Polynomial Learning of Positive ReLU Networks under Gaussian Inputs

We study the fundamental problem of PAC learning positive linear combinations of $k$ arbitrary ReLU activations on $\mathbb{R}^d$ with respect to the Gaussian distribution, in $L_2$ loss. We give the first fully-polynomial time algorithm for this learning task. Specifically, our algorithm draws $\mathrm{poly}(d, k, 1/ε)$ samples, runs in $\mathrm{poly}(d, k, 1/ε)$ time, and outputs a hypothesis that is $ε$-close to the target concept with high probability. All prior algorithms for this problem require super-polynomial time in $k$ or $1/ε$, or leverage strong well-conditioned assumptions on the weight matrix. On a technical level, our approach develops a novel boosting procedure of a weak'' learner that recovers the hidden weight vectors to constant error, in a list-decodable manner. To develop this weak'' learning procedure, we leverage implicit tensor computation techniques to construct a low-degree filtering procedure based on Hermite tensors of the target function. This procedure amplifies separation among (clusters of) weight vectors to learn each of them to constant error. We then argue that the same procedure can be used for improving the estimation of hidden weight vectors, using gradient descent-like updates on the Euclidean sphere. A crucial ingredient of our algorithm is a pruning approach that ensures only relevant vectors that approximate the hidden weight vectors at target accuracy are maintained by the algorithm.

We study exact verification of ReLU networks in an adversarial smoothed model. Every network weight and bias is independently perturbed by Gaussian noise, clipped to $[-2,2]$, and rounded to the exact dyadic grid determined by the input bit complexity. We show that, under the standard assumption $\mathrm{NP}\not\subseteq\mathrm{BPP}$, there is no sound and complete verifier whose expected running time is polynomial in network size, bit complexity, and inverse noise level for every base instance. The conclusion already holds at the fixed noise level $σ_\star=2^{-11}$ for one-hidden-layer networks over a unit box, with hidden fan-in at most three and base coefficients in $[-1,1]$. The proof combines an exact gap embedding with a quantitative robustness argument. For every E3SAT formula $Φ$ with $m$ clauses, a four-ReLU-per-clause construction satisfies $\max_{x\in[0,1]^n} g_Φ(x)=(m-\operatorname{unsat}(Φ))/3$, and coordinatewise threshold rounding never decreases the objective. A weighted parameter-sensitivity inequality and Gaussian concentration then show that a verification gap linear in $m$ survives the aggregate perturbation of all coefficients with probability at least $1-e^{-m/8}$. The proof includes clipping, exact dyadic rounding, output-layer perturbations, polynomial-bit sampling of the rounded Gaussian law, and the conversion from expected smoothed running time to a BPP algorithm. Computational checks test the exact identity and illustrate the different scaling of extensive and constant gaps; they are diagnostics rather than evidence for the complexity theorem. The result concerns worst-case base networks in the stated absolute-noise model, but it shows that parameter nondegeneracy alone does not yield a universal smoothed-polynomial guarantee for exact verification.

This paper studies the $\ell^p$-Lipschitz constants of ReLU neural networks $Φ: \mathbb{R}^d \to \mathbb{R}$ with random parameters for $p \in [1,\infty]$. The distribution of the weights follows a variant of the He initialization. In the case of zero-bias networks, we derive matching high-probability upper and lower bounds for wide networks that differ at most by an absolute constant. The bounds reveal two remarkable phenomena. First, the bounds grow root-logarithmically in terms of the depth of the net. Second, the behavior of the $\ell^p$-Lipschitz constants varies significantly between the regimes $p \in [1,2)$ and $p \in [2,\infty]$. For $p \in [2,\infty]$, the $\ell^p$-Lipschitz constant behaves similarly to $\|g\|_{p'}$, where $g \in \mathbb{R}^d$ is a standard Gaussian vector and $1/p + 1/p' = 1$. In contrast, for $p \in [1,2)$, the $\ell^p$-Lipschitz constant aligns more closely with $\|g\|_2$. For networks with possibly nonzero biases, we obtain the same lower bounds without assumptions on the bias distributions. For symmetric bias distributions with bounded densities, we also derive high-probability upper bounds for the $\ell^p$-Lipschitz constants. For fixed sub-exponential bias distributions, these upper bounds differ from the lower bounds by a factor that has square-root dependence on the depth and logarithmic dependence on the width.