New polynomial-time solutions found for training ReLU networks, mirroring Max-Cut complexity.
problem Training two-layer ReLU neural networks with weight decay regularization.
method Developed a convex formulation and randomized algorithm to find approximate global optimizers.
result First polynomial-time approximation guarantees and hardness of approximation results for regularized ReLU networks.
Efficiently finds sparse solutions to max-plus equations for convex regression.
problem Finding sparse solutions to max-plus equations for convex multivariate regression.
method Polynomial-time algorithm for sparse approximate solutions.
result Optimal piecewise-linear fitting with minimum number of regions.
Strongly polynomial algorithm for approximate Forster transforms and halfspace learning.
problem Computing approximate Forster transforms and halfspace learning.
method Strongly polynomial time algorithm for approximate Forster transforms and halfspace learning.
result First strongly polynomial time algorithm for distribution-free PAC learning of halfspaces.
New algorithms improve approximation of matrix norms, with applications in statistics and machine learning.
problem Improving approximation of matrix norms for 2ightarrowq in polynomial time. method Polynomial-time multiplicative approximation algorithms for 2ightarrowq norm, leveraging sum-of-squares certificates. result Achieved polynomially improved approximation factors, notably d1/8 for q=4. Two new algorithms improve robust PCA and Schatten packing.
problem Robustly estimating the top eigenvector of corrupted sub-Gaussian data.
method Two iterative filtering and nearly-linear time algorithms.
result First polynomial-time algorithms for non-trivial covariance estimation.
The problem of high-dimensional path-dependent optimal stopping (OS) is important to multiple academic communities and applications. Modern OS tasks often have a large number of decision epochs, and complicated non-Markovian dynamics, making them especially challenging. Standard approaches, often relying on ADP, dualit…
Polynomial-time RL algorithm for constant actions under linear Bellman completeness.
problem Efficient online reinforcement learning with few actions.
method Polynomial-time algorithm based on linear function approximation.
result First computationally efficient algorithm for RL with constant actions under linear Bellman completeness.
Optimal transport is #P-hard when components are independent, even with approximate solutions.
problem Computational complexity of optimal transport with independent marginals.
method Proved #P-hardness and developed a pseudo-polynomial time approximation algorithm.
result Optimal transport is #P-hard even with independent components and approximate solutions.
The paper studies randomized approximations of Tukey's depth for log-concave isotropic data.
problem The challenge of approximating Tukey's depth in high dimensions.
method The study examines randomized algorithms for approximating Tukey's depth for log-concave isotropic data.
result Randomized algorithms correctly approximate maximal depth and close to zero depths but not intermediate depths.
Designs efficient algorithms to maximize the expectation of Gaussian random variables.
problem Maximizing the expectation of the supremum of Gaussian random variables.
method Polynomial time approximation scheme and O(logn) approximation algorithm for general m>1. result Characterizes optimal variance allocation and provides approximation algorithms.
The problem of explaining the behavior of deep neural networks has recently gained a lot of attention. While several attribution methods have been proposed, most come without strong theoretical foundations, which raises questions about their reliability. On the other hand, the literature on cooperative game theory sugg…
Hard to approximate critical points for simple nonconvex functions.
problem Approximating critical points of nonconvex functions.
method Proving hardness results for polynomial-time approximation of critical points.
result Proving that approximating critical points is intractable for simple nonconvex functions.
EKM solves the K-medoids problem in polynomial time.
problem The K-medoids problem in data analysis. method EKM is a novel algorithm using transformational programming and combinatorial generation.
result EKM solves the K-medoids problem in worst-case $O\left(N^{K+1}
ight)$ time complexity. Paper improves likelihood estimation for discrete distributions.
problem Computing profile maximum likelihood for discrete distributions.
method New bounds on Bethe and Sinkhorn permanents for low rank matrices.
result Achieves an approximation factor of exp(-O(sqrt(n) log n)) in polynomial time.
Algorithm learns arbitrary ReLU neurons under Gaussian inputs.
problem Learn an arbitrary ReLU activation over Gaussian marginals.
method Statistical Query (SQ) algorithm that outputs a ReLU activation achieving O(OPT)+ε loss. result First constant factor approximation for arbitrary bias in polynomial time.
The experimental design problem concerns the selection of k points from a potentially large design pool of p-dimensional vectors, so as to maximize the statistical efficiency regressed on the selected k design points. Statistical efficiency is measured by optimality criteria, including A(verage), D(eterminant), T(race)…
New algorithm samples from Ising models efficiently, even with outliers.
problem Sampling from Ising models with general interaction matrices.
method Combines MCMC and variational inference techniques.
result First polynomial time sampling algorithms for low-rank Ising models.
Estimating symmetric properties of a distribution, e.g. support size, coverage, entropy, distance to uniformity, are among the most fundamental problems in algorithmic statistics. While each of these properties have been studied extensively and separate optimal estimators are known for each, in striking recent work, Ac…
Paper shows Burer-Monteiro method can solve SDPs in polynomial time under smoothed analysis.
problem Solving large-scale semidefinite programs (SDPs) efficiently.
method Perturbing SDP to create a nonconvex program in Y where Y is an nimesp matrix. result The Burer-Monteiro method can solve SDPs to any desired accuracy in polynomial time under certain conditions.
Polynomial-time reachability for LTI systems with TLL NN controllers is achieved.
problem Bounding the reachable set of LTI systems controlled by TLL NN controllers.
method Polynomial-time computation of exact one-step reachable set and tight bounding box via two methods.
result Exact reachability computation in polynomial time for TLL NN controllers.
Paper proposes efficient SHAP computation methods.
problem Efficient computation of SHAP values for machine learning models.
method Develops polynomial time methods for SHAP computation based on model structure.
result Exact SHAP computation in polynomial time for various model structures.
Principal components analysis (PCA) is the optimal linear auto-encoder of data, and it is often used to construct features. Enforcing sparsity on the principal components can promote better generalization, while improving the interpretability of the features. We study the problem of constructing optimal sparse linear a…
We study approximations of the partition function of dense graphical models. Partition functions of graphical models play a fundamental role is statistical physics, in statistics and in machine learning. Two of the main methods for approximating the partition function are Markov Chain Monte Carlo and Variational Method…
We study the problem of stochastic combinatorial pure exploration (CPE), where an agent sequentially pulls a set of single arms (a.k.a. a super arm) and tries to find the best super arm. Among a variety of problem settings of the CPE, we focus on the full-bandit setting, where we cannot observe the reward of each singl…
We show that the existence of a computationally efficient calibration algorithm, with a low weak calibration rate, would imply the existence of an efficient algorithm for computing approximate Nash equilibria - thus implying the unlikely conclusion that every problem in PPAD is solvable in polynomial time.
The aim of this short note is to draw attention to a method by which the partition function and marginal probabilities for a certain class of random fields on complete graphs can be computed in polynomial time. This class includes Ising models with homogeneous pairwise potentials but arbitrary (inhomogeneous) unary pot…
We develop an efficient algorithm to find confidence ellipsoids with volume guarantees in high dimensions.
problem Finding robust confidence ellipsoids in high-dimensional data.
method Polynomial time algorithm using primal-dual structure and geometric Brascamp-Lieb inequality.
result Algorithm finds ellipsoids within a O(β)γd volume factor of best β-conditioned ellipsoid. Combining diffusion models with Langevin dynamics improves posterior sampling efficiency.
problem Sampling from noisy posterior distributions efficiently.
method Annealed Langevin dynamics combined with diffusion models.
result Achieves posterior sampling in polynomial time with a weaker score error bound.
Polynomial-time algorithm estimates mean with bounded covariance using differential privacy.
problem Estimating mean of a d-variate distribution with differential privacy constraints.
method Sum of Squares (SoS) exponential mechanism for polynomial-time differentially private estimation.
result First polynomial-time algorithm with O(d) samples for mean estimation under pure differential privacy. New algorithm reduces combinatorial semi-bandit regret efficiently.
problem Optimizing rewards from uncorrelated items in combinatorial semi-bandits.
method Developed an approximate version of ESCB with polynomial complexity.
result Achieved statistically efficient and polynomial time algorithm for combinatorial semi-bandits.
Polynomial-time method solves complex combinatorial semi-bandits.
problem Optimal strategies for combinatorial semi-bandits with uncorrelated Gaussian rewards.
method Proposes a polynomial-time method to solve the Graves-Lai optimization problem for various combinatorial structures.
result First known approach to implement asymptotically optimal algorithms in polynomial time for combinatorial semi-bandits.
Optimal intervention in economic networks modeled as influence maximization, with hard computational problems.
problem Optimal intervention in economic networks modeled as influence maximization.
method Transformed into influence maximization-like form, with theoretical and practical implications.
result Optimal intervention is NP-hard and cannot be approximated to a constant factor in polynomial time.
Polynomial time algorithm matches correlated Gaussian matrices without vanishing correlation.
problem Matching vertices in two correlated Erdős-Rényi graphs.
method Iterative matching algorithm for correlated Gaussian Wigner matrices.
result First polynomial time algorithm for graph matching with arbitrarily small constant correlation.
Efficient algorithms learn causal graphs with minimal interventions.
problem Learning causal relationships between observed variables in the presence of latents.
method Bi-criteria approximation goal combining intervention design and graph property testing.
result Achieve intervention cost within a small constant factor of the optimal.
We propose practical algorithms for entrywise ℓp-norm low-rank approximation, for p=1 or p=∞. The proposed framework, which is non-convex and gradient-based, is easy to implement and typically attains better approximations, faster, than state of the art. From a theoretical standpoint, we show that th…
Polynomial-time algorithm for estimating covariance in corrupted Gaussian data.
problem Estimating covariance in data with up to 1-α fraction of adversarial corruptions.
method Uses low-degree sum-of-squares certificates for anti-concentration and hypercontractivity.
result Outputs a list of candidate parameters with high probability containing a nearly correct covariance.
Optimal analysis of subset-selection based L_p low rank approximation.
problem Finding a rank-k matrix X to minimize the entry-wise L_p loss of matrix A.
method Column subset selection algorithm with improved approximation ratio using Riesz-Thorin interpolation theorem.
result Improved approximation ratio for subset selection based L_p low rank approximation.
This paper finds efficient algorithms for approximating Markov networks with k-tree topologies.
problem Efficiently approximating Markov networks with complex topologies.
method Developed O(n^{k+1})-time algorithms for finding maximum spanning k-trees (MSkT) that retain certain subgraphs.
result Optimal approximation of Markov networks with k-tree topology is achieved in polynomial time.
Efficient algorithms for WRT invariants of torus bundles using algebraic structures.
problem Computing topological invariants of 3-manifolds is generally intractable.
method Embedding skein algebra into symmetric subalgebra at roots of unity for polynomial-time classical computation and using quantum algorithms for exponential space advantage.
result Polynomial-time classical computation and quantum algorithms for WRT invariants of torus bundles.
Massively parallel RWS improves inference in complex models.
problem Exponential sample requirement for effective importance weighting.
method Draws K samples of all n latent variables and individually reasons through all combinations. result Significant improvements over standard RWS.
Polynomial-time methods count and sample DAGs from equivalence classes.
problem Counting and sampling DAGs from Markov equivalence classes.
method Polynomial-time algorithms for DAGs.
result Counting and sampling can be done in polynomial time.
We present the strongest known knot invariant that can be computed effectively (in polynomial time).
Efficiently plans large MDPs with weak function approximations.
problem Planning in large MDPs with limited function approximation capabilities.
method Uses linear value function approximation with weak requirements and a generative oracle.
result Produces almost-optimal actions for any state with polynomial computation time.
Smooth Schrödinger Bridges improve trajectory inference by smoothing Gaussian processes.
problem Improving trajectory inference in applications like particle tracking.
method Generalizes Schrödinger Bridge problem to smooth Gaussian processes, solving the problem on phase space.
result The method outperforms existing methods on real datasets.
Investigates polynomial time algorithms for computing Khovanov homology of braids.
problem Computing Khovanov homology for general braids is intractable.
method Examines polynomial time algorithms for 3-braids and a variation of the scanning algorithm for more general braids.
result Shows that for 3-braids, Khovanov homology can be computed in polynomial time, while for more general braids, it can be computed in polynomial time for bounded homological degrees.
New lower bound shows RL with linear approximations is computationally hard.
problem Computational hardness in reinforcement learning with linear function approximation.
method Reduction from Unique-Sat to MDPs with linear optimal value functions.
result No randomized polynomial time algorithm exists for RL with linear function approximation unless NP=RP.
This work studies the linear approximation of high-dimensional dynamical systems using low-rank dynamic mode decomposition (DMD). Searching this approximation in a data-driven approach is formalised as attempting to solve a low-rank constrained optimisation problem. This problem is non-convex and state-of-the-art algor…
This paper establishes for the first time the predictive performance of speed priors and their computational complexity. A speed prior is essentially a probability distribution that puts low probability on strings that are not efficiently computable. We propose a variant to the original speed prior (Schmidhuber, 2002),…