# Learning Deep ReLU Networks Is Fixed-Parameter Tractable

@article{Chen2022LearningDR, title={Learning Deep ReLU Networks Is Fixed-Parameter Tractable}, author={Sitan Chen and Adam R. Klivans and Raghu Meka}, journal={2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)}, year={2022}, pages={696-707} }

We consider the problem of learning an unknown ReLU network with respect to Gaussian inputs and obtain the first nontrivial results for networks of depth more than two. We give an algorithm whose running time is a fixed polynomial in the ambient dimension and some (exponentially large) function of only the network's parameters. Our results provably cannot be obtained using gradient-based methods and give the first example of a class of efficiently learnable neural networks that gradient descent…

## 19 Citations

### The Computational Complexity of ReLU Network Training Parameterized by Data Dimensionality

- Computer ScienceJ. Artif. Intell. Res.
- 2022

This work provides running time lower bounds in terms of W[1]-hardness for parameter d and proves that known brute-force strategies are essentially optimal (assuming the Exponential Time Hypothesis).

### Learning (Very) Simple Generative Models Is Hard

- Computer ScienceArXiv
- 2022

The key ingredient in the proof is an ODE-based construction of a compactly supported, piecewise-linear function f with polynomially-bounded slopes such that the pushforward of N under f matches all low-degree moments of N (0, 1).

### Bounding the Width of Neural Networks via Coupled Initialization - A Worst Case Analysis

- Computer ScienceICML
- 2022

This work shows how to signiﬁcantly reduce the number of neurons required for two-layer ReLU networks, both in the under-parameterized setting with logistic loss and with squared loss, and proves new lower bounds that improve upon prior work, and that under certain assumptions, are best possible.

### Algorithms for Efficiently Learning Low-Rank Neural Networks

- Computer ScienceArXiv
- 2022

This work presents a provably efficient algorithm which learns an optimal low-rank approximation to a single-hidden-layer ReLU network up to additive error with probability ≥ 1 − δ, given access to noiseless samples with Gaussian marginals in polynomial time and samples.

### Training Fully Connected Neural Networks is ∃R-Complete

- Computer ScienceArXiv
- 2022

The algorithmic problem of finding the optimal weights and biases for a two-layer fully connected neural network to a given set of data points is considered and it is shown that even very simple networks are difficult to train.

### Hardness of Noise-Free Learning for Two-Hidden-Layer Neural Networks

- Computer ScienceArXiv
- 2022

Superpolynomial statistical query lower bounds for learning two-hidden-layer ReLU networks with respect to Gaussian inputs in the standard (noise-free) model are given and a lifting procedure due to Daniely and Vardi is shown that reduces Boolean PAC learning problems toGaussian ones.

### Training Fully Connected Neural Networks is $\exists\mathbb{R}$-Complete

- Computer Science
- 2022

The algorithmic problem of finding the optimal weights and biases for a two-layer fully connected neural network to a given set of data points is considered and it is shown that even very simple networks are difficult to train.

### Efficient Algorithms for Learning Depth-2 Neural Networks with General ReLU Activations

- Computer ScienceNeurIPS
- 2021

This work considers learning an unknown network of the form f(x) = aσ(Wx+b), where x is drawn from the Gaussian distribution, and σ(t) := max(t, 0) is the ReLU activation.

### A Convergence Analysis of Gradient Descent on Graph Neural Networks

- Computer ScienceNeurIPS
- 2021

It is proved that for the case of deep linear GNNs gradient descent provably recovers solutions up to error in O(log(1/ )) iterations, under natural assumptions on the data distribution.

### Efficiently Learning Any One Hidden Layer ReLU Network From Queries

- Computer ScienceArXiv
- 2021

This work gives the first polynomial-time algorithm for learning one hidden layer neural networks provided black-box access to the network, and it is shown that if F is an arbitrary onehidden layer neural network with ReLU activations, there is an algorithm with query complexity and running time that outputs a network F achieving low square loss relative to F with respect to the Gaussian measure.

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