We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank $3$): (a) The number of extreme points in an $n$-point order type, chosen uniformly at random from all such order types, is on average $4+o(1)$. For labeled order types, this number has average $4- \frac{8}{n^2 - n +2}$ and variance at most $3$. (b) The (labeled) order types read off a set of $n$ points sampled independently from the uniform measure on a convex planar domain, smooth or polygonal, or from a Gaussian distribution are concentrated, i.e. such sampling typically encounters only a vanishingly small fraction of all order types of the given size. Result (a) generalizes to arbitrary dimension $d$ for labeled order types with the average number of extreme points $2d+o(1)$ and constant variance. We also discuss to what extent our methods generalize to the abstract setting of uniform acyclic oriented matroids. Moreover, our methods allow to show the following relative of the Erd\H{o}s-Szekeres theorem: for any fixed $k$, as $n \to \infty$, a proportion $1 - O(1/n)$ of the $n$-point simple order types contain a triangle enclosing a convex $k$-chain over an edge. For the unlabeled case in (a), we prove that for any antipodal, finite subset of the $2$-dimensional sphere, the group of orientation preserving bijections is cyclic, dihedral or one of $A_4$, $S_4$ or $A_5$ (and each case is possible). These are the finite subgroups of $SO(3)$ and our proof follows the lines of their characterization by Felix Klein.
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Stochastic gradient algorithms are widely used for both optimization and sampling in large-scale learning and inference problems. However, in practice, tuning these algorithms is typically done using heuristics and trial-and-error rather than rigorous, generalizable theory. To address this gap between theory and practice, we novel insights into the effect of tuning parameters by characterizing the large-sample behavior of iterates of a very general class of preconditioned stochastic gradient algorithms with fixed step size. In the optimization setting, our results show that iterate averaging with a large fixed step size can result in statistically efficient approximation of the (local) M-estimator. In the sampling context, our results show that with appropriate choices of tuning parameters, the limiting stationary covariance can match either the Bernstein--von Mises limit of the posterior, adjustments to the posterior for model misspecification, or the asymptotic distribution of the MLE; and that with a naive tuning the limit corresponds to none of these. Moreover, we argue that an essentially independent sample from the stationary distribution can be obtained after a fixed number of passes over the dataset. We validate our asymptotic results in realistic finite-sample regimes via several experiments using simulated and real data. Overall, we demonstrate that properly tuned stochastic gradient algorithms with constant step size offer a computationally efficient and statistically robust approach to obtaining point estimates or posterior-like samples.
Neural networks are high-dimensional nonlinear dynamical systems that process information through the coordinated activity of many interconnected units. Understanding how biological and machine-learning networks function and learn requires knowledge of the structure of this coordinated activity, information contained in cross-covariances between units. Although dynamical mean field theory (DMFT) has elucidated several features of random neural networks -- in particular, that they can generate chaotic activity -- existing DMFT approaches do not support the calculation of cross-covariances. We solve this longstanding problem by extending the DMFT approach via a two-site cavity method. This reveals, for the first time, several spatial and temporal features of activity coordination, including the effective dimension, defined as the participation ratio of the spectrum of the covariance matrix. Our results provide a general analytical framework for studying the structure of collective activity in random neural networks and, more broadly, in high-dimensional nonlinear dynamical systems with quenched disorder.
We establish optimal convergence rates up to a log-factor for a class of deep neural networks in a classification setting under a restraint sometimes referred to as the Tsybakov noise condition. We construct classifiers in a general setting where the boundary of the bayes-rule can be approximated well by neural networks. Corresponding rates of convergence are proven with respect to the misclassification error. It is then shown that these rates are optimal in the minimax sense if the boundary satisfies a smoothness condition. Non-optimal convergence rates already exist for this setting. Our main contribution lies in improving existing rates and showing optimality, which was an open problem. Furthermore, we show almost optimal rates under some additional restraints which circumvent the curse of dimensionality. For our analysis we require a condition which gives new insight on the restraint used. In a sense it acts as a requirement for the "correct noise exponent" for a class of functions.
The random batch method provides an efficient algorithm for computing statistical properties of a canonical ensemble of interacting particles. In this work, we study the error estimates of the fully discrete random batch method, especially in terms of approximating the invariant distribution. Using a triangle inequality framework, we show that the long-time error of the method is $O(\sqrt{\tau} + e^{-\gamma t})$, where $\tau$ is the time step and $\gamma$ is the convergence rate which does not depend on the time step $\tau$ or the number of particles $N$. Our results also apply to the McKean-Vlasov process, which is the mean-field limit of the interacting particle system as the number of particles $N\rightarrow\infty$.
We consider stochastic gradient descent and its averaging variant for binary classification problems in a reproducing kernel Hilbert space. In the traditional analysis using a consistency property of loss functions, it is known that the expected classification error converges more slowly than the expected risk even when assuming a low-noise condition on the conditional label probabilities. Consequently, the resulting rate is sublinear. Therefore, it is important to consider whether much faster convergence of the expected classification error can be achieved. In recent research, an exponential convergence rate for stochastic gradient descent was shown under a strong low-noise condition but provided theoretical analysis was limited to the squared loss function, which is somewhat inadequate for binary classification tasks. In this paper, we show an exponential convergence of the expected classification error in the final phase of the stochastic gradient descent for a wide class of differentiable convex loss functions under similar assumptions. As for the averaged stochastic gradient descent, we show that the same convergence rate holds from the early phase of training. In experiments, we verify our analyses on the $L_2$-regularized logistic regression.
In this paper we find an integer $h=h(n)$ such that the minimum number of variables of a first order sentence that distinguishes between two independent uniformly distributed random graphs of size $n$ with the asymptotically largest possible probability $\frac{1}{4}-o(1)$ belongs to $\{h,h+1,h+2,h+3\}$. We also prove that the minimum (random) $k$ such that two independent random graphs are distinguishable by a first order sentence with $k$ variables belongs to $\{h,h+1,h+2\}$ with probability $1-o(1)$.
It is a well-known fact that there is no complete and discrete invariant on the collection of all multiparameter persistence modules. Nonetheless, many invariants have been proposed in the literature to study multiparameter persistence modules, though each invariant will lose some amount of information. One such invariant is the generalized rank invariant. This invariant is known to be complete on the class of interval decomposable persistence modules in general, under mild assumptions on the indexing poset $P$. There is often a trade-off, where the stronger an invariant is, the more expensive it is to compute in practice. The generalized rank invariant on its own is difficult to compute, whereas the standard rank invariant is readily computable through software implementations such as RIVET. We can interpolate between these two to induce new invariants via restricting the domain of the generalized rank invariant, and this family exhibits the aforementioned trade-off. This work studies the tension which exists between computational efficiency and retaining strength when restricting the domain of the generalized rank invariant. We provide a characterization result on where such restrictions are complete invariants in the setting where $P$ is finite, and furthermore show that such restricted generalized rank invariants are stable.
The symmetric circulant TSP is a special case of the traveling salesman problem in which edge costs are symmetric and obey circulant symmetry. Despite the substantial symmetry of the input, remarkably little is known about the symmetric circulant TSP, and the complexity of the problem has been an often-cited open question. Considerable effort has been made to understand the case in which only edges of two lengths are allowed to have finite cost: the two-stripe symmetric circulant TSP. In this paper, we resolve the complexity of the two-stripe symmetric circulant TSP. To do so, we reduce two-stripe symmetric circulant TSP to the problem of finding certain minimum-cost Hamiltonian paths on cylindrical graphs. We then solve this Hamiltonian path problem. Our results show that the two-stripe symmetric circulant TSP is in P. Note that a two-stripe symmetric circulant TSP instance consists of a constant number of inputs (including $n$, the number of cities), so that a polynomial-time algorithm for the decision problem must run in time polylogarithmic in $n$, and a polynomial-time algorithm for the optimization problem cannot output the tour. We address this latter difficulty by showing that the optimal tour must fall into one of two parameterized classes of tours, and that we can output the class and the parameters in polynomial time. Thus we make a substantial contribution to the set of polynomial-time solvable special cases of the TSP, and take an important step towards resolving the complexity of the general symmetric circulant TSP.
The seminal paper by Mazumdar and Saha \cite{MS17a} introduced an extensive line of work on clustering with noisy queries. Yet, despite significant progress on the problem, the proposed methods depend crucially on knowing the exact probabilities of errors of the underlying fully-random oracle. In this work, we develop robust learning methods that tolerate general semi-random noise obtaining qualitatively the same guarantees as the best possible methods in the fully-random model. More specifically, given a set of $n$ points with an unknown underlying partition, we are allowed to query pairs of points $u,v$ to check if they are in the same cluster, but with probability $p$, the answer may be adversarially chosen. We show that information theoretically $O\left(\frac{nk \log n} {(1-2p)^2}\right)$ queries suffice to learn any cluster of sufficiently large size. Our main result is a computationally efficient algorithm that can identify large clusters with $O\left(\frac{nk \log n} {(1-2p)^2}\right) + \text{poly}\left(\log n, k, \frac{1}{1-2p} \right)$ queries, matching the guarantees of the best known algorithms in the fully-random model. As a corollary of our approach, we develop the first parameter-free algorithm for the fully-random model, answering an open question by \cite{MS17a}.
Motivated by applications to COVID dynamics, we describe a branching process in random environments model $\{Z_n\}$ whose path behavior changes when crossing upper and lower thresholds. This introduces a cyclical path behavior involving periods of increase and decrease leading to supercritical and subcritical regimes. Even though the process is not Markov, we identify subsequences at random time points $\{(\tau_j, \nu_j)\}$ -- specifically the values of the process at crossing times, viz., $\{(Z_{\tau_j}, Z_{\nu_j})\}$ -- along which the process retains the Markov structure. Under mild moment and regularity conditions, we establish that the subsequences possess a regenerative structure and prove that the limiting normal distribution of the growth rates of the process in supercritical and subcritical regimes decouple. For this reason, we establish limit theorems concerning the length of supercritical and subcritical regimes and the proportion of time the process spends in these regimes. As a byproduct of our analysis, we explicitly identify the limiting variances in terms of the functionals of the offspring distribution, threshold distribution, and environmental sequences.
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