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1. Statistical-Query Lower Bounds via Functional Gradients

   Goel, S; Gollakota, A; Klivans, A. (January 2020, Advances in neural information processing systems)

   null (Ed.)

   We give the first statistical-query lower bounds for agnostically learning any non-polynomial activation with respect to Gaussian marginals (e.g., ReLU, sigmoid, sign). For the specific problem of ReLU regression (equivalently, agnostically learning a ReLU), we show that any statistical-query algorithm with tolerance n−(1/ϵ)b must use at least 2ncϵ queries for some constant b,c>0, where n is the dimension and ϵ is the accuracy parameter. Our results rule out general (as opposed to correlational) SQ learning algorithms, which is unusual for real-valued learning problems. Our techniques involve a gradient boosting procedure for "amplifying" recent lower bounds due to Diakonikolas et al. (COLT 2020) and Goel et al. (ICML 2020) on the SQ dimension of functions computed by two-layer neural networks. The crucial new ingredient is the use of a nonstandard convex functional during the boosting procedure. This also yields a best-possible reduction between two commonly studied models of learning: agnostic learning and probabilistic concepts. 

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2. Superpolynomial Lower Bounds for Learning One-Layer Neural Networks using Gradient Descent

   Goel, Surbhi; Gollakota, Aravind; Jin, Zhihan; Karmalkar, Sushrut; Klivans, Adam (July 2020, International Conference on Machine Learning)

   We prove the first superpolynomial lower bounds for learning one-layer neural networks with respect to the Gaussian distribution using gradient descent. We show that any classifier trained using gradient descent with respect to square-loss will fail to achieve small test error in polynomial time given access to samples labeled by a one-layer neural network. For classification, we give a stronger result, namely that any statistical query (SQ) algorithm (including gradient descent) will fail to achieve small test error in polynomial time. Prior work held only for gradient descent run with small batch sizes, required sharp activations, and applied to specific classes of queries. Our lower bounds hold for broad classes of activations including ReLU and sigmoid. The core of our result relies on a novel construction of a simple family of neural networks that are exactly orthogonal with respect to all spherically symmetric distributions. 

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3. Learning Neural Networks with Two Nonlinear Layers in Polynomial Time

   Goel, Surbhi; Klivans, Adam (June 2019, Conference on Learning Theory)

   We give a polynomial-time algorithm for learning neural networks with one layer of sigmoids feeding into any Lipschitz, monotone activation function (e.g., sigmoid or ReLU). We make no assumptions on the structure of the network, and the algorithm succeeds with respect to {\em any} distribution on the unit ball in n dimensions (hidden weight vectors also have unit norm). This is the first assumption-free, provably efficient algorithm for learning neural networks with two nonlinear layers. Our algorithm-- Alphatron-- is a simple, iterative update rule that combines isotonic regression with kernel methods. It outputs a hypothesis that yields efficient oracle access to interpretable features. It also suggests a new approach to Boolean learning problems via real-valued conditional-mean functions, sidestepping traditional hardness results from computational learning theory. Along these lines, we subsume and improve many longstanding results for PAC learning Boolean functions to the more general, real-valued setting of {\em probabilistic concepts}, a model that (unlike PAC learning) requires non-i.i.d. noise-tolerance. 

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4. Tight Hardness Results for Training Depth-2 ReLU Networks

   Goel, Surbhi; Klivans, Adam; Manurangsi, Pasin; Reichman, Daniel (January 2021, ITCS 2021)

   We prove several hardness results for training depth-2 neural networks with the ReLU activation function; these networks are simply weighted sums (that may include negative coefficients) of ReLUs. Our goal is to output a depth-2 neural network that minimizes the square loss with respect to a given training set. We prove that this problem is NP-hard already for a network with a single ReLU. We also prove NP-hardness for outputting a weighted sum of k ReLUs minimizing the squared error (for k>1) even in the realizable setting (i.e., when the labels are consistent with an unknown depth-2 ReLU network). We are also able to obtain lower bounds on the running time in terms of the desired additive error ϵ. To obtain our lower bounds, we use the Gap Exponential Time Hypothesis (Gap-ETH) as well as a new hypothesis regarding the hardness of approximating the well known Densest k-Subgraph problem in subexponential time (these hypotheses are used separately in proving different lower bounds). For example, we prove that under reasonable hardness assumptions, any proper learning algorithm for finding the best fitting ReLU must run in time exponential in (1/epsilon)^2. Together with a previous work regarding improperly learning a ReLU (Goel et al., COLT'17), this implies the first separation between proper and improper algorithms for learning a ReLU. We also study the problem of properly learning a depth-2 network of ReLUs with bounded weights giving new (worst-case) upper bounds on the running time needed to learn such networks both in the realizable and agnostic settings. Our upper bounds on the running time essentially matches our lower bounds in terms of the dependency on epsilon. 

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5. On the Cryptographic Hardness of Learning Single Periodic Neurons

   Min Jae Song; Ilias Zadik; Joan Bruna (December 2021, Advances in neural information processing systems)

   We show a simple reduction which demonstrates the cryptographic hardness of learning a single periodic neuron over isotropic Gaussian distributions in the pres- ence of noise. More precisely, our reduction shows that any polynomial-time algorithm (not necessarily gradient-based) for learning such functions under small noise implies a polynomial-time quantum algorithm for solving worst-case lattice problems, whose hardness form the foundation of lattice-based cryptography. Our core hard family of functions, which are well-approximated by one-layer neural networks, take the general form of a univariate periodic function applied to an affine projection of the data. These functions have appeared in previous seminal works which demonstrate their hardness against gradient-based (Shamir’18), and Statisti- cal Query (SQ) algorithms (Song et al.’17). We show that if (polynomially) small noise is added to the labels, the intractability of learning these functions applies to all polynomial-time algorithms, beyond gradient-based and SQ algorithms, under the aforementioned cryptographic assumptions. Moreover, we demonstrate the necessity of noise in the hardness result by designing a polynomial-time algorithm for learning certain families of such functions under exponentially small adversarial noise. Our proposed algorithm is not a gradient-based or an SQ algorithm, but is rather based on the celebrated Lenstra-Lenstra-Lovász (LLL) lattice basis reduction algorithm. Furthermore, in the absence of noise, this algorithm can be directly applied to solve CLWE detection (Bruna et al.’21) and phase retrieval with an optimal sample complexity of d + 1 samples. In the former case, this improves upon the quadratic-in-d sample complexity required in (Bruna et al.’21). 

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