem Reward-free learning in high-dimensional, continuous-control domains.
method Maximum Entropy POLicy optimization (MEPOL) algorithm that maximizes a non-parametric state entropy estimate.
result MEPOL learns a maximum-entropy exploration policy that facilitates learning various reward-based tasks.
The paper develops a new probabilistic framework for denoising diffusion models using free entropy and stochastic analysis.
problem Developing a mathematical framework for denoising diffusion models in noncommutative settings.
method Formulating diffusion and reverse processes governed by operator-valued stochastic dynamics, using tools from free stochastic analysis.
result Establishing an information-geometric link between entropy production, transport, and deconvolution.
Study minimal volume entropy for free-by-cyclic groups and 2D right-angled Artin groups.
problem Characterize minimal volume entropy for aspherical simplicial complexes with these groups as fundamental groups.
method Algebraic and geometric characterization, using fiber π1-growth collapse and non-collapsing assumptions. result Provide bounds and criteria for minimal volume entropy in aspherical simplicial complexes.
We translate the problem of calculating the entropy of a set of binary configurations/signals into a sequence of supervised classification tasks. Subsequently, one can use virtually any machine learning classification algorithm for computing entropy. This procedure can be used to compute entropy, and consequently the f…
Study curve shortening flow in high dimensions with boundary constraints.
problem Understanding the behavior of curves in high-dimensional spaces with boundary conditions.
method Used curvature and higher-derivative estimates, Stahl-type maximum principle, and blow-up analysis.
result Flow converges to a shrinking semicircle model or has only semicircle boundary singularities in low entropy regimes.
Maximizes Rényi entropy for efficient exploration in reward-free RL.
problem Challenges of exploration in reward-free reinforcement learning.
method Maximizes Rényi entropy over state-action space in exploration phase; uses batch RL for planning phase.
result Effective and sample-efficient exploration leading to superior policies.
We prove a Margulis' Lemma à la Besson Courtois Gallot, for manifolds whose fundamental group is a nontrivial free product A*B, without 2-torsion. Moreover, if A*B is torsion-free we give a lower bound for the homotopy systole in terms of upper bounds on the diameter and the volume entropy. We also provide examples and…
The study shows a finite number of groups acting on hyperbolic spaces with bounded entropy and compact quotient.
problem Finite number of groups acting on hyperbolic spaces with bounded entropy and compact quotient.
method Analyzing torsion-free groups acting by isometries on hyperbolic metric spaces with bounded entropy and compact quotient.
result The set of such groups is finite and can be estimated based on hyperbolicity constant, entropy, and quotient diameter.
This study proposes hidden state curiosity to enhance RL models' resilience against noise.
problem Curiosity traps in RL models distract agents from discovering novel experiences.
method Proposed hidden state curiosity based on the Free Energy Principle to reward agents for KL divergence between predictive priors and posteriors.
result Agents with hidden state curiosity are more resilient against curiosity traps compared to those with prediction error curiosity.
New RL method nearly optimally learns policies with generative models.
problem Finding optimal policies in reinforcement learning with generative models.
method Mirror descent value iteration with KL divergence and entropy regularization.
result The method is nearly minimax-optimal for small ε-optimal policies. Sharp inequality in spaces with non-negative Ricci curvature.
problem Proving a sharp isoperimetric inequality in metric measure spaces.
method Using volume entropy in non-compact metric measure spaces with non-negative synthetic Ricci curvature.
result Proved a sharp dimension-free isoperimetric inequality.
Tree-AMP simplifies inference in complex tree-structured models.
problem Inference in high-dimensional tree-structured models.
method Approximate Message Passing algorithms for various machine learning tasks.
result Theoretical performance predictions and automated entropy estimation.
Characterizes sample complexity for outcome indistinguishability in machine learning.
problem Outcome indistinguishability in machine learning, focusing on distinguishers and predictors.
method Sample complexity characterized by metric entropy of predictor and distinguisher classes, using dual Minkowski norms.
result Equivalence and tightness of sample complexity characterizations in distribution-specific and distribution-free settings.
The paper studies entropy and free energy for harmonic metrics on cyclic Higgs bundles.
problem Quantifying the degree of mutual misalignment of metrics on Higgs bundles.
method Introduced entropy and free energy to quantify mutual misalignment; provided conditions for entropy and free energy to change.
result Extended work on boundedness of functions related to entropy and free energy on the unit disc.
We extend recent work (Brehmer, et. al., 2018) that use neural networks as surrogate models for likelihood-free inference. As in the previous work, we exploit the fact that the joint likelihood ratio and joint score, conditioned on both observed and latent variables, can often be extracted from an implicit generative m…
EntProp increases entropy of clean samples to generate out-of-distribution data for better DNN performance.
problem Improving deep neural networks' accuracy and robustness to out-of-distribution data.
method High entropy propagation using data augmentation and free adversarial training.
result EntProp achieves higher standard accuracy and robustness with lower training cost.
This paper presents a curvature-free version of the Log(2k-1) Theorem of Anderson, Canary, Culler & Shalen [ACCS96]. It generalizes a result by Hou [Hou01] and its proof is rather straightforward once we know the work by Lim [Lim08] on volume entropy for graphs. As a byproduct we obtain a curvature-free version of the …
The paper finds a short graph incompressible in a complex with specific properties.
problem Finding a short graph in a complex with specific properties.
method Analyzing a finite connected 2-complex with a piecewise Riemannian metric and showing the existence of a 2-incompressible graph.
result The existence of a 2-incompressible graph with a length satisfying a curvature-free inequality.
Adaptive approximations improve variational inference for complex models.
problem Efficiently approximate marginal distributions and partition functions in complex probabilistic models.
method Two classes of adaptive approximations that include Bethe, tree-reweighted, and convex free energies.
result Proposed approximations automatically adapt to a given model and outperform existing methods.
We explore a new method for discrete-time control problems using randomization and entropy.
problem Discrete-time linear-exponential quadratic Gaussian (LEQG) control problem.
method Introduce exploration through randomization and apply duality between free energy and relative entropy.
result Reduced LEQG problem to equivalent risk-neutral LQG control problem with entropy regularization.
Reinforcement learning for continuous-time risk-sensitive asset allocation
problem Continuous-time risk-sensitive asset allocation
method Free energy-entropy duality reformulation and q-learning actor-critic method result Optimal policy learning with high accuracy
New bandit algorithm maximizes information gain.
problem Optimizing decision-making in uncertain environments.
method Approximates information maximization using entropy and free energy principles.
result Asymptotic optimality proven for two-armed bandit problem.
We prove the following entropy-rigidity result in finite volume: if X is a negatively curved manifold with curvature −b2≤KX≤−1, then Enttop(X)=n−1 if and only if X is hyperbolic. In particular, if X has the same length spectrum of a hyperbolic manifold X0, the it is isometric to X0 (we a…
New bound limits generalization gap for large models, independent of model complexity.
problem Understanding generalization gap in large-scale machine learning models.
method Established a model-independent upper bound for generalization gap using Rényi entropy.
result Generalization gap can be maintained with arbitrarily large models if data entropy is sufficient.
In this paper, we present a new class of Markov decision processes (MDPs), called Tsallis MDPs, with Tsallis entropy maximization, which generalizes existing maximum entropy reinforcement learning (RL). A Tsallis MDP provides a unified framework for the original RL problem and RL with various types of entropy, includin…
EDD uses entropy of distance distributions to cluster unlabeled data.
problem Challenges in clustering unlabeled high-dimensional data.
method EDD employs Shannon entropy to quantify distance distribution peaks.
result EDD detects varying degrees of clustering sensitivity.
Improved exploration methods for reinforcement learning with reduced sample complexity.
problem Challenges in reinforcement learning exploration in unknown environments.
method Proposed game-theoretic and trajectory entropy algorithms with improved sample complexity.
result Established statistical advantage of entropy-regularized MDPs for exploration and reduced sample complexity.
The main result of this article is that if a 3-manifold M supports an Anosov flow, then the number of conjugacy classes in the fundamental group of M grows exponentially fast with the length of the shortest orbit representative, hereby answering a question raised by Plante and Thurston in 1972. In fact we show th…
The study counts geodesics on special manifolds without focusing points.
problem Counting geodesics on specific types of manifolds.
method Margulis-type asymptotic estimates and analysis of geodesic flow.
result The geodesic flow on these manifolds has a unique measure of maximal entropy with the Bernoulli property.
We propose a new policy iteration theory as an important extension of soft policy iteration and Soft Actor-Critic (SAC), one of the most efficient model free algorithms for deep reinforcement learning. Supported by the new theory, arbitrary entropy measures that generalize Shannon entropy, such as Tsallis entropy and R…
Variable selection is of significant importance for classification and regression tasks in machine learning and statistical applications where both predictability and explainability are needed. In this paper, a Copula Entropy (CE) based method for variable selection which use CE based ranks to select variables is propo…
A \emph{geodesic current} on a free group F is an F-invariant measure on the set ∂2F of pairs of distinct points of ∂F. The space of geodesic currents on F is a natural companion of Culler-Vogtmann's Outer space cv(F) and studying them together yields new information about both spaces as we…
Paper improves Frank-Wolfe algorithm's efficiency bounds.
problem Establishing efficient iteration complexity for Frank-Wolfe algorithm.
method Using metric entropy to provide lower bounds.
result Frank-Wolfe requires many iterations for certain problems.
Entropy rate of sequential data-streams naturally quantifies the complexity of the generative process. Thus entropy rate fluctuations could be used as a tool to recognize dynamical perturbations in signal sources, and could potentially be carried out without explicit background noise characterization. However, state of…
Tent adapts models during testing by minimizing entropy of predictions.
problem Adapting models to new data during testing with limited information.
method Test entropy minimization (tent) and online channel-wise affine transformations.
result Reduces generalization error on various datasets and benchmarks.
The aim of this paper is to provide new theoretical and computational understanding on two loss regularizations employed in deep learning, known as local entropy and heat regularization. For both regularized losses we introduce variational characterizations that naturally suggest a two-step scheme for their optimizatio…
A framework uses free probability to analyze Transformer models.
problem Understanding the dynamics and complexity of Transformer-based language models.
method Formal operator-theoretic analysis using free probability theory.
result Entropy-based generalization bounds derived under freeness assumptions.
ELBO converges to a sum of entropies for many generative models.
problem Understanding the convergence of variational lower bounds in unsupervised learning.
method Analyzing the ELBO for a broad class of generative models, showing it equals a sum of entropies.
result The ELBO is equal to a sum of entropies at stationary points for many generative models.
Modernizes Thurston's proof of entropy theorem for traintrack maps.
problem Proving the entropy theorem for traintrack maps using Thurston's methods.
method Modernizes Thurston's original proof, fills gaps, and proves ergodicity.
result A cohesive proof of the traintrack theorem, including ergodicity.
We review some approaches to the understanding of fluctuations in some models used to describe socio and economic systems. Our approach builds on the development of a simple Langevin equation that characterises stochastic processes. This provides a unifying approach that allows first a straightforward description of th…
Study on existence of ground states for free energy on hyperbolic space.
problem Existence of ground states for a free energy functional on hyperbolic space.
method Derived HLS-type inequalities on Cartan-Hadamard manifolds to prove existence.
result Established conditions for the existence of ground states on hyperbolic space.
The paper applies math and physics to language models, introducing entropy and geometric concepts.
problem Understanding and improving language models to approximate intelligent language.
method Formal definitions, functional analysis, topology, thermodynamics, and set theory.
result Entropy function reveals key obstacles for LLMs and offers insights into language models.
We investigate the position of the Buchen-Kelly density in a family of entropy maximising densities which all match European call option prices for a given maturity observed in the market. Using the Legendre transform which links the entropy function and the cumulant generating function, we show that it is both the uni…
Maximum entropy deep reinforcement learning (RL) methods have been demonstrated on a range of challenging continuous tasks. However, existing methods either suffer from severe instability when training on large off-policy data or cannot scale to tasks with very high state and action dimensionality such as 3D humanoid l…
Proves multiplicity one for boundary minimal hypersurfaces in compact manifolds.
problem Proving multiplicity one for min-max free boundary minimal hypersurfaces in compact manifolds with boundary.
method Developed existence and regularity theory for free boundary hypersurfaces with prescribed mean curvature, including Morse index bounds.
result Proved multiplicity one theorem for min-max free boundary minimal hypersurfaces in compact manifolds with boundary.
Solves risk-sensitive investment via duality, entropic regularization, and RL.
problem Risk-sensitive portfolio management in a factor-based setting.
method Free energy-entropy duality, Kuroda-Nagai change-of-measure, RL algorithm.
result Direct analytical solution, explicit controls, two interpretations of optimal allocation.
In this paper, we generalize White's regularity and structure theory for mean-convex mean curvature flow to the setting with free boundary. A major new challenge in the free boundary setting is to derive an a priori bound for the ratio between the norm of the second fundamental form and the mean curvature. We establish…
In this paper, we prove the characterization of the (K,∞)-super Perelman Ricci flows by various functional inequalities and gradient estimate for the heat semigroup generated by the Witten Laplacian on manifolds equipped with time dependent metrics and potentials. As a byproduct, we derive the Hamilton type dim…