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.
This study shows the moment-SOS hierarchy converges in polynomial optimization over product of spheres.
problem Minimizing multihomogeneous polynomials over product of spheres.
method Moment-SOS hierarchy, local optimality conditions, differential geometry, Morse theory.
result The moment-SOS hierarchy has finite convergence for generic multihomogeneous objective functions.
Shallow neural networks can represent polynomials efficiently.
problem Representing polynomials using shallow neural networks.
method Using shallow neural networks of width 2(R+d)d to represent d-variate polynomials of degree R. result Derives minimax optimal convergence rate for shallow networks to unknown univariate regression functions.
Paper develops exact convex optimization for neural networks with polynomial activations.
problem Training two-layer neural networks with nonlinear polynomial activations.
method Exact convex optimization using semidefinite programming.
result Global optimization of neural networks is polynomial-time computable.
The paper provides an almost optimal learning and testing algorithm for sparse polynomials.
problem Learning and testing sparse multivariate polynomials efficiently.
method The paper presents an algorithm with sublinear query complexity in 1/ε and almost linear in s for learning and testing s-sparse polynomials. result The algorithm achieves almost optimal query complexity, making it the first of its kind.
LiPopt uses polynomial optimization to estimate neural network Lipschitz constants efficiently.
problem Estimating the Lipschitz constant of neural networks efficiently.
method Sparse polynomial optimization, leveraging network connectivity to reduce complexity.
result Superior estimates of the ℓ∞-Lipschitz constant compared to existing methods. Optimizes algorithms for non-concave bandit problems.
problem Optimizing algorithms for non-concave bandit problems.
method Unified zeroth-order optimization paradigm.
result Minimax-optimal algorithms in the dimension for low-rank generalized linear bandit problems.
New bounds for learning polynomial surrogates with L∞ guarantees.
problem Learning polynomial surrogates for bounded binary functions with L∞ error guarantees. method Characterized minimax sample complexity for two classes of polynomials under subgaussian noise.
result Sample complexity rates differ from noiseless case, scaling as nd+1 for degree d polynomials and ns2 for sparse polynomials. 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…
We develop a polynomial method to optimize trading in markets with transaction costs.
problem Optimizing trading strategies in markets with proportional transaction costs.
method Polynomial approximation of the residual value function to determine optimal trading strategies.
result Identify the trade-off between trading frequency and trade sizes for satisfactory agreement with theoretically optimal strategies.
The study optimizes polynomial regression for learning under Gaussian distributions.
problem Agnostic learning of Boolean and real-valued functions under Gaussian distributions.
method LP duality and polynomial degree analysis for L1-regression. result Optimal SQ lower bounds for various function classes.
We consider the problem of decomposing a multivariate polynomial as the difference of two convex polynomials. We introduce algebraic techniques which reduce this task to linear, second order cone, and semidefinite programming. This allows us to optimize over subsets of valid difference of convex decompositions (dcds) a…
Polynomial-time convex optimization for CNNs with ReLU activations.
problem Training Convolutional Neural Networks (CNNs) with ReLU activations.
method Developed a convex analytic framework using semi-infinite duality to formulate equivalent convex optimization problems for CNN architectures.
result Proved that two-layer CNNs can be globally optimized via an ℓ2 norm regularized convex program. Efficiently optimizes boolean functions using multilinear polynomials and exponential weight updates.
problem Optimizing boolean functions over the boolean hypercube with high computational cost.
method Proposes a computationally efficient algorithm using multilinear polynomials and exponential weight updates.
result Improves computational time up to several orders of magnitude compared to state-of-the-art algorithms.
This work explores representation complexity in RL paradigms, revealing model-based RL as the easiest task.
problem Investigating the representation complexity gap among model-based, policy-based, and value-based RL.
method Demonstrated through analysis of Markov decision processes (MDPs) and introduced new classes of MDPs.
result Representation complexity hierarchy: model-based RL > policy-based RL > value-based RL.
Optimizes privacy-preserving optimization for heavy-tailed data.
problem Privacy-preserving optimization with heavy-tailed gradients.
method Pure ε-differential privacy framework for Lipschitz extensions.
result Minimax optimal excess-risk rate for pure ε-DP heavy-tailed SCO.
SURF simplifies distribution estimation with simple, robust, and fast algorithms.
problem Efficient and accurate distribution estimation in statistics and machine learning.
method Piecewise polynomial approximation using empirical probability interpolation and divide-and-conquer merging.
result Surpassing state-of-the-art algorithms in efficiency and accuracy, SURF estimates distributions robustly and quickly.
Machine learning improves Buchberger's algorithm for polynomial systems.
problem Optimizing S-pair selection in Buchberger's algorithm for polynomial systems.
method Reinforcement learning agents trained with PPO to select S-pairs.
result Trained model outperforms existing heuristics in polynomial additions.
Gradient Descent with Projection learns low-degree polynomials efficiently.
problem Learning low-degree spherical polynomials with neural networks.
method Over-parameterized two-layer neural network with Gradient Descent with Projection.
result Achieves nearly minimax optimal sample complexity and risk bound.
New algorithm reduces dynamic regret for noisy gradient feedback with piecewise polynomial comparators.
problem Online estimation of piecewise polynomial trends with noisy feedback.
method Introduces variational constraint for piecewise polynomial comparators, designs adaptive algorithm.
result Achieves nearly optimal dynamic regret of $ ilde{O}(n^{ rac{1}{2k+3}}C_n^{ rac{2}{2k+3}})$.
This paper explores BDL hyperparameters for robust polynomial mapping with noise.
problem Designing BDL hyperparameters for robust function mapping with uncertainty quantification.
method Mapping Bayesian connectionist representations to polynomials of varying orders and noise types.
result Optimal network depth and ensemble size for prediction and uncertainty quantification.
New algorithm optimizes matrix reordering for noisy disordered matrices.
problem Optimizing matrix reordering for noisy disordered matrices in single-cell biology and metagenomics.
method Proposed a polynomial-time adaptive sorting algorithm to improve upon spectral seriation.
result Our algorithm achieves superior performance compared to existing methods in real datasets.
Paper identifies reductive MDPs, solving them in polynomial time.
problem Computational hardness of general MDPs and tractability of finite-horizon MDPs.
method Defines reductivity, a new class of SSPs, and develops a polynomial-time solution.
result Optimal policies can be found in polynomial time for reductive SSPs and MDPs.
Polynomial-time algorithm for near-optimal community detection in graphs.
problem Node-private community estimation in stochastic block models.
method Explicit Lipschitz surrogate and accept-reject algorithm for sampling community labels.
result Achieves minimax rates for exact recovery with polynomial-time runtime and logarithmic privacy parameter.
For any manifold with polynomial volume growth, we show: The dimension of the space of ancient caloric functions with polynomial growth is bounded by the degree of growth times the dimension of harmonic functions with the same growth. As a consequence, we get a sharp bound for the dimension of ancient caloric functions…
Develops exact convex optimization formulations for neural networks.
problem Training two-layer neural networks with rectified linear units.
method Uses semi-infinite duality and minimum norm regularization to develop exact convex optimization formulations.
result Shows equivalence of ReLU networks trained with weight decay to block ℓ1 penalized convex models. We propose a method called ideal regression for approximating an arbitrary system of polynomial equations by a system of a particular type. Using techniques from approximate computational algebraic geometry, we show how we can solve ideal regression directly without resorting to numerical optimization. Ideal regression…
The paper solves the optimal transport problem between algebraic hypersurfaces.
problem Optimal deformation of projective hypersurfaces.
method Measure theory and optimal transport, embedding into measure space, constrained dynamical formulation.
result Introduction of an inner Wasserstein distance finer than the Fubini-Study distance.
We tackle tensor denoising with unknown permutations, achieving optimal recovery with polynomial estimators.
problem Structured tensor denoising with unknown permutations in recommendation systems, neuroimaging, etc.
method Developed a constrained least-squares estimator in a block-wise polynomial family.
result Achieved the minimax error bound with polynomial estimators of degree up to (m−2)(m+1)/2. 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 algorithms find half-optimal independent sets in sparse graphs.
problem Finding large independent sets in sparse random graphs.
method Low-degree polynomial algorithms.
result Low-degree polynomial algorithms can find independent sets of half-optimal size.
Unified approach to experimental design using interlacing polynomials.
problem Experimental design problems, especially D/A/E-design and E-design.
method Unified deterministic approach using interlacing polynomials.
result Improved approximation guarantees for various experimental design objectives.
New algorithms improve privacy in statistical estimation by making them robust.
problem Improving privacy in statistical estimation methods.
method Black-box reduction from privacy to robustness, using Sum-of-Squares method.
result Design of polynomial-time private estimators with optimal tradeoffs among sample complexity, accuracy, and privacy.
This article describes a multivariate polynomial regression method where the uncertainty of the input parameters are approximated with Gaussian distributions, derived from the central limit theorem for large weighted sums, directly from the training sample. The estimated uncertainties can be propagated into the optimal…
New method tackles geodesically convex optimization with polynomial convergence.
problem Designing an efficient algorithm for geodesically convex optimization.
method Ellipsoid-like algorithm with polynomial query and per-query complexity.
result Achieves polynomial convergence for geodesically convex functions.
Study non-linear combinatorial bandits with polynomial rewards, finding significant differences from linear cases.
problem Adversarial combinatorial bandits with general non-linear reward functions.
method Extending existing work on adversarial linear combinatorial bandits, analyzing minimax optimal regret for polynomial and non-polynomial reward functions.
result Minimax optimal regret bounds for adversarial combinatorial bandits with general non-linear reward functions.
We study deep neural networks with polynomial activations, particularly their expressive power. For a fixed architecture and activation degree, a polynomial neural network defines an algebraic map from weights to polynomials. The image of this map is the functional space associated to the network, and it is an irreduci…
Random covers of hyperbolic surfaces have a spectral gap with polynomial rate.
problem Finding spectral gaps in random covers of hyperbolic surfaces.
method Applying recent work on spectral gaps to uniformly random covers of closed hyperbolic surfaces.
result Uniformly random degree-n covers of a closed hyperbolic surface have no new Laplacian eigenvalues below a specific threshold with high probability.
We prove near-tight concentration of measure for polynomial functions of the Ising model under high temperature. For any degree d, we show that a degree-d polynomial of a n-spin Ising model exhibits exponential tails that scale as exp(−r2/d) at radius r=Ω~d(nd/2). Our concentration radius is opti…
This work finds a point with small test error in polynomial time for mildly overparameterized neural nets.
problem Achieving small test error in mildly overparameterized neural networks.
method The work shows that the landscape of loss functions with explicit regularization has a property that all local minima and certain stationary points achieve small test error. It also proves the existence of polynomial time algorithms for finding such points in convolutional and fully connected neural nets.
result Polynomial time algorithms exist for finding points with small test error in mildly overparameterized neural nets.
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.
Predicts the number of polynomial additions in Buchberger's algorithm using machine learning.
problem Predict the number of polynomial additions in Buchberger's algorithm.
method Multiple linear regression and recursive neural network models trained on ideal generator statistics.
result Machine learning can predict the number of polynomial additions in Buchberger's algorithm.
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)…
Computer simulation has become the standard tool in many engineering fields for designing and optimizing systems, as well as for assessing their reliability. To cope with demanding analysis such as optimization and reliability, surrogate models (a.k.a meta-models) have been increasingly investigated in the last decade.…
Enhanced PC2 improves surrogate modeling for high-dimensional problems.
problem Degrading performance and efficiency of PC2 in high-dimensional parameter spaces. method Integrates SULM solver and D-optimal sampling strategy into PC2 framework. result Enhanced PC2 demonstrates better comprehensive capability and efficiency. This work improves polynomial approximations for functions with asymmetric behavior.
problem Efficiently approximating functions with asymmetric behavior, especially those growing unbounded on one side.
method Introduces weighted deep polynomial approximants that combine learnable deep polynomials with one-sided weights.
result Weighted deep polynomial approximants outperform existing methods in approximating functions with asymmetric behavior.
The paper explores how polynomial roots and operator eigenvalues change with parameters.
problem How do roots of polynomials and eigenvalues of operators vary with parameter changes?
method Analyzes parameter dependence of polynomials and linear operators, covering real analytic to differentiable of finite order.
result Definitive optimal results for perturbation theory of polynomials and linear operators, including hyperbolic polynomials.
We show that the class of strongly connected graphical models with treewidth at most k can be properly efficiently PAC-learnt with respect to the Kullback-Leibler Divergence. Previous approaches to this problem, such as those of Chow ([1]), and Ho gen ([7]) have shown that this class is PAC-learnable by reducing it to …