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Santosh S. Vempala

Publications and source records attributed to Santosh S. Vempala.

At least 37 records · Page 2Linked to original sources
arXiv2023

Contrastive Moments: Unsupervised Halfspace Learning in Polynomial Time

We give a polynomial-time algorithm for learning high-dimensional halfspaces with margins in $d$-dimensional space to within desired TV distance when the ambient distribution is an unknown affine transformation of the $d$-fold product of an (unknown) symmetric one-dimensional logconcave distribution, and the halfspace is introduced by deleting at least an $ε$ fraction of the data in one of the component distributions. Notably, our algorithm does not need labels and establishes the unique (and efficient) identifiability of the hidden halfspace under this distributional assumption. The sample and time complexity of the algorithm are polynomial in the dimension and $1/ε$. The algorithm uses only the first two moments of suitable re-weightings of the empirical distribution, which we call contrastive moments; its analysis uses classical facts about generalized Dirichlet polynomials and relies crucially on a new monotonicity property of the moment ratio of truncations of logconcave distributions. Such algorithms, based only on first and second moments were suggested in earlier work, but hitherto eluded rigorous guarantees. Prior work addressed the special case when the underlying distribution is Gaussian via Non-Gaussian Component Analysis. We improve on this by providing polytime guarantees based on Total Variation (TV) distance, in place of existing moment-bound guarantees that can be super-polynomial. Our work is also the first to go beyond Gaussians in this setting.

cs.LG↗
arXiv2023

Computation with Sequences in a Model of the Brain

Even as machine learning exceeds human-level performance on many applications, the generality, robustness, and rapidity of the brain's learning capabilities remain unmatched. How cognition arises from neural activity is a central open question in neuroscience, inextricable from the study of intelligence itself. A simple formal model of neural activity was proposed in Papadimitriou [2020] and has been subsequently shown, through both mathematical proofs and simulations, to be capable of implementing certain simple cognitive operations via the creation and manipulation of assemblies of neurons. However, many intelligent behaviors rely on the ability to recognize, store, and manipulate temporal sequences of stimuli (planning, language, navigation, to list a few). Here we show that, in the same model, time can be captured naturally as precedence through synaptic weights and plasticity, and, as a result, a range of computations on sequences of assemblies can be carried out. In particular, repeated presentation of a sequence of stimuli leads to the memorization of the sequence through corresponding neural assemblies: upon future presentation of any stimulus in the sequence, the corresponding assembly and its subsequent ones will be activated, one after the other, until the end of the sequence. Finally, we show that any finite state machine can be learned in a similar way, through the presentation of appropriate patterns of sequences. Through an extension of this mechanism, the model can be shown to be capable of universal computation. We support our analysis with a number of experiments to probe the limits of learning in this model in key ways. Taken together, these results provide a concrete hypothesis for the basis of the brain's remarkable abilities to compute and learn, with sequences playing a vital role.

cs.NE↗
arXiv2023

Sampling with Barriers: Faster Mixing via Lewis Weights

We analyze Riemannian Hamiltonian Monte Carlo (RHMC) for sampling a polytope defined by $m$ inequalities in $\R^n$ endowed with the metric defined by the Hessian of a convex barrier function. The advantage of RHMC over Euclidean methods such as the ball walk, hit-and-run and the Dikin walk is in its ability to take longer steps. However, in all previous work, the mixing rate has a linear dependence on the number of inequalities. We introduce a hybrid of the Lewis weights barrier and the standard logarithmic barrier and prove that the mixing rate for the corresponding RHMC is bounded by $\tilde O(m^{1/3}n^{4/3})$, improving on the previous best bound of $\tilde O(mn^{2/3})$ (based on the log barrier). This continues the general parallels between optimization and sampling, with the latter typically leading to new tools and more refined analysis. To prove our main results, we have to overcomes several challenges relating to the smoothness of Hamiltonian curves and the self-concordance properties of the barrier. In the process, we give a general framework for the analysis of Markov chains on Riemannian manifolds, derive new smoothness bounds on Hamiltonian curves, a central topic of comparison geometry, and extend self-concordance to the infinity norm, which gives sharper bounds; these properties appear to be of independent interest.

cs.DS↗
arXiv2023

The Bit Complexity of Efficient Continuous Optimization

We analyze the bit complexity of efficient algorithms for fundamental optimization problems, such as linear regression, $p$-norm regression, and linear programming (LP). State-of-the-art algorithms are iterative, and in terms of the number of arithmetic operations, they match the current time complexity of multiplying two $n$-by-$n$ matrices (up to polylogarithmic factors). However, previous work has typically assumed infinite precision arithmetic, and due to complicated inverse maintenance techniques, the actual running times of these algorithms are unknown. To settle the running time and bit complexity of these algorithms, we demonstrate that a core common subroutine, known as \emph{inverse maintenance}, is backward-stable. Additionally, we show that iterative approaches for solving constrained weighted regression problems can be accomplished with bounded-error pre-conditioners. Specifically, we prove that linear programs can be solved approximately in matrix multiplication time multiplied by polylog factors that depend on the condition number $κ$ of the matrix and the inner and outer radius of the LP problem. $p$-norm regression can be solved approximately in matrix multiplication time multiplied by polylog factors in $κ$. Lastly, linear regression can be solved approximately in input-sparsity time multiplied by polylog factors in $κ$. Furthermore, we present results for achieving lower than matrix multiplication time for $p$-norm regression by utilizing faster solvers for sparse linear systems.

cs.DS↗
arXiv2023

Is Planted Coloring Easier than Planted Clique?

We study the computational complexity of two related problems: recovering a planted $q$-coloring in $G(n,1/2)$, and finding efficiently verifiable witnesses of non-$q$-colorability (a.k.a. refutations) in $G(n,1/2)$. Our main results show hardness for both these problems in a restricted-but-powerful class of algorithms based on computing low-degree polynomials in the inputs. The problem of recovering a planted $q$-coloring is equivalent to recovering $q$ disjoint planted cliques that cover all the vertices -- a potentially easier variant of the well-studied planted clique problem. Our first result shows that this variant is as hard as the original planted clique problem in the low-degree polynomial model of computation: each clique needs to have size $k \gg \sqrt{n}$ for efficient recovery to be possible. For the related variant where the cliques cover a $(1-ε)$-fraction of the vertices, we also show hardness by reduction from planted clique. Our second result shows that refuting $q$-colorability of $G(n,1/2)$ is hard in the low-degree polynomial model when $q \gg n^{2/3}$ but easy when $q \lesssim n^{1/2}$, and we leave closing this gap for future work. Our proof is more subtle than similar results for planted clique and involves constructing a non-standard distribution over $q$-colorable graphs. We note that while related to several prior works, this is the first work that explicitly formulates refutation problems in the low-degree polynomial model. The proofs of our main results involve showing low-degree hardness of hypothesis testing between an appropriately constructed pair of distributions. For refutation, we show completeness of this approach: in the low-degree model, the refutation task is precisely as hard as the hardest associated testing problem, i.e., proving hardness of refutation amounts to finding a "hard" distribution.

cs.CC↗
arXiv2023

Beyond Moments: Robustly Learning Affine Transformations with Asymptotically Optimal Error

We present a polynomial-time algorithm for robustly learning an unknown affine transformation of the standard hypercube from samples, an important and well-studied setting for independent component analysis (ICA). Specifically, given an $ε$-corrupted sample from a distribution $D$ obtained by applying an unknown affine transformation $x \rightarrow Ax+s$ to the uniform distribution on a $d$-dimensional hypercube $[-1,1]^d$, our algorithm constructs $\hat{A}, \hat{s}$ such that the total variation distance of the distribution $\hat{D}$ from $D$ is $O(ε)$ using poly$(d)$ time and samples. Total variation distance is the information-theoretically strongest possible notion of distance in our setting and our recovery guarantees in this distance are optimal up to the absolute constant factor multiplying $ε$. In particular, if the columns of $A$ are normalized to be unit length, our total variation distance guarantee implies a bound on the sum of the $\ell_2$ distances between the column vectors of $A$ and $A'$, $\sum_{i =1}^d \|a_i-\hat{a}_i\|_2 = O(ε)$. In contrast, the strongest known prior results only yield a $ε^{O(1)}$ (relative) bound on the distance between individual $a_i$'s and their estimates and translate into an $O(dε)$ bound on the total variation distance. Our key innovation is a new approach to ICA (even to outlier-free ICA) that circumvents the difficulties in the classical method of moments and instead relies on a new geometric certificate of correctness of an affine transformation. Our algorithm is based on a new method that iteratively improves an estimate of the unknown affine transformation whenever the requirements of the certificate are not met.

cs.DS↗
arXiv2023

Condition-number-independent convergence rate of Riemannian Hamiltonian Monte Carlo with numerical integrators

We study the convergence rate of discretized Riemannian Hamiltonian Monte Carlo on sampling from distributions in the form of $e^{-f(x)}$ on a convex body $\mathcal{M}\subset\mathbb{R}^{n}$. We show that for distributions in the form of $e^{-α^{\top}x}$ on a polytope with $m$ constraints, the convergence rate of a family of commonly-used integrators is independent of $\left\Vert α\right\Vert _{2}$ and the geometry of the polytope. In particular, the implicit midpoint method (IMM) and the generalized Leapfrog method (LM) have a mixing time of $\widetilde{O}\left(mn^{3}\right)$ to achieve $ε$ total variation distance to the target distribution. These guarantees are based on a general bound on the convergence rate for densities of the form $e^{-f(x)}$ in terms of parameters of the manifold and the integrator. Our theoretical guarantee complements the empirical results of [KLSV22], which shows that RHMC with IMM can sample ill-conditioned, non-smooth and constrained distributions in very high dimension efficiently in practice.

cs.DS↗
arXiv2022

The $k$-Cap Process on Geometric Random Graphs

The $k$-cap (or $k$-winners-take-all) process on a graph works as follows: in each iteration, exactly $k$ vertices of the graph are in the cap (i.e., winners); the next round winners are the vertices that have the highest total degree to the current winners, with ties broken randomly. This natural process is a simple model of firing activity in the brain. We study its convergence on geometric random graphs, revealing rather surprising behavior.

math.PR↗
arXiv2022

Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Space

We demonstrate for the first time that ill-conditioned, non-smooth, constrained distributions in very high dimension, upwards of 100,000, can be sampled efficiently $\textit{in practice}$. Our algorithm incorporates constraints into the Riemannian version of Hamiltonian Monte Carlo and maintains sparsity. This allows us to achieve a mixing rate independent of smoothness and condition numbers. On benchmark data sets in systems biology and linear programming, our algorithm outperforms existing packages by orders of magnitude. In particular, we achieve a 1,000-fold speed-up for sampling from the largest published human metabolic network (RECON3D). Our package has been incorporated into the COBRA toolbox.

cs.LG↗
arXiv2022

Assemblies of neurons learn to classify well-separated distributions

An assembly is a large population of neurons whose synchronous firing is hypothesized to represent a memory, concept, word, and other cognitive categories. Assemblies are believed to provide a bridge between high-level cognitive phenomena and low-level neural activity. Recently, a computational system called the Assembly Calculus (AC), with a repertoire of biologically plausible operations on assemblies, has been shown capable of simulating arbitrary space-bounded computation, but also of simulating complex cognitive phenomena such as language, reasoning, and planning. However, the mechanism whereby assemblies can mediate learning has not been known. Here we present such a mechanism, and prove rigorously that, for simple classification problems defined on distributions of labeled assemblies, a new assembly representing each class can be reliably formed in response to a few stimuli from the class; this assembly is henceforth reliably recalled in response to new stimuli from the same class. Furthermore, such class assemblies will be distinguishable as long as the respective classes are reasonably separated -- for example, when they are clusters of similar assemblies. To prove these results, we draw on random graph theory with dynamic edge weights to estimate sequences of activated vertices, yielding strong generalizations of previous calculations and theorems in this field over the past five years. These theorems are backed up by experiments demonstrating the successful formation of assemblies which represent concept classes on synthetic data drawn from such distributions, and also on MNIST, which lends itself to classification through one assembly per digit. Seen as a learning algorithm, this mechanism is entirely online, generalizes from very few samples, and requires only mild supervision -- all key attributes of learning in a model of the brain.

cs.NE↗
arXiv2022

Constant-Factor Approximation Algorithms for Socially Fair $k$-Clustering

We study approximation algorithms for the socially fair $(\ell_p, k)$-clustering problem with $m$ groups, whose special cases include the socially fair $k$-median ($p=1$) and socially fair $k$-means ($p=2$) problems. We present (1) a polynomial-time $(5+2\sqrt{6})^p$-approximation with at most $k+m$ centers (2) a $(5+2\sqrt{6}+ε)^p$-approximation with $k$ centers in time $n^{2^{O(p)}\cdot m^2}$, and (3) a $(15+6\sqrt{6})^p$ approximation with $k$ centers in time $k^{m}\cdot\text{poly}(n)$. The first result is obtained via a refinement of the iterative rounding method using a sequence of linear programs. The latter two results are obtained by converting a solution with up to $k+m$ centers to one with $k$ centers using sparsification methods for (2) and via an exhaustive search for (3). We also compare the performance of our algorithms with existing bicriteria algorithms as well as exactly $k$ center approximation algorithms on benchmark datasets, and find that our algorithms also outperform existing methods in practice.

cs.DS↗
arXiv2022

A Unified Approach to Discrepancy Minimization

We study a unified approach and algorithm for constructive discrepancy minimization based on a stochastic process. By varying the parameters of the process, one can recover various state-of-the-art results. We demonstrate the flexibility of the method by deriving a discrepancy bound for smoothed instances, which interpolates between known bounds for worst-case and random instances.

cs.DS↗
arXiv2022

Convergence of the Riemannian Langevin Algorithm

We study the Riemannian Langevin Algorithm for the problem of sampling from a distribution with density $ν$ with respect to the natural measure on a manifold with metric $g$. We assume that the target density satisfies a log-Sobolev inequality with respect to the metric and prove that the manifold generalization of the Unadjusted Langevin Algorithm converges rapidly to $ν$ for Hessian manifolds. This allows us to reduce the problem of sampling non-smooth (constrained) densities in ${\bf R}^n$ to sampling smooth densities over appropriate manifolds, while needing access only to the gradient of the log-density, and this, in turn, to sampling from the natural Brownian motion on the manifold. Our main analytic tools are (1) an extension of self-concordance to manifolds, and (2) a stochastic approach to bounding smoothness on manifolds. A special case of our approach is sampling isoperimetric densities restricted to polytopes by using the metric defined by the logarithmic barrier.

cs.LG↗
arXiv2022

How and When Random Feedback Works: A Case Study of Low-Rank Matrix Factorization

The success of gradient descent in ML and especially for learning neural networks is remarkable and robust. In the context of how the brain learns, one aspect of gradient descent that appears biologically difficult to realize (if not implausible) is that its updates rely on feedback from later layers to earlier layers through the same connections. Such bidirected links are relatively few in brain networks, and even when reciprocal connections exist, they may not be equi-weighted. Random Feedback Alignment (Lillicrap et al., 2016), where the backward weights are random and fixed, has been proposed as a bio-plausible alternative and found to be effective empirically. We investigate how and when feedback alignment (FA) works, focusing on one of the most basic problems with layered structure -- low-rank matrix factorization. In this problem, given a matrix $Y_{n\times m}$, the goal is to find a low rank factorization $Z_{n \times r}W_{r \times m}$ that minimizes the error $\|ZW-Y\|_F$. Gradient descent solves this problem optimally. We show that FA converges to the optimal solution when $r\ge \mbox{rank}(Y)$. We also shed light on how FA works. It is observed empirically that the forward weight matrices and (random) feedback matrices come closer during FA updates. Our analysis rigorously derives this phenomenon and shows how it facilitates convergence of FA*, a closely related variant of FA. We also show that FA can be far from optimal when $r < \mbox{rank}(Y)$. This is the first provable separation result between gradient descent and FA. Moreover, the representations found by gradient descent and FA can be almost orthogonal even when their error $\|ZW-Y\|_F$ is approximately equal. As a corollary, these results also hold for training two-layer linear neural networks when the training input is isotropic, and the output is a linear function of the input.

cs.NE↗
arXiv2022

Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices

We study the Unadjusted Langevin Algorithm (ULA) for sampling from a probability distribution $ν= e^{-f}$ on $\mathbb{R}^n$. We prove a convergence guarantee in Kullback-Leibler (KL) divergence assuming $ν$ satisfies a log-Sobolev inequality and the Hessian of $f$ is bounded. Notably, we do not assume convexity or bounds on higher derivatives. We also prove convergence guarantees in Rényi divergence of order $q > 1$ assuming the limit of ULA satisfies either the log-Sobolev or Poincaré inequality. We also prove a bound on the bias of the limiting distribution of ULA assuming third-order smoothness of $f$, without requiring isoperimetry.

cs.DS↗
arXiv2022

Provable Lifelong Learning of Representations

In lifelong learning, tasks (or classes) to be learned arrive sequentially over time in arbitrary order. During training, knowledge from previous tasks can be captured and transferred to subsequent ones to improve sample efficiency. We consider the setting where all target tasks can be represented in the span of a small number of unknown linear or nonlinear features of the input data. We propose a lifelong learning algorithm that maintains and refines the internal feature representation. We prove that for any desired accuracy on all tasks, the dimension of the representation remains close to that of the underlying representation. The resulting sample complexity improves significantly on existing bounds. In the setting of linear features, our algorithm is provably efficient and the sample complexity for input dimension $d$, $m$ tasks with $k$ features up to error $ε$ is $\tilde{O}(dk^{1.5}/ε+km/ε)$. We also prove a matching lower bound for any lifelong learning algorithm that uses a single task learner as a black box. We complement our analysis with an empirical study, including a heuristic lifelong learning algorithm for deep neural networks. Our method performs favorably on challenging realistic image datasets compared to state-of-the-art continual learning methods.

cs.LG↗
arXiv2022

Approximating Sparse Graphs: The Random Overlapping Communities Model

How can we approximate sparse graphs and sequences of sparse graphs (with unbounded average degree)? We consider convergence in the first $k$ moments of the graph spectrum (equivalent to the numbers of closed $k$-walks) appropriately normalized. We introduce a simple, easy to sample, random graph model that captures the limiting spectra of many sequences of interest, including the sequence of hypercube graphs. The Random Overlapping Communities (ROC) model is specified by a distribution on pairs $(s,q)$, $s \in \mathbb{Z}_+, q \in (0,1]$. A graph on $n$ vertices with average degree $d$ is generated by repeatedly picking pairs $(s,q)$ from the distribution, adding an Erdős-Rényi random graph of edge density $q$ on a subset of vertices chosen by including each vertex with probability $s/n$, and repeating this process so that the expected degree is $d$. Our proof of convergence to a ROC random graph is based on the Stieltjes moment condition. We also show that the model is an effective approximation for individual graphs. For almost all possible triangle-to-edge and four-cycle-to-edge ratios, there exists a pair $(s,q)$ such that the ROC model with this single community type produces graphs with both desired ratios, a property that cannot be achieved by stochastic block models of bounded description size. Moreover, ROC graphs exhibit an inverse relationship between degree and clustering coefficient, a characteristic of many real-world networks.

math.CO↗