The paper shows how sublinear biLipschitz equivalences affect Morse boundaries of metric spaces.
problem Understanding how sublinear biLipschitz equivalences affect Morse boundaries of metric spaces.
method Defining sublinear biLipschitz equivalence and Morse boundaries, proving invariance under SBEs, using sublinear rays.
result κ-Morse boundaries of proper geodesic metric spaces are invariant under suitable sublinear biLipschitz equivalences.
The aim of this paper is to introduce the sublinear Higson corona and show that the sublinear Higson corona of Euclidean cone of P and X is decomposed into the product of P and that of X. Here P is a compact metric space and X is unbounded proper metric space. For example, the sublinear Higson corona of n-dimensional E…
We provide a general construction of time-consistent sublinear expectations on the space of continuous paths. It yields the existence of the conditional G-expectation of a Borel-measurable (rather than quasi-continuous) random variable, a generalization of the random G-expectation, and an optional sampling theorem that…
New method connects CAT(0) spaces to hyperbolic spaces.
problem Injecting sublinear Morse boundaries into Gromov boundaries.
method Developed curtain machinery to characterize sublinear Morse properties.
result Continuous injection of sublinear Morse boundaries into Gromov boundaries.
We give a proof of the sublinear tracking property for sample paths of random walks on various groups acting on spaces with hyperbolic-like properties. As an application, we prove sublinear tracking in Teichmueller distance for random walks on mapping class groups, and on Cayley graphs of a large class of finitely gene…
New model for Knightian uncertainty with jumps.
problem Knightian uncertainty and non-linear jumps.
method Probabilistic construction of non-linear affine processes with jumps.
result Tractable model for Knightian uncertainty with sublinear expectations.
Sublinear functionals of random variables are known as sublinear expectations; they are convex homogeneous functionals on infinite-dimensional linear spaces. We extend this concept for set-valued functionals defined on measurable set-valued functions (which form a nonlinear space), equivalently, on random closed sets. …
Develops geometric foundations for sublinear Morse boundaries in mapping class groups and Teichmüller spaces.
problem Capturing generic directions in mapping class groups and Teichmüller spaces.
method Develops tools for modeling hulls of median rays in hierarchically hyperbolic spaces via CAT(0) cube complexes.
result Sublinear Morse boundaries are visibility spaces and admit continuous equivariant injections into the boundary of the curve graph.
Develops sublinear Morse theory in symmetric spaces.
problem Understanding sublinear Morse properties in symmetric spaces.
method Theory of sublinearly Morse boundary and lemma in higher rank symmetric spaces.
result Proves sublinear Morse lemma in higher rank symmetric spaces.
Memory-limited learning tackles adversarial bandits with reduced storage.
problem Adversarial bandit problem with limited memory storage.
method Hierarchical learning policy with sublinear memory requirement.
result Established sublinear regret bounds for weak and shifting regrets.
GP-UCB resolves sublinear regret for kernelized bandits.
problem Minimizing regret in kernelized bandit problems.
method Using a new regularization technique for kernel ridge estimators, improving GP-UCB's sublinear regret rate.
result GP-UCB achieves nearly optimal sublinear regret for the Matérn kernel.
New sublinear sketches improve ANN and KDE for massive data streams.
problem Efficiently approximate nearest neighbors and kernel density estimation in large datasets.
method Developed sublinear space and query time algorithms for ANN and A-KDE in streaming and sliding-window models.
result Achieved near-optimal trade-offs between memory size and approximation error for ANN.
Study online learning in unknown Markov games with sublinear regret.
problem Online learning in unknown Markov games with unobservable opponents.
method Introduced an algorithm achieving sublinear regret against the minimax value.
result First sublinear regret bound for unknown Markov games, independent of action spaces size.
New algorithm optimizes functions in Matérn kernel RKHS with noisy feedback.
problem Optimizing functions in RKHS of Matérn kernel with noisy bandit feedback.
method π-GP-UCB algorithm with guaranteed sublinear regret for all ν > 1 and d ≥ 1.
result First practical approach with guaranteed sublinear regret for all ν > 1 and d ≥ 1.
Sublinear LSVI via LSH reduces runtime to sublinear in actions.
problem Efficiently estimating value functions in reinforcement learning with sublinear runtime.
method Formulated as approximate maximum inner product search, used LSH to solve with sublinear time complexity.
result Sublinear runtime while maintaining LSVI's regret.
FPP preserves sublinear Morse boundaries in geodesic graphs.
problem Preserving sublinear Morse boundaries in FPP.
method First passage percolation on geodesic graphs with i.i.d. passage times.
result Sublinear Morse boundaries are invariant under FPP.
This paper studies convergence of horospheres in CAT(0) spaces.
problem Analysis of convergence of horospheres in CAT(0) spaces.
method Examines horofunctions associated with sublinearly contracting geodesic rays.
result Horospheres associated with sublinearly contracting horofunctions are convergent.
New method for computing terminal embeddings in sublinear time.
problem Efficiently computing terminal embeddings with sublinear time complexity.
method Developed a data structure to compute terminal embeddings in sublinear time.
result Achieved sublinear time computation of terminal embeddings.
Inference in log-linear models scales linearly in the size of output space in the worst-case. This is often a bottleneck in natural language processing and computer vision tasks when the output space is feasibly enumerable but very large. We propose a method to perform inference in log-linear models with sublinear amor…
For a large class of metric space X including discrete groups we prove that the asymptotic Assouad-Nagata dimension AN-asdim X of X coincides with the covering dimension dim ( ν L X ) \dim(ν_L X) dim ( ν L X ) of the Higson corona of X with respect to the sublinear coarse structure on X. Then we apply this fact to prove the equality AN-asdim(X …
Probabilistic programming languages can simplify the development of machine learning techniques, but only if inference is sufficiently scalable. Unfortunately, Bayesian parameter estimation for highly coupled models such as regressions and state-space models still scales poorly; each MCMC transition takes linear time i…
This work addresses dynamic KDE data structures with robustness to adversarial queries.
problem Efficient KDE data structures for dynamic changing data distributions.
method Developed a theoretical framework for KDE data structures that support subquadratic space, sublinear updates, and adaptive queries.
result Theoretical framework and practical implementation of KDE data structures robust to adversarial queries.
The purpose of the paper is to characterize the dimension of sublinear Higson corona ν L ( X ) ν_L(X) ν L ( X ) of X X X in terms of Lipschitz extensions of functions: Theorem: Suppose ( X , d ) (X,d) ( X , d ) is a proper metric space. The dimension of the sublinear Higson corona ν L ( X ) ν_L(X) ν L ( X ) of X X X is the smallest integer m ≥ 0 m\ge 0 m ≥ 0 with the following property…
Study efficient policy value estimation with sublinear samples.
problem Estimating optimal policy value in stochastic disjoint linear bandits.
method Sublinear sample estimation of optimal policy value.
result Achieves near optimal estimation error with sublinear samples.
A new method solves convex optimization problems on manifolds efficiently.
problem Optimization on Hadamard manifolds with convex objectives.
method Intrinsic Riemannian proximal gradient method.
result Sublinear and linear convergence rates for convex and strongly convex problems, respectively.
GP-PSRL achieves sublinear regret for continuous control with unbounded state space.
problem Analyzing regret bounds for GP-PSRL in continuous control with unbounded state space.
method Recursive application of Borell-Tsirelson-Ibragimov-Sudakov inequality and chaining method.
result Sublinear regret bound of O ~ ( H γ T T ) \widetilde{\mathcal{O}}(H\sqrt{γ_TT}) O ( H γ T T ) for GP-PSRL. The paper develops methods for time-varying constrained online convex optimization.
problem Time-varying loss and constraint functions in online convex optimization.
method Model-based augmented Lagrangian methods (MALM) for time-varying and delayed feedback.
result Sublinear regret and constraint violation for both time-varying and delayed feedback scenarios.
New algorithm tackles delayed feedback in Lipschitz bandits with sublinear regret.
problem Delayed feedback in Lipschitz bandits.
method Design of algorithms for bounded and unbounded stochastic delays.
result Sublinear regret guarantees for both bounded and unbounded delays.
For α ∈ ( 1 , 2 ) α\in (1,2) α ∈ ( 1 , 2 ) , we present a generalized central limit theorem for α α α -stable random variables under sublinear expectation. The foundation of our proof is an interior regularity estimate for partial integro-differential equations (PIDEs). A classical generalized central limit theorem is recovered as a special case, p…
Minimal graphs grow slowly on curved spaces, proving constant solutions.
problem Characterizing minimal graphs with sublinear growth on manifolds.
method New technique to get gradient bounds by integral estimates, no further geometric assumptions.
result Entire solutions are constant when negative part grows like r / log r r/\log r r / log r . New algorithms for constrained online optimization with memory and predictions.
problem Control of constrained dynamical systems and scheduling with reconfiguration budgets.
method Proposed algorithms achieving sublinear regret and constraint violation under time-varying constraints, both with and without predictions.
result First algorithms achieving sublinear regret and constraint violation in constrained online optimization with memory.
Sharp Liouville theorem for minimal graphs on manifolds with nonnegative Ricci curvature.
problem Characterizing smooth solutions to minimal hypersurface equations on manifolds with nonnegative Ricci curvature.
method Gradient estimate for minimal graphs over Σ Σ Σ with small linear growth of the negative parts of graphic functions via iteration. result Every smooth solution u u u to minimal hypersurface equation on Σ Σ Σ is a constant provided u u u has sublinear growth for its negative part. We prove a Liouville property for any f f f -harmonic function with polynomial growth on a complete noncompact smooth metric measure space ( M , g , e − f d v ) (M,g,e^{-f}dv) ( M , g , e − f d v ) when the Bakry-Émery Ricci curvature is nonnegative and its diameter of geodesic sphere has sublinear growth.
A new algorithm reduces memory usage for long token attention in streaming applications.
problem Memory inefficiency in computing attention for long documents.
method One-pass streaming algorithm using sublinear space storage.
result Super-efficient memory usage for long token attention.
Open problem seeks an online learning algorithm for binary classification.
problem Existence of an online learning algorithm for binary classification with sublinear mistakes.
method Assumption of sequence allowing learning algorithm's existence.
result Specific condition determines sequence's learnability.
Efficiently trains large GMMs with millions to billions of parameters.
problem Training large Gaussian Mixture Models (GMMs) is computationally expensive.
method Derives a variational approximation integrated with mixtures of factor analyzers (MFAs) to reduce complexity.
result Sublinear scaling in training GMMs, achieving significant speed-ups.
Average signature of 2-bridge knots approximates sqrt(2c/π).
problem Estimating the average signature and 4-genus of 2-bridge knots.
method Developed a model for 2-bridge knot diagrams indexed by crossing number, and used it to derive upper bounds for the average 4-genus.
result Upper bound for the average 4-genus of a 2-bridge knot is 9.75c/log c.
Online learning is a powerful tool for analyzing iterative algorithms. However, the classic adversarial setup sometimes fails to capture certain regularity in online problems in practice. Motivated by this, we establish a new setup, called Continuous Online Learning (COL), where the gradient of online loss function cha…
New algorithm achieves sublinear regret in CMDPs without error cancellations.
problem Safety constraints in reinforcement learning with error cancellations.
method Model-based primal-dual algorithm for CMDPs with multiple constraints.
result Achieves sublinear regret without error cancellations.
Extends tracking guarantees for time-varying variational inequalities.
problem Tracking solutions of time-varying variational inequalities.
method Extends existing results to sublinear solution paths and periodic problems.
result Discrete dynamical systems of periodic time-varying VI can exhibit chaotic behavior or converge to the solution.
Geodesic loops escape from balls at a sublinear rate imply virtually abelian fundamental group.
problem Understanding fundamental groups of open manifolds with nonnegative Ricci curvature.
method Generalizing the Cheeger-Gromoll splitting theorem to sublinear escape rates.
result Fundamental groups of open manifolds with nonnegative Ricci curvature are virtually abelian if geodesic loops escape sublinearly.
LaPSRL achieves optimal regret for isoperimetric RL distributions.
problem Designing RL algorithms with sublinear regret for non-log-concave distributions.
method Posterior Sampling (PSRL) and Langevin sampling (LaPSRL) for isoperimetric distributions.
result LaPSRL achieves order-optimal regret and subquadratic complexity.
Study shows how repetition affects learning in bandit settings, providing algorithms with sublinear regret.
problem Effect of persistence of engagement on learning in stochastic multi-armed bandit settings.
method Novel algorithms that achieve sublinear regret under temporal constraints.
result Additive effect of priming on regret upper bound, matching popular algorithms in absence of priming.
Quantum algorithm speeds up Gibbs partition function estimation.
problem Estimating partition functions in sublinear time.
method Sublinear-time quantum algorithm using quantum phase and amplitude estimation.
result First sublinear-time speed-up for partition function estimation.
Proposes H-UCRL for efficient model-based RL with sublinear regret.
problem Greedy policy exploration in model-based RL ignores epistemic uncertainty.
method Reparameterizes plausible models, hallucinates control, augments input space, solves with greedy planners.
result H-UCRL achieves provably sublinear regret for Gaussian Process models.
New approach for distributed online optimization of non-convex losses with sublinear regret.
problem Regret evaluation and consensus in distributed, multi-agent systems with non-convex losses.
method Composite regret metric and consensus-based online normalized gradient (CONGD) approach for pseudo-convex losses; offline optimization oracle for general non-convex losses.
result First sublinear regret bound for general distributed online non-convex learning.
LIBO optimizes repeated bandit tasks without prior knowledge or regret.
problem Optimizing repeated bandit tasks without prior knowledge or regret.
method LIBO sequentially meta-learns a kernel to adapt to the environment and solve tasks with the latest estimate.
result LIBO achieves sublinear lifelong regret, converging to oracle performance as more tasks are solved.
New data structure identifies close match from multiple distributions.
problem Identify the closest distribution to a given sample.
method Developed a sublinear-time data structure for identifying the closest distribution.
result First data structure that identifies the closest distribution in sublinear time.