arXiv subjects

Guy Bresler

Publications and source records attributed to Guy Bresler.

At least 37 records · Page 2Linked to original sources

arXiv2023

Random Algebraic Graphs and Their Convergence to Erdos-Renyi

A random algebraic graph is defined by a group $G$ with a uniform distribution over it and a connection $σ:G\longrightarrow[0,1]$ with expectation $p,$ satisfying $σ(g)=σ(g^{-1}).$ The random graph $\mathsf{RAG}(n,G,p,σ)$ with vertex set $[n]$ is formed as follows. First, $n$ independent vectors $x_1,\ldots,x_n$ are sampled uniformly from $G.$ Then, vertices $i,j$ are connected with probability $σ(x_ix_j^{-1}).$ This model captures random geometric graphs over the sphere and the hypercube, certain regimes of the stochastic block model, and random subgraphs of Cayley graphs. The main question of interest to the current paper is: when is a random algebraic graph statistically and/or computationally distinguishable from $\mathsf{G}(n,p)$? Our results fall into two categories. 1) Geometric. We focus on the case $G =\{\pm1\}^d$ and use Fourier-analytic tools. For hard threshold connections, we match [LMSY22b] for $p = ω(1/n)$ and for $1/(r\sqrt{d})$-Lipschitz connections we extend the results of [LR21b] when $d = Ω(n\log n)$ to the non-monotone setting. We study other connections such as indicators of interval unions and low-degree polynomials. 2) Algebraic. We provide evidence for an exponential statistical-computational gap. Consider any finite group $G$ and let $A\subseteq G$ be a set of elements formed by including each set of the form $\{g, g^{-1}\}$ independently with probability $1/2.$ Let $Γ_n(G,A)$ be the distribution of random graphs formed by taking a uniformly random induced subgraph of size $n$ of the Cayley graph $Γ(G,A).$ Then, $Γ_n(G,A)$ and $\mathsf{G}(n,1/2)$ are statistically indistinguishable with high probability over $A$ if and only if $\log|G|\gtrsim n.$ However, low-degree polynomial tests fail to distinguish $Γ_n(G,A)$ and $\mathsf{G}(n,1/2)$ with high probability over $A$ when $\log |G|=\log^{Ω(1)}n.$

math.PR↗

arXiv2022

Metastable Mixing of Markov Chains: Efficiently Sampling Low Temperature Exponential Random Graphs

In this paper we consider the problem of sampling from the low-temperature exponential random graph model (ERGM). The usual approach is via Markov chain Monte Carlo, but Bhamidi et al. showed that any local Markov chain suffers from an exponentially large mixing time due to metastable states. We instead consider metastable mixing, a notion of approximate mixing relative to the stationary distribution, for which it turns out to suffice to mix only within a collection of metastable states. We show that the Glauber dynamics for the ERGM at any temperature -- except at a lower-dimensional critical set of parameters -- when initialized at $G(n,p)$ for the right choice of $p$ has a metastable mixing time of $O(n^2\log n)$ to within total variation distance $\exp(-Ω(n))$.

math.PR↗

arXiv2022

Threshold for Detecting High Dimensional Geometry in Anisotropic Random Geometric Graphs

In the anisotropic random geometric graph model, vertices correspond to points drawn from a high-dimensional Gaussian distribution and two vertices are connected if their distance is smaller than a specified threshold. We study when it is possible to hypothesis test between such a graph and an Erdős-Rényi graph with the same edge probability. If $n$ is the number of vertices and $α$ is the vector of eigenvalues, Eldan and Mikulincer show that detection is possible when $n^3 \gg (\|α\|_2/\|α\|_3)^6$ and impossible when $n^3 \ll (\|α\|_2/\|α\|_4)^4$. We show detection is impossible when $n^3 \ll (\|α\|_2/\|α\|_3)^6$, closing this gap and affirmatively resolving the conjecture of Eldan and Mikulincer.

math.ST↗

arXiv2021

Chow-Liu++: Optimal Prediction-Centric Learning of Tree Ising Models

We consider the problem of learning a tree-structured Ising model from data, such that subsequent predictions computed using the model are accurate. Concretely, we aim to learn a model such that posteriors $P(X_i|X_S)$ for small sets of variables $S$ are accurate. Since its introduction more than 50 years ago, the Chow-Liu algorithm, which efficiently computes the maximum likelihood tree, has been the benchmark algorithm for learning tree-structured graphical models. A bound on the sample complexity of the Chow-Liu algorithm with respect to the prediction-centric local total variation loss was shown in [BK19]. While those results demonstrated that it is possible to learn a useful model even when recovering the true underlying graph is impossible, their bound depends on the maximum strength of interactions and thus does not achieve the information-theoretic optimum. In this paper, we introduce a new algorithm that carefully combines elements of the Chow-Liu algorithm with tree metric reconstruction methods to efficiently and optimally learn tree Ising models under a prediction-centric loss. Our algorithm is robust to model misspecification and adversarial corruptions. In contrast, we show that the celebrated Chow-Liu algorithm can be arbitrarily suboptimal.

cs.LG↗

arXiv2021

The staircase property: How hierarchical structure can guide deep learning

This paper identifies a structural property of data distributions that enables deep neural networks to learn hierarchically. We define the "staircase" property for functions over the Boolean hypercube, which posits that high-order Fourier coefficients are reachable from lower-order Fourier coefficients along increasing chains. We prove that functions satisfying this property can be learned in polynomial time using layerwise stochastic coordinate descent on regular neural networks -- a class of network architectures and initializations that have homogeneity properties. Our analysis shows that for such staircase functions and neural networks, the gradient-based algorithm learns high-level features by greedily combining lower-level features along the depth of the network. We further back our theoretical results with experiments showing that staircase functions are also learnable by more standard ResNet architectures with stochastic gradient descent. Both the theoretical and experimental results support the fact that staircase properties have a role to play in understanding the capabilities of gradient-based learning on regular networks, in contrast to general polynomial-size networks that can emulate any SQ or PAC algorithms as recently shown.

cs.LG↗

arXiv2021

The Algorithmic Phase Transition of Random $k$-SAT for Low Degree Polynomials

Let $Φ$ be a uniformly random $k$-SAT formula with $n$ variables and $m$ clauses. We study the algorithmic task of finding a satisfying assignment of $Φ$. It is known that satisfying assignments exist with high probability up to clause density $m/n = 2^k \log 2 - \frac12 (\log 2 + 1) + o_k(1)$, while the best polynomial-time algorithm known, the Fix algorithm of Coja-Oghlan, finds a satisfying assignment at the much lower clause density $(1 - o_k(1)) 2^k \log k / k$. This prompts the question: is it possible to efficiently find a satisfying assignment at higher clause densities? We prove that the class of low degree polynomial algorithms cannot find a satisfying assignment at clause density $(1 + o_k(1)) κ^* 2^k \log k / k$ for a universal constant $κ^* \approx 4.911$. This class encompasses Fix, message passing algorithms including Belief and Survey Propagation guided decimation (with bounded or mildly growing number of rounds), and local algorithms on the factor graph. This is the first hardness result for any class of algorithms at clause density within a constant factor of that achieved by Fix. Our proof establishes and leverages a new many-way overlap gap property tailored to random $k$-SAT.

cs.CC↗

arXiv2021

The Average-Case Complexity of Counting Cliques in Erdos-Renyi Hypergraphs

We consider the problem of counting $k$-cliques in $s$-uniform Erdos-Renyi hypergraphs $G(n,c,s)$ with edge density $c$, and show that its fine-grained average-case complexity can be based on its worst-case complexity. We prove the following: 1. Dense Erdos-Renyi graphs and hypergraphs: Counting $k$-cliques on $G(n,c,s)$ with $k$ and $c$ constant matches its worst-case time complexity up to a $\mathrm{polylog}(n)$ factor. Assuming randomized ETH, it takes $n^{Ω(k)}$ time to count $k$-cliques in $G(n,c,s)$ if $k$ and $c$ are constant. 2. Sparse Erdos-Renyi graphs and hypergraphs: When $c = Θ(n^{-α})$, we give several algorithms exploiting the sparsity of $G(n, c, s)$ that are faster than the best known worst-case algorithms. Complementing this, based on a fine-grained worst-case assumption, our results imply a different average-case phase diagram for each fixed $α$ depicting a tradeoff between a runtime lower bound and $k$. Surprisingly, in the hypergraph case ($s \ge 3$), these lower bounds are tight against our algorithms exactly when $c$ is above the Erdős-Rényi $k$-clique percolation threshold. This is the first worst-case-to-average-case hardness reduction for a problem on Erdős-Rényi hypergraphs that we are aware of. We also give a variant of our result for computing the parity of the $k$-clique count that tolerates higher error probability.

cs.CC↗

arXiv2021

Statistical Query Algorithms and Low-Degree Tests Are Almost Equivalent

Researchers currently use a number of approaches to predict and substantiate information-computation gaps in high-dimensional statistical estimation problems. A prominent approach is to characterize the limits of restricted models of computation, which on the one hand yields strong computational lower bounds for powerful classes of algorithms and on the other hand helps guide the development of efficient algorithms. In this paper, we study two of the most popular restricted computational models, the statistical query framework and low-degree polynomials, in the context of high-dimensional hypothesis testing. Our main result is that under mild conditions on the testing problem, the two classes of algorithms are essentially equivalent in power. As corollaries, we obtain new statistical query lower bounds for sparse PCA, tensor PCA and several variants of the planted clique problem.

cs.CC↗

arXiv2021

The EM Algorithm is Adaptively-Optimal for Unbalanced Symmetric Gaussian Mixtures

This paper studies the problem of estimating the means $\pmθ_{*}\in\mathbb{R}^{d}$ of a symmetric two-component Gaussian mixture $δ_{*}\cdot N(θ_{*},I)+(1-δ_{*})\cdot N(-θ_{*},I)$ where the weights $δ_{*}$ and $1-δ_{*}$ are unequal. Assuming that $δ_{*}$ is known, we show that the population version of the EM algorithm globally converges if the initial estimate has non-negative inner product with the mean of the larger weight component. This can be achieved by the trivial initialization $θ_{0}=0$. For the empirical iteration based on $n$ samples, we show that when initialized at $θ_{0}=0$, the EM algorithm adaptively achieves the minimax error rate $\tilde{O}\Big(\min\Big\{\frac{1}{(1-2δ_{*})}\sqrt{\frac{d}{n}},\frac{1}{\|θ_{*}\|}\sqrt{\frac{d}{n}},\left(\frac{d}{n}\right)^{1/4}\Big\}\Big)$ in no more than $O\Big(\frac{1}{\|θ_{*}\|(1-2δ_{*})}\Big)$ iterations (with high probability). We also consider the EM iteration for estimating the weight $δ_{*}$, assuming a fixed mean $θ$ (which is possibly mismatched to $θ_{*}$). For the empirical iteration of $n$ samples, we show that the minimax error rate $\tilde{O}\Big(\frac{1}{\|θ_{*}\|}\sqrt{\frac{d}{n}}\Big)$ is achieved in no more than $O\Big(\frac{1}{\|θ_{*}\|^{2}}\Big)$ iterations. These results robustify and complement recent results of Wu and Zhou obtained for the equal weights case $δ_{*}=1/2$.

math.ST↗

arXiv2021

De Finetti-Style Results for Wishart Matrices: Combinatorial Structure and Phase Transitions

A recent line of work has studied the relationship between the Wishart matrix $X^\top X$, where $X\in \mathbb{R}^{d\times n}$ has i.i.d. standard Gaussian entries, and the corresponding Gaussian matrix with independent entries above the diagonal. Jiang and Li (2015) and Bubeck et al. (2016) showed that these two matrix ensembles converge in total variation whenever $d/n^3\to \infty$, and Bubeck et al. (2016) showed this to be sharp. In this paper we aim to identify the precise threshold for $d$ in terms of $n$ for subsets of Wishart matrices to converge in total variation to independent Gaussians. It turns out that the combinatorial structure of the revealed entries, viewed as the adjacency matrix of a graph $G$, characterizes the distance from fully independent. Specifically, we show that the threshold for $d$ depends on the number of various small subgraphs in $G$. So, even when the number of revealed entries is fixed, the threshold can vary wildly depending on their configuration. Convergence of masked Wishart to independent Gaussians thus inherently involves an interplay between both probabilistic and combinatorial phenomena. Our results determine the sharp threshold for a large family of $G$, including Erdős-Rényi $G\sim \mathcal{G}(n,p)$ at all values $p\gtrsim n^{-2}\mathrm{polylog}(n)$. Our proof techniques are both combinatorial and information theoretic, which together allow us to carefully unravel the dependencies in the masked Wishart ensemble.

math.PR↗

arXiv2021

Sharp Representation Theorems for ReLU Networks with Precise Dependence on Depth

We prove sharp dimension-free representation results for neural networks with $D$ ReLU layers under square loss for a class of functions $\mathcal{G}_D$ defined in the paper. These results capture the precise benefits of depth in the following sense: 1. The rates for representing the class of functions $\mathcal{G}_D$ via $D$ ReLU layers is sharp up to constants, as shown by matching lower bounds. 2. For each $D$, $\mathcal{G}_{D} \subseteq \mathcal{G}_{D+1}$ and as $D$ grows the class of functions $\mathcal{G}_{D}$ contains progressively less smooth functions. 3. If $D^{\prime} < D$, then the approximation rate for the class $\mathcal{G}_D$ achieved by depth $D^{\prime}$ networks is strictly worse than that achieved by depth $D$ networks. This constitutes a fine-grained characterization of the representation power of feedforward networks of arbitrary depth $D$ and number of neurons $N$, in contrast to existing representation results which either require $D$ growing quickly with $N$ or assume that the function being represented is highly smooth. In the latter case similar rates can be obtained with a single nonlinear layer. Our results confirm the prevailing hypothesis that deeper networks are better at representing less smooth functions, and indeed, the main technical novelty is to fully exploit the fact that deep networks can produce highly oscillatory functions with few activation functions.

stat.ML↗

arXiv2020

Information Storage in the Stochastic Ising Model

Most information storage devices write data by modifying the local state of matter, in the hope that sub-atomic local interactions stabilize the state for sufficiently long time, thereby allowing later recovery. Motivated to explore how temporal evolution of physical states in magnetic storage media affects their capacity, this work initiates the study of information retention in locally-interacting particle systems. The system dynamics follow the stochastic Ising model (SIM) over a 2-dimensional $\sqrt{n}\times\sqrt{n}$ grid. The initial spin configuration $X_0$ serves as the user-controlled input. The output configuration $X_t$ is produced by running $t$ steps of Glauber dynamics. Our main goal is to evaluate the information capacity $I_n(t):=\max_{p_{X_0}}I(X_0;X_t)$ when time $t$ scales with the system's size $n$. While the positive (but low) temperature regime is our main interest, we start by exploring the simpler zero-temperature dynamics. We first show that at zero temperature, order of $\sqrt{n}$ bits can be stored in the system indefinitely by coding over stable, striped configurations. While $\sqrt{n}$ is order optimal for infinite time, backing off to $t<\infty$, higher orders of $I_n(t)$ are achievable. First, linear coding arguments imply that $I_n(t) = Θ(n)$ for $t=O(n)$. To go beyond the linear scale, we develop a droplet-based achievability scheme that reliably stores $Ω\left(n/\log n\right)$ for $t=O(n\log n)$ time ($\log n$ can be replaced with any $o(n)$ function). Moving to the positive but low temperature regime, two main results are provided. First, we show that an initial configuration drawn from the Gibbs measure cannot retain more than a single bit for $t\geq \exp(Cβn^{1/4+ε})$ time. On the other hand, when scaling time with the inverse temperature $β$, the stripe-based coding scheme is shown to retain its bits for $e^{cβ}$.

cs.IT↗

arXiv2020

Learning Restricted Boltzmann Machines with Sparse Latent Variables

Restricted Boltzmann Machines (RBMs) are a common family of undirected graphical models with latent variables. An RBM is described by a bipartite graph, with all observed variables in one layer and all latent variables in the other. We consider the task of learning an RBM given samples generated according to it. The best algorithms for this task currently have time complexity $\tilde{O}(n^2)$ for ferromagnetic RBMs (i.e., with attractive potentials) but $\tilde{O}(n^d)$ for general RBMs, where $n$ is the number of observed variables and $d$ is the maximum degree of a latent variable. Let the MRF neighborhood of an observed variable be its neighborhood in the Markov Random Field of the marginal distribution of the observed variables. In this paper, we give an algorithm for learning general RBMs with time complexity $\tilde{O}(n^{2^s+1})$, where $s$ is the maximum number of latent variables connected to the MRF neighborhood of an observed variable. This is an improvement when $s < \log_2 (d-1)$, which corresponds to RBMs with sparse latent variables. Furthermore, we give a version of this learning algorithm that recovers a model with small prediction error and whose sample complexity is independent of the minimum potential in the Markov Random Field of the observed variables. This is of interest because the sample complexity of current algorithms scales with the inverse of the minimum potential, which cannot be controlled in terms of natural properties of the RBM.

cs.LG↗

arXiv2020

Phase Transitions for Detecting Latent Geometry in Random Graphs

Random graphs with latent geometric structure are popular models of social and biological networks, with applications ranging from network user profiling to circuit design. These graphs are also of purely theoretical interest within computer science, probability and statistics. A fundamental initial question regarding these models is: when are these random graphs affected by their latent geometry and when are they indistinguishable from simpler models without latent structure, such as the Erdős-Rényi graph $\mathcal{G}(n, p)$? We address this question for two of the most well-studied models of random graphs with latent geometry -- the random intersection and random geometric graph. Our results are as follows: (1) we prove that the random intersection graph converges in total variation to $\mathcal{G}(n, p)$ when $d = \tildeω(n^3)$, and does not if $d = o(n^3)$, resolving an open problem in Fill et al. (2000), Rybarczyk (2011) and Kim et al. (2018); (2) we provide conditions under which the matrix of intersection sizes of random family of sets converges in total variation to a symmetric matrix with independent Poisson entries, yielding the first total variation convergence result for $τ$-random intersection graphs to $\mathcal{G}(n, p)$; and (3) we show that the random geometric graph on $\mathbb{S}^{d - 1}$ with edge density $p$ converges in total variation to $\mathcal{G}(n, p)$ when $d = \tildeω\left(\min\{ pn^3, p^2 n^{7/2} \} \right)$, yielding the first progress towards a conjecture of Bubeck et al. (2016). The first of these three results was obtained simultaneously and independently by Bubeck, Racz and Richey.

math.PR↗

arXiv2020

Reducibility and Statistical-Computational Gaps from Secret Leakage

Inference problems with conjectured statistical-computational gaps are ubiquitous throughout modern statistics, computer science and statistical physics. While there has been success evidencing these gaps from the failure of restricted classes of algorithms, progress towards a more traditional reduction-based approach to computational complexity in statistical inference has been limited. Existing reductions have largely been limited to inference problems with similar structure -- primarily mapping among problems representable as a sparse submatrix signal plus a noise matrix, which are similar to the common hardness assumption of planted clique. The insight in this work is that a slight generalization of the planted clique conjecture -- secret leakage planted clique -- gives rise to a variety of new average-case reduction techniques, yielding a web of reductions among problems with very different structure. Using variants of the planted clique conjecture for specific forms of secret leakage planted clique, we deduce tight statistical-computational tradeoffs for a diverse range of problems including robust sparse mean estimation, mixtures of sparse linear regressions, robust sparse linear regression, tensor PCA, variants of dense $k$-block stochastic block models, negatively correlated sparse PCA, semirandom planted dense subgraph, detection in hidden partition models and a universality principle for learning sparse mixtures. In particular, a $k$-partite hypergraph variant of the planted clique conjecture is sufficient to establish all of our computational lower bounds. Our techniques also reveal novel connections to combinatorial designs and to random matrix theory. This work gives the first evidence that an expanded set of hardness assumptions, such as for secret leakage planted clique, may be a key first step towards a more complete theory of reductions among statistical problems.

cs.CC↗

arXiv2020

A Corrective View of Neural Networks: Representation, Memorization and Learning

We develop a corrective mechanism for neural network approximation: the total available non-linear units are divided into multiple groups and the first group approximates the function under consideration, the second group approximates the error in approximation produced by the first group and corrects it, the third group approximates the error produced by the first and second groups together and so on. This technique yields several new representation and learning results for neural networks. First, we show that two-layer neural networks in the random features regime (RF) can memorize arbitrary labels for arbitrary points under under Euclidean distance separation condition using $\tilde{O}(n)$ ReLUs which is optimal in $n$ up to logarithmic factors. Next, we give a powerful representation result for two-layer neural networks with ReLUs and smoothed ReLUs which can achieve a squared error of at most $ε$ with $O(C(a,d)ε^{-1/(a+1)})$ for $a \in \mathbb{N}\cup\{0\}$ when the function is smooth enough (roughly when it has $Θ(ad)$ bounded derivatives). In certain cases $d$ can be replaced with effective dimension $q \ll d$. Previous results of this type implement Taylor series approximation using deep architectures. We also consider three-layer neural networks and show that the corrective mechanism yields faster representation rates for smooth radial functions. Lastly, we obtain the first $O(\mathrm{subpoly}(1/ε))$ upper bound on the number of neurons required for a two layer network to learn low degree polynomials up to squared error $ε$ via gradient descent. Even though deep networks can express these polynomials with $O(\mathrm{polylog}(1/ε))$ neurons, the best learning bounds on this problem require $\mathrm{poly}(1/ε)$ neurons.

cs.LG↗

arXiv2020

Least Squares Regression with Markovian Data: Fundamental Limits and Algorithms

We study the problem of least squares linear regression where the data-points are dependent and are sampled from a Markov chain. We establish sharp information theoretic minimax lower bounds for this problem in terms of $τ_{\mathsf{mix}}$, the mixing time of the underlying Markov chain, under different noise settings. Our results establish that in general, optimization with Markovian data is strictly harder than optimization with independent data and a trivial algorithm (SGD-DD) that works with only one in every $\tildeΘ(τ_{\mathsf{mix}})$ samples, which are approximately independent, is minimax optimal. In fact, it is strictly better than the popular Stochastic Gradient Descent (SGD) method with constant step-size which is otherwise minimax optimal in the regression with independent data setting. Beyond a worst case analysis, we investigate whether structured datasets seen in practice such as Gaussian auto-regressive dynamics can admit more efficient optimization schemes. Surprisingly, even in this specific and natural setting, Stochastic Gradient Descent (SGD) with constant step-size is still no better than SGD-DD. Instead, we propose an algorithm based on experience replay--a popular reinforcement learning technique--that achieves a significantly better error rate. Our improved rate serves as one of the first results where an algorithm outperforms SGD-DD on an interesting Markov chain and also provides one of the first theoretical analyses to support the use of experience replay in practice.

cs.LG↗

arXiv2020

Average-Case Lower Bounds for Learning Sparse Mixtures, Robust Estimation and Semirandom Adversaries

This paper develops several average-case reduction techniques to show new hardness results for three central high-dimensional statistics problems, implying a statistical-computational gap induced by robustness, a detection-recovery gap and a universality principle for these gaps. A main feature of our approach is to map to these problems via a common intermediate problem that we introduce, which we call Imbalanced Sparse Gaussian Mixtures. We assume the planted clique conjecture for a version of the planted clique problem where the position of the planted clique is mildly constrained, and from this obtain the following computational lower bounds: (1) a $k$-to-$k^2$ statistical-computational gap for robust sparse mean estimation, providing the first average-case evidence for a conjecture of Li (2017) and Balakrishnan et al. (2017); (2) a tight lower bound for semirandom planted dense subgraph, which shows that a semirandom adversary shifts the detection threshold in planted dense subgraph to the conjectured recovery threshold; and (3) a universality principle for $k$-to-$k^2$ gaps in a broad class of sparse mixture problems that includes many natural formulations such as the spiked covariance model. Our main approach is to introduce several average-case techniques to produce structured and Gaussianized versions of an input graph problem, and then to rotate these high-dimensional Gaussians by matrices carefully constructed from hyperplanes in $\mathbb{F}_r^t$. For our universality result, we introduce a new method to perform an algorithmic change of measure tailored to sparse mixtures. We also provide evidence that the mild promise in our variant of planted clique does not change the complexity of the problem.

cs.CC↗