We prove optimal subspace embedding conjecture up to sub-polylogarithmic factors.
problem Optimal dimension and sparsity of subspace embeddings.
method Iterative decoupling technique to analyze higher-order trace moment bounds.
result Sub-polylogarithmic factors in dimension and sparsity of subspace embeddings.
Mirzakhani volumes of moduli spaces are polylogarithmic.
problem Understanding the volume of moduli spaces of hyperbolic surfaces.
method Expressed as a sum of polylogarithms evaluated at specific points.
result Mirzakhani volumes are polylogarithmic.
Study higher genus polylogarithms under Riemann surface degenerations.
problem Understanding higher genus polylogarithms under degenerations.
method Investigate the Enriquez connection for polylogarithms and show it becomes a known connection for families of Riemann surfaces.
result Higher genus polylogarithms can be described explicitly as power series in deformation parameters and logarithms of families.
Investigates webs related to cluster algebras and polylogarithms.
problem Understanding webs associated with cluster algebras and polylogarithms.
method Introducing AMP webs and analyzing their properties, proving results and conjectures.
result Many webs associated with polylogarithms and cluster algebras are AMP webs.
Paper solves no-swap regret minimization for combinatorial bandits with polylogarithmic dependence on N.
problem Design efficient no-swap regret algorithms for combinatorial bandits with exponentially large action space.
method Introduces a no-swap-regret learning algorithm with polylogarithmic dependence on N and demonstrates efficient implementation.
result Achieves no-swap regret with polylogarithmic dependence on N, resolving an open problem.
New algorithm reduces regret in online portfolio and quantum state learning.
problem Efficiently learning portfolios and quantum states online with minimal regret.
method BISONS algorithm for online portfolio selection, SCHRODINGER'S BISONS for quantum states, with polylogarithmic regret.
result First efficient algorithm with polylogarithmic regret for online portfolio selection and quantum states.
Novel parallelization simplifies machine learning algorithms.
problem Adapting machine learning algorithms to growing data and needs.
method A novel parallelization scheme that applies to broad learning algorithms.
result Reduces runtime to polylogarithmic time on quasi-polynomially many units.
Gradient descent with polylogarithmic width achieves arbitrarily low test error for shallow ReLU networks.
problem Achieving low test error with shallow ReLU networks using gradient descent.
method Gradient descent with polylogarithmic width and polylogarithmic number of samples.
result Gradient descent achieves arbitrarily low test error with shallow ReLU networks of polylogarithmic width.
Quantum machine learning can't achieve polylogarithmic runtimes, even with quantum data access.
problem Bounding the minimum number of samples required for supervised quantum learning.
method Statistical learning theory and quantum machine learning algorithms.
result Quantum machine learning algorithms for supervised learning have at most polynomial speedups over classical algorithms.
New method proves fast regret bounds for online RLHF with generalized preferences.
problem Minimizing max-regret in online RLHF with general preferences and bandit feedback.
method Adopted Generalized Bilinear Preference Model (GBPM) to investigate polylogarithmic regret guarantees.
result Proved polylogarithmic regret bounds for Greedy Sampling and Explore-Then-Commit policies under GBPM.
Efficiently estimates binary product distributions with privacy.
problem Estimating means of binary product distributions privately and accurately.
method Polynomial time, pure differential privacy approach.
result Optimal sample complexity with polylogarithmic factors.
New algorithm selects best distribution privately in nearly-linear time.
problem Estimating the best distribution from samples under differential privacy constraints.
method Differentially private algorithm with nearly-linear time complexity and optimal approximation factor.
result Achieves optimal approximation factor of 3 with modest sample complexity increase.
New method uses higher-order Langevin dynamics for efficient parallel sampling.
problem Efficient parallel sampling from high-dimensional log-concave distributions.
method Combines higher-order Langevin dynamics with blockwise Lagrange polynomial interpolation.
result Reduces the number of parallel points required for a target accuracy.
We present an efficient and practical algorithm for the online prediction of discrete-time linear dynamical systems with a symmetric transition matrix. We circumvent the non-convex optimization problem using improper learning: carefully overparameterize the class of LDSs by a polylogarithmic factor, in exchange for con…
Kähler information manifolds for signal filters in weighted Hardy spaces are explored.
problem Developing a geometric framework for signal processing filters in weighted Hardy spaces.
method Introducing weighted Hardy spaces and smooth transformations of transfer functions, demonstrating the Kähler manifold structure.
result The Riemannian geometry of weighted Hardy norms for transfer functions forms a Kähler manifold.
New research shows deep ReLU networks can be learned with polylogarithmic width.
problem Learning deep ReLU networks with limited over-parameterization.
method Using gradient descent, the study establishes learning guarantees for networks with polylogarithmic width.
result Deep ReLU networks can be learned with a polylogarithmic width condition, not just a high degree polynomial.
New algorithms solve linear algebra problems in sublinear time.
problem Numerical linear algebra problems, especially with structured matrices.
method Sublinear time algorithms using matrix-vector multiplications.
result Solve problems like least squares regression and low rank approximation in sublinear time.
New analysis shows GD and SGD avoid saddle points efficiently in high dimensions.
problem Gradient descent and SGD struggle with saddle points in high-dimensional nonconvex optimization problems.
method Perturbed versions of GD and SGD analyzed for efficiency in high dimensions.
result Perturbed GD and SGD converge to second-order stationary points efficiently, avoiding saddle points.
Machine learning has recently emerged as a fruitful area for finding potential quantum computational advantage. Many of the quantum enhanced machine learning algorithms critically hinge upon the ability to efficiently produce states proportional to high-dimensional data points stored in a quantum accessible memory. Eve…
New study reveals a polynomial penalty for adapting to unknown margin parameters in batched nonparametric bandits.
problem Adapting to an unknown margin parameter in batched nonparametric bandits.
method Introduces the regret inflation criterion and develops RoBIN algorithm to achieve optimal regret inflation.
result The optimal regret inflation grows polynomially with the horizon T, characterized by a convex optimization problem.
QATS efficiently decodes HMMs with polylogarithmic complexity.
problem Efficiently decoding hidden Markov models from noisy observations.
method Divide-and-conquer procedure with polylogarithmic sequence complexity and cubic state space complexity.
result QATS outperforms Viterbi and PMAP in speed and accuracy.
New method controls linear systems with adversarial disturbances.
problem Controlling linear dynamical systems under adversarial conditions.
method Novel convex relaxation using spectral filters from Hankel matrix eigenvectors.
result Polylogarithmic running time improvement over prior methods.
In this paper, we study local solutions F=(F1,..,Fn) of a general functional equation of the form F1(U1(x,y))+....+Fn(Un(x,y))=0. A such equation will be called an ``abelian functional equation'' (Afe). We will restrict ourselves to the case when the inner functions Ui's are real rational functions. First we prove that…
New algorithm speeds up sampling from logconcave densities.
problem Sampling from logconcave functions in statistics and ML.
method Solves ODEs to improve HMC and other sampling methods.
result Nearly linear runtime for polylogarithmic depth.
Paper presents a faster classical algorithm for principal component regression.
problem Efficiently solving principal component regression problems.
method Uses quantum-inspired linear algebra techniques.
result Achieves polylogarithmic runtime, significantly faster than state-of-the-art.
Study shows sample complexity for multicalibration is Θ(ε^-3) with polylogarithmic factors.
problem Minimizing Expected Calibration Error (ECE) for predictors with respect to a family of groups.
method Proved necessary and sufficient sample complexity of Θ(ε^-3) for multicalibration, using online-to-batch reduction and lower bounds.
result Sample complexity of multicalibration is Θ(ε^-3) with polylogarithmic factors, distinguishing it from marginal calibration.
Quantum algorithm speeds up Lasso regression by quadratically faster per iteration.
problem Efficiently solving high-dimensional linear regression with L1-penalty.
method Pathwise LARS algorithm adapted for quantum computing, using minimum-finding subroutines.
result Quadratic speedup in computation time for both number of features and observations.
Neural networks can achieve optimal sample complexity for learning single-index models.
problem Achieving optimal computational-statistical tradeoff in learning Gaussian single-index models.
method Unified gradient-based algorithm for training a two-layer neural network, adaptable to various loss and activation functions.
result Sample complexity of d s ⋆ / 2 ∨ d d^{s^\star/2} \lor d d s ⋆ /2 ∨ d matches the SQ lower bound up to a polylogarithmic factor. A new algorithm estimates mean adaptively to covariance, faster and more flexible than existing methods.
problem Estimating mean of a distribution with unknown covariance efficiently and privately.
method Adaptive differentially private algorithm with optimal convergence rates and near-linear sample complexity.
result Achieves optimal rates of convergence with respect to the Mahalanobis norm ∣ ∣ ⋅ ∣ ∣ Σ ||\cdot||_Σ ∣∣ ⋅ ∣ ∣ Σ . Improved GNN simulation of WL test with exponentially lower complexity.
problem Improving the complexity of simulating the Weisfeiler-Lehman test with GNNs.
method Exponentially lower complexity simulation of WL test using GNNs with polylogarithmic parameters and O(log n) bits feature vectors.
result Near-optimal construction with logarithmic lower bounds for feature vector length and neural network size.
New algorithm reduces bandit regret to log^3(T).
problem Noise model for linear stochastic bandits with vanishing noise.
method Weighted least-squares estimation, leveraging eigenvalue relation.
result Minimax regret scaling as log^3(T) for time horizon T.
New algorithms minimize regret in both adversarial and stochastic contexts.
problem Minimizing regret in linear contextual bandits.
method Best-of-both-worlds algorithms using FTRL with Shannon entropy regularizer.
result Achieves near-optimal regret bounds in both adversarial and stochastic regimes.
The paper explores the geometry of the Spence-Kummer trilogarithm equation and its Galois analogue.
problem Investigating the geometry and functional equation of the Spence-Kummer trilogarithm.
method Using algebraic relations between polylogarithm generating series and path systems, along with tensor and homotopy criteria for functional equations.
result Derives a precise form of the Spence-Kummer equation and its Galois analogue.
Constructs manifolds from quantum codes with novel geometric properties.
problem Creating manifolds with specific geometric constraints.
method Reverse engineering manifolds from quantum code chain complexes.
result First examples of power law Z 2 \mathbb{Z}_2 Z 2 systolic freedom. New algorithms learn from untrusted batches with improved efficiency.
problem Learning from untrusted batches with adversarial responses.
method Sum-of-Squares hierarchy applied to robust mean estimation.
result Reduces sample complexity to polylogarithmic in n n n for most natural distributions. The Fock-Bargmann-Hartogs domain D n , m ( μ ) D_{n,m}(μ) D n , m ( μ ) ( μ > 0 μ>0 μ > 0 ) in C n + m \mathbf{C}^{n+m} C n + m is defined by the inequality ∥ w ∥ 2 < e − μ ∥ z ∥ 2 , \|w\|^2<e^{-μ\|z\|^2}, ∥ w ∥ 2 < e − μ ∥ z ∥ 2 , where ( z , w ) ∈ C n × C m (z,w)\in \mathbf{C}^n\times \mathbf{C}^m ( z , w ) ∈ C n × C m , which is an unbounded non-hyperbolic domain in C n + m \mathbf{C}^{n+m} C n + m . Recently, Yamamori gave an explicit formula for the Bergman kernel of the…
New algorithm speeds up polynomial kernel approximations.
problem Efficiently approximating polynomial kernels of high degree.
method Oblivious sketching combined with novel sampling.
result Polynomial factor slowdown removed in running time.
Given a matrix A ∈ R n × d \mathbf{A}\in\mathbb{R}^{n\times d} A ∈ R n × d and a vector b ∈ R d b \in\mathbb{R}^{d} b ∈ R d , we show how to compute an ε ε ε -approximate solution to the regression problem min x ∈ R d 1 2 ∥ A x − b ∥ 2 2 \min_{x\in\mathbb{R}^{d}}\frac{1}{2} \|\mathbf{A} x - b\|_{2}^{2} min x ∈ R d 2 1 ∥ A x − b ∥ 2 2 in time O ~ ( ( n + d ⋅ κ sum ) ⋅ s ⋅ log ε − 1 ) \tilde{O} ((n+\sqrt{d\cdotκ_{\text{sum}}})\cdot s\cdot\logε^{-1}) O ~ (( n + d ⋅ κ sum ) ⋅ s ⋅ log ε − 1 ) where …
Improved bounds for estimating discrete distributions in KL divergence.
problem Estimating discrete distributions in KL divergence with accuracy.
method Used Laplace estimator and established concentration bounds.
result Deviation from mean scales as k / n \sqrt{k}/n k / n for n ≥ k n \ge k n ≥ k . This work proves that large models can be compressed significantly without losing performance.
problem Achieving comparable performance with smaller models and less data.
method Developed a universal compression theory for neural networks and datasets.
result Proved that a generic permutation-invariant function can be compressed into a function of polylogarithmic size with vanishing error.
Protocol learns pure quantum states with minimal disturbance.
problem Efficiently learn quantum states with minimal disturbance.
method Sequential measurements with minimal disturbance.
result Achieves maximal precision with polylogarithmic regret.
Improved guarantees for misspecified kernelized bandit optimization.
problem Misspecification in kernelized bandit optimization.
method Localization and domain splitting techniques.
result Logarithmic or polylogarithmic growth of misspecification amplification.
The paper analyzes GD for KANs, deriving bounds for training, generalization, and privacy.
problem Training dynamics, generalization, and privacy properties of KANs.
method Gradient Descent (GD) analysis for two-layer KANs under logistic loss and NTK-separable assumption.
result Polylogarithmic width suffices for GD to achieve optimization and generalization rates under DP.
New algorithms tackle heavy-tailed payoffs in linear stochastic bandits, matching lower bounds up to polylogarithmic factors.
problem Linear stochastic bandits with heavy-tailed payoffs.
method Median of means with well-designed allocation and truncation based on historical information.
result Regret upper bounds match the lower bound up to polylogarithmic factors.
New algorithms optimize without tuning, matching tuned SGD performance.
problem Optimizing machine learning models without manual hyperparameter tuning.
method Formalizes tuning-free algorithms for matching SGD performance with loose hints.
result Tuning-free algorithms can match SGD performance, but not optimal convergence rates.
We introduce a new approach for designing computationally efficient learning algorithms that are tolerant to noise, and demonstrate its effectiveness by designing algorithms with improved noise tolerance guarantees for learning linear separators. We consider both the malicious noise model and the adversarial label nois…
A new method for efficient Gaussian process inference using sparse approximations.
problem Scalable and accurate inference for latent Gaussian processes.
method Variational approximation with sparse inverse Cholesky factors and double Kullback-Leibler minimization.
result The proposed method can achieve highly accurate approximations with polylogarithmic time complexity.
New sampling method improves accuracy for constrained spaces.
problem Sampling from constrained convex subsets of R^d.
method Metropolis-adjusted Preconditioned Langevin Algorithm.
result High-accuracy sampling with polylogarithmic error dependence.