Dan Mikulincer

Dan Mikulincer, Daniel Reichman

We study monotone neural networks with threshold gates where all the weights (other than the biases) are non-negative. We focus on the expressive power and efficiency of representation of such networks. Our first result establishes that every monotone function over $[0,1]^d$ can be approximated within arbitrarily small additive error by a depth-4 monotone network. When $d > 3$, we improve upon the previous best-known construction which has depth $d+1$. Our proof goes by solving the monotone interpolation problem for monotone datasets using a depth-4 monotone threshold network. In our second main result we compare size bounds between monotone and arbitrary neural networks with threshold gates. We find that there are monotone real functions that can be computed efficiently by networks with no restriction on the gates whereas monotone networks approximating these functions need exponential size in the dimension.

Dan Mikulincer, Youngtak Sohn

We study the mixing time of Glauber dynamics for Ising models in which the interaction matrix contains a single negative spectral outlier. This class includes the anti-ferromagnetic Curie-Weiss model, the anti-ferromagnetic Ising model on expander graphs, and the Sherrington-Kirkpatrick model with disorder of negative mean. Existing approaches to rapid mixing rely crucially on log-concavity or spectral width bounds and therefore can break down in the presence of a negative outlier. To address this difficulty, we develop a new covariance approximation method based on Gaussian approximation. This method is implemented via an iterative application of Stein's method to quadratic tilts of sums of bounded random variables, which may be of independent interest. The resulting analysis provides an operator-norm control of the full correlation structure under arbitrary external fields. Combined with the localization schemes of Eldan and Chen, these estimates lead to a modified logarithmic Sobolev inequality and near-optimal mixing time bounds in regimes where spectral width bounds fail. We complement these results by proving exponential lower bounds on the mixing time for low temperature anti-ferromagnetic Ising models on sparse random regular graphs and Erdös-Rényi graphs, based on the existence of gapped states as in the recent work of Sellke.

Elisabetta Cornacchia, Dan Mikulincer, Elchanan Mossel

We study how temporal correlations in the data can make certain sparse learning problems efficiently learnable by gradient-based methods. Our focus is on Boolean k-juntas, a canonical sparse learning problem known to pose barriers for gradient-based methods under independent uniform samples. We show that this picture changes when the samples are generated by a lazy random walk on the hypercube. In this setting, the temporal dependencies can be exploited by a two-layer ReLU network trained using stylized-SGD with a temporal-difference loss, which compares target and predicted increments across consecutive samples. For every fixed k, the resulting sample complexity is essentially linear in the ambient dimension d. By contrast, we show that for large-batch gradient methods using standard convex pointwise losses, temporal correlations do not provide the same advantage.

Elisabetta Cornacchia, Dan Mikulincer, Elchanan Mossel · mit

The problem of learning single index and multi index models has gained significant interest as a fundamental task in high-dimensional statistics. Many recent works have analysed gradient-based methods, particularly in the setting of isotropic data distributions, often in the context of neural network training. Such studies have uncovered precise characterisations of algorithmic sample complexity in terms of certain analytic properties of the target function, such as the leap, information, and generative exponents. These properties establish a quantitative separation between low and high complexity learning tasks. In this work, we show that high complexity cases are rare. Specifically, we prove that introducing a small random perturbation to the data distribution--via a random shift in the first moment--renders any Gaussian single index model as easy to learn as a linear function. We further extend this result to a class of multi index models, namely sparse Boolean functions, also known as Juntas.

Ronen Eldan, Dan Mikulincer, Tselil Schramm

We study the extent to which wide neural networks may be approximated by Gaussian processes when initialized with random weights. It is a well-established fact that as the width of a network goes to infinity, its law converges to that of a Gaussian process. We make this quantitative by establishing explicit convergence rates for the central limit theorem in an infinite-dimensional functional space, metrized with a natural transportation distance. We identify two regimes of interest; when the activation function is polynomial, its degree determines the rate of convergence, while for non-polynomial activations, the rate is governed by the smoothness of the function.

Ronen Eldan, Dan Mikulincer, Hester Pieters

We make the first steps towards generalizing the theory of stochastic block models, in the sparse regime, towards a model where the discrete community structure is replaced by an underlying geometry. We consider a geometric random graph over a homogeneous metric space where the probability of two vertices to be connected is an arbitrary function of the distance. We give sufficient conditions under which the locations can be recovered (up to an isomorphism of the space) in the sparse regime. Moreover, we define a geometric counterpart of the model of flow of information on trees, due to Mossel and Peres, in which one considers a branching random walk on a sphere and the goal is to recover the location of the root based on the locations of leaves. We give some sufficient conditions for percolation and for non-percolation of information in this model.

Sébastien Bubeck, Ronen Eldan, Yin Tat Lee et al.

In 1988, Eric B. Baum showed that two-layers neural networks with threshold activation function can perfectly memorize the binary labels of $n$ points in general position in $\mathbb{R}^d$ using only $\ulcorner n/d \urcorner$ neurons. We observe that with ReLU networks, using four times as many neurons one can fit arbitrary real labels. Moreover, for approximate memorization up to error $ε$, the neural tangent kernel can also memorize with only $O\left(\frac{n}{d} \cdot \log(1/ε) \right)$ neurons (assuming that the data is well dispersed too). We show however that these constructions give rise to networks where the magnitude of the neurons' weights are far from optimal. In contrast we propose a new training procedure for ReLU networks, based on complex (as opposed to real) recombination of the neurons, for which we show approximate memorization with both $O\left(\frac{n}{d} \cdot \frac{\log(1/ε)}ε\right)$ neurons, as well as nearly-optimal size of the weights.

Sébastien Bubeck, Dan Mikulincer

We consider the problem of finding an $\varepsilon$-approximate stationary point of a smooth function on a compact domain of $\mathbb{R}^d$. In contrast with dimension-free approaches such as gradient descent, we focus here on the case where $d$ is finite, and potentially small. This viewpoint was explored in 1993 by Vavasis, who proposed an algorithm which, for any fixed finite dimension $d$, improves upon the $O(1/\varepsilon^2)$ oracle complexity of gradient descent. For example for $d=2$, Vavasis' approach obtains the complexity $O(1/\varepsilon)$. Moreover for $d=2$ he also proved a lower bound of $Ω(1/\sqrt{\varepsilon})$ for deterministic algorithms (we extend this result to randomized algorithms). Our main contribution is an algorithm, which we call gradient flow trapping (GFT), and the analysis of its oracle complexity. In dimension $d=2$, GFT closes the gap with Vavasis' lower bound (up to a logarithmic factor), as we show that it has complexity $O\left(\sqrt{\frac{\log(1/\varepsilon)}{\varepsilon}}\right)$. In dimension $d=3$, we show a complexity of $O\left(\frac{\log(1/\varepsilon)}{\varepsilon}\right)$, improving upon Vavasis' $O\left(1 / \varepsilon^{1.2} \right)$. In higher dimensions, GFT has the remarkable property of being a logarithmic parallel depth strategy, in stark contrast with the polynomial depth of gradient descent or Vavasis' algorithm. In this higher dimensional regime, the total work of GFT improves quadratically upon the only other known polylogarithmic depth strategy for this problem, namely naive grid search. We augment this result with another algorithm, named \emph{cut and flow} (CF), which improves upon Vavasis' algorithm in any fixed dimension.