Learning sum of diverse features: computational hardness and efficient gradient-based training for ridge combinations

• URL: http://arxiv.org/abs/2406.11828v1
• Date: Mon, 17 Jun 2024 17:59:17 GMT
• Title: Learning sum of diverse features: computational hardness and efficient gradient-based training for ridge combinations
• Authors: Kazusato Oko, Yujin Song, Taiji Suzuki, Denny Wu,
• Abstract summary: We study the computational complexity of learning a target function $f_*:mathbbRdtomathbbR$ with additive structure. We prove that a large subset of $f_*$ can be efficiently learned by gradient training of a two-layer neural network.
• Score: 40.77319247558742
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the computational and sample complexity of learning a target function $f_*:\mathbb{R}^d\to\mathbb{R}$ with additive structure, that is, $f_*(x) = \frac{1}{\sqrt{M}}\sum_{m=1}^M f_m(\langle x, v_m\rangle)$, where $f_1,f_2,...,f_M:\mathbb{R}\to\mathbb{R}$ are nonlinear link functions of single-index models (ridge functions) with diverse and near-orthogonal index features $\{v_m\}_{m=1}^M$, and the number of additive tasks $M$ grows with the dimensionality $M\asymp d^\gamma$ for $\gamma\ge 0$. This problem setting is motivated by the classical additive model literature, the recent representation learning theory of two-layer neural network, and large-scale pretraining where the model simultaneously acquires a large number of "skills" that are often localized in distinct parts of the trained network. We prove that a large subset of polynomial $f_*$ can be efficiently learned by gradient descent training of a two-layer neural network, with a polynomial statistical and computational complexity that depends on the number of tasks $M$ and the information exponent of $f_m$, despite the unknown link function and $M$ growing with the dimensionality. We complement this learnability guarantee with computational hardness result by establishing statistical query (SQ) lower bounds for both the correlational SQ and full SQ algorithms.

Related papers

• On the Complexity of Learning Sparse Functions with Statistical and Gradient Queries [25.03801392644285]
  We provide a tight characterization of the query complexity of $mathsfDLQ$ for learning the support of a sparse function over generic product distributions. For the squared loss, $mathsfDLQ$ matches the complexity of Correlation Statistical Queries $(mathsfCSQ)$--potentially much worse than $mathsfSQ$. We also provide evidence that $mathsfDLQ$ can indeed capture learning with (stochastic) gradient descent by showing it correctly describes the complexity of learning with a two-layer
  arXiv  Detail & Related papers  (2024-07-08T05:30:34Z)
• Neural network learns low-dimensional polynomials with SGD near the information-theoretic limit [75.4661041626338]
  We study the problem of gradient descent learning of a single-index target function $f_*(boldsymbolx) = textstylesigma_*left(langleboldsymbolx,boldsymbolthetarangleright)$ under isotropic Gaussian data. We prove that a two-layer neural network optimized by an SGD-based algorithm learns $f_*$ of arbitrary link function with a sample and runtime complexity of $n asymp T asymp C(q) cdot d
  arXiv  Detail & Related papers  (2024-06-03T17:56:58Z)
• Learning Hierarchical Polynomials with Three-Layer Neural Networks [56.71223169861528]
  We study the problem of learning hierarchical functions over the standard Gaussian distribution with three-layer neural networks. For a large subclass of degree $k$s $p$, a three-layer neural network trained via layerwise gradientp descent on the square loss learns the target $h$ up to vanishing test error. This work demonstrates the ability of three-layer neural networks to learn complex features and as a result, learn a broad class of hierarchical functions.
  arXiv  Detail & Related papers  (2023-11-23T02:19:32Z)
• Efficiently Learning One-Hidden-Layer ReLU Networks via Schur Polynomials [50.90125395570797]
  We study the problem of PAC learning a linear combination of $k$ ReLU activations under the standard Gaussian distribution on $mathbbRd$ with respect to the square loss. Our main result is an efficient algorithm for this learning task with sample and computational complexity $(dk/epsilon)O(k)$, whereepsilon>0$ is the target accuracy.
  arXiv  Detail & Related papers  (2023-07-24T14:37:22Z)
• On Function Approximation in Reinforcement Learning: Optimism in the Face of Large State Spaces [208.67848059021915]
  We study the exploration-exploitation tradeoff at the core of reinforcement learning. In particular, we prove that the complexity of the function class $mathcalF$ characterizes the complexity of the function. Our regret bounds are independent of the number of episodes.
  arXiv  Detail & Related papers  (2020-11-09T18:32:22Z)
• Learning Over-Parametrized Two-Layer ReLU Neural Networks beyond NTK [58.5766737343951]
  We consider the dynamic of descent for learning a two-layer neural network. We show that an over-parametrized two-layer neural network can provably learn with gradient loss at most ground with Tangent samples.
  arXiv  Detail & Related papers  (2020-07-09T07:09:28Z)
• Sample Efficient Reinforcement Learning via Low-Rank Matrix Estimation [30.137884459159107]
  We consider the question of learning $Q$-function in a sample efficient manner for reinforcement learning with continuous state and action spaces. We develop a simple, iterative learning algorithm that finds $epsilon$-Schmidt $Q$-function with sample complexity of $widetildeO(frac1epsilonmax(d1), d_2)+2)$ when the optimal $Q$-function has low rank $r$ and the factor $$ is below a certain threshold.
  arXiv  Detail & Related papers  (2020-06-11T00:55:35Z)
• A Corrective View of Neural Networks: Representation, Memorization and Learning [26.87238691716307]
  We develop a corrective mechanism for neural network approximation. We show that two-layer neural networks in the random features regime (RF) can memorize arbitrary labels. We also consider three-layer neural networks and show that the corrective mechanism yields faster representation rates for smooth radial functions.
  arXiv  Detail & Related papers  (2020-02-01T20:51:09Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.