arXiv research

Paper establishes universal lower bounds and optimal rates for clustering sub-exponential mixture models.

We prove a general connection between the communication complexity of two-player games and the sample complexity of their multi-player locally private analogues. We use this connection to prove sample complexity lower bounds for locally differentially private protocols as straightforward corollaries of results from com…

In three-dimensional computational topology, the theory of normal surfaces is a tool of great theoretical and practical significance. Although this theory typically leads to exponential time algorithms, very little is known about how these algorithms perform in "typical" scenarios, or how far the best known theoretical…

We prove a new lower bound for the dilatation of an arbitrary pseudo-Anosov map on a surface of genus g with n punctures. Our bound improves the former super-exponential dependence on the genus by a polynomial dependence.

Improved sample complexity for identifying best policies in risk-sensitive reinforcement learning.

Paper provides a new lower bound on MMSE using Poincaré inequality.

New study shows exponential lower bound for RL even with constant suboptimality gap.

For n > 2, the Dehn functions of Aut(F_n) and Out(F_n) are exponential. Hatcher and Vogtmann proved that they are at most exponential, and the complementary lower bound in the case n=3 was established by Bridson and Vogtmann. Handel and Mosher completed the proof by reducing the lower bound for n>4 to the case n=3. In …

New lower bounds for linear classification problems in high dimensions.

TensorPlan shows an exponential lower bound for planning in MDPs with linearly realizable value functions.

Improved GNN simulation of WL test with exponentially lower complexity.

Improved lower bound for parallel tempering's mixing time.

We give a lower bound on the number of non-simple closed curves on a hyperbolic surface, given upper bounds on both length and self-intersection number. In particular, we carefully show how to construct closed geodesics on pairs of pants, and give a lower bound on the number of curves in this case. The lower bound for …

Policy iteration is a family of algorithms that are used to find an optimal policy for a given Markov Decision Problem (MDP). Simple Policy iteration (SPI) is a type of policy iteration where the strategy is to change the policy at exactly one improvable state at every step. Melekopoglou and Condon [1990] showed an exp…

We establish new strong lower bounds on the (subnormal) subgroup growth of a large class of groups. This includes the fundamental groups of all finite-volume hyperbolic 3-manifolds and all (free non-abelian)-by-cyclic groups. The lower bound is nearly exponential, which should be compared with the fastest possible subg…

Local differential privacy (LDP) is a model where users send privatized data to an untrusted central server whose goal it to solve some data analysis task. In the non-interactive version of this model the protocol consists of a single round in which a server sends requests to all users then receives their responses. Th…

We provide a lower bound for the uniform exponential growth rate of closed nonflat nonpositively curved 3-manifold groups. A detailed study of the uniform exponential growth rate of closed 3-manifold groups is also presented.

Study shows exponential gap in sample complexity between noisy and non-noisy recurrent neural networks.

Improved risk-sensitive RL with exponential Bellman equation and better regret bounds.

The paper shows that relaxing assumptions about causal graphs can lead to exponentially large equivalence classes.

New research shows exponential lower bounds for planning in MDPs with linearly-realizable optimal action-value functions.

We provide linear lower bounds for $f_ρ(L)$, the smallest integer so that every curve on a fixed hyperbolic surface $(S,ρ)$ of length at most $L$ lifts to a simple curve on a cover of degree at most $f_ρ(L)$. This bound is independent of hyperbolic structure $ρ$, and improves on a recent bound of Gupta-Kapovich. When $…

The paper studies scaling limits of hedging prices in financial models.

We propose an online convex optimization algorithm (RescaledExp) that achieves optimal regret in the unconstrained setting without prior knowledge of any bounds on the loss functions. We prove a lower bound showing an exponential separation between the regret of existing algorithms that require a known bound on the los…

The paper develops algorithms to restore monotonicity in non-monotone functions.

New combinatorial dimension VCL refines learning curve theory.

New research sets the minimax lower bound for KSD estimation at sqrt(n).

Extends online learning to metric spaces using exponential weights.

New lower bounds for private covariance estimation of Gaussian distributions are proven.

We present a general method for deriving collapsed variational inference algo- rithms for probabilistic models in the conjugate exponential family. Our method unifies many existing approaches to collapsed variational inference. Our collapsed variational inference leads to a new lower bound on the marginal likelihood. W…

Learning to control linear systems is statistically hard, especially for underactuated systems.

The paper proves a quantitative rigidity result for spaces with specific curvature bounds.

This paper assesses Gaussian and Exponential mechanisms for certifying adversarial robustness.

New method for tensor completion using nonconvex dual total variation.

Improved algorithm reduces excess risk in selective learning.

We analyze exponential integrability properties of the Cox-Ingersoll-Ross (CIR) process and its Euler discretizations with various types of truncation and reflection at 0. These properties play a key role in establishing the finiteness of moments and the strong convergence of numerical approximations for a class of sto…

Lower bounds on MALA and HMC for well-conditioned distributions.

Study on biharmonic heat equation on manifolds with curvature constraints.

Algorithm approximates functions into manifolds with curvature bounds.

Lower bounds show OLS outperforms basis pursuit in overparameterized linear regression.

Unified framework for understanding TVO and improving model learning.

The paper finds lower bounds for volumes of complex geometric structures.

Proves depth 2 neural networks can't approximate certain functions as well as depth 3 networks.

This paper considers the growth in the length of one-dimensional trajectories as they are passed through deep ReLU neural networks, which, among other things, is one measure of the expressivity of deep networks. We generalise existing results, providing an alternative, simpler method for lower bounding expected traject…

We show that a smooth unknotted curve in R^3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length L of the curve and the thickness r (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length, the expression giv…

Machine learning algorithms typically perform optimization over a class of non-convex functions. In this work, we provide bounds on the fundamental hardness of identifying the global minimizer of a non convex function. Specifically, we design a family of parametrized non-convex functions and employ statistical lower bo…

This paper explores the computational hardness of generating latent vectors for generative models.

Adaptive algorithm identifies best arm with abstention, showing phase transition from polynomial to exponential error probability.