We consider the problem of allocating $m$ balls into $n$ bins with incomplete information. In the classical two-choice process, a ball first queries the load of $\textit{two}$ randomly chosen bins and is then placed in the least loaded bin. In our setting, each ball also samples two random bins but can only estimate a bin's load by sending $\textit{binary queries}$ of the form "Is the load at least the median?" or "Is the load at least $100$?". For the lightly loaded case $m=O(n)$, one can achieve an $O(\sqrt{\log n/\log \log n})$ maximum load with one query per chosen bin using an oblivious strategy, as shown by Feldheim and Gurel-Gurevich (2018). For the case $m=\Omega(n)$, the authors conjectured that the same strategy achieves a maximum load of $m/n+O(\sqrt{\log n/\log \log n})$. In this work, we disprove this conjecture by showing a lower bound of $m/n+\Omega( \sqrt{\log n})$ for a fixed $m=\Theta(n \sqrt{\log n})$, and a lower bound of $m/n+\Omega(\log n/\log\log n)$ for some $m$ depending on the used strategy. Surprisingly, these lower bounds hold even for any $\textit{adaptive strategy}$ with one query, i.e., queries may depend on the full history of the process. We complement this negative result by proving a positive result for multiple queries. In particular, we show that with only two binary queries per chosen bin, there is an oblivious strategy which ensures a maximum load of $m/n+O(\sqrt{\log n})$ whp for any $m \geq 1$. For any $k=O(\log \log n)$ binary queries, the upper bound on the maximum load improves to $m/n+O(k(\log n)^{1/k})$ whp for any $m \geq 1$. Hence for $k=\Theta(\log\log n)$, we recover the two-choice result up to a constant multiplicative factor, including the heavily loaded case where $m=\Omega(n)$. One novel aspect of our proof techniques is the use of multiple super-exponential potential functions, which might be of use in future work.
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We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let $f$ be an $m$-bit Boolean function and consider an $n$-bit function $F$ obtained by applying $f$ to conjunctions of possibly overlapping subsets of $n$ variables. If $f$ has quantum query complexity $Q(f)$, we give an algorithm for evaluating $F$ using $\tilde{O}(\sqrt{Q(f) \cdot n})$ quantum queries. This improves on the bound of $O(Q(f) \cdot \sqrt{n})$ that follows by treating each conjunction independently, and our bound is tight for worst-case choices of $f$. Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of $f$. By recursively applying our composition theorems, we obtain a nearly optimal $\tilde{O}(n^{1-2^{-d}})$ upper bound on the quantum query complexity and approximate degree of linear-size depth-$d$ AC$^0$ circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC$^0$ circuits. As an additional consequence, we show that AC$^0 \circ \oplus$ circuits of depth $d+1$ require size $\tilde{\Omega}(n^{1/(1- 2^{-d})}) \geq \omega(n^{1+ 2^{-d}} )$ to compute the Inner Product function even on average. The previous best size lower bound was $\Omega(n^{1+4^{-(d+1)}})$ and only held in the worst case (Cheraghchi et al., JCSS 2018).
Multitask Gaussian processes (MTGP) are the Gaussian process (GP) framework's solution for multioutput regression problems in which the $T$ elements of the regressors cannot be considered conditionally independent given the observations. Standard MTGP models assume that there exist both a multitask covariance matrix as a function of an intertask matrix, and a noise covariance matrix. These matrices need to be approximated by a low rank simplification of order $P$ in order to reduce the number of parameters to be learnt from $T^2$ to $TP$. Here we introduce a novel approach that simplifies the multitask learning by reducing it to a set of conditioned univariate GPs without the need for any low rank approximations, therefore completely eliminating the requirement to select an adequate value for hyperparameter $P$. At the same time, by extending this approach with both a hierarchical and an approximate model, the proposed extensions are capable of recovering the multitask covariance and noise matrices after learning only $2T$ parameters, avoiding the validation of any model hyperparameter and reducing the overall complexity of the model as well as the risk of overfitting. Experimental results over synthetic and real problems confirm the advantages of this inference approach in its ability to accurately recover the original noise and signal matrices, as well as the achieved performance improvement in comparison to other state of art MTGP approaches. We have also integrated the model with standard GP toolboxes, showing that it is computationally competitive with state of the art options.
We consider the goodness-of fit testing problem for H\"older smooth densities over $\mathbb{R}^d$: given $n$ iid observations with unknown density $p$ and given a known density $p_0$, we investigate how large $\rho$ should be to distinguish, with high probability, the case $p=p_0$ from the composite alternative of all H\"older-smooth densities $p$ such that $\|p-p_0\|_t \geq \rho$ where $t \in [1,2]$. The densities are assumed to be defined over $\mathbb{R}^d$ and to have H\"older smoothness parameter $\alpha>0$. In the present work, we solve the case $\alpha \leq 1$ and handle the case $\alpha>1$ using an additional technical restriction on the densities. We identify matching upper and lower bounds on the local minimax rates of testing, given explicitly in terms of $p_0$. We propose novel test statistics which we believe could be of independent interest. We also establish the first definition of an explicit cutoff $u_B$ allowing us to split $\mathbb{R}^d$ into a bulk part (defined as the subset of $\mathbb{R}^d$ where $p_0$ takes only values greater than or equal to $u_B$) and a tail part (defined as the complementary of the bulk), each part involving fundamentally different contributions to the local minimax rates of testing.
We revisit the discrete heterogeneous two-facility location problem, in which there is a set of agents that occupy nodes of a line graph, and have private approval preferences over two facilities. When the facilities are located at some nodes of the line, each agent derives a cost that is equal to her total distance from the facilities she approves. The goal is to decide where to locate the two facilities, so as to (a) incentivize the agents to truthfully report their preferences, and (b) achieve a good approximation of the minimum total (social) cost or the maximum cost among all agents. For both objectives, we design deterministic strategyproof mechanisms with approximation ratios that significantly outperform the state-of-the-art, and complement these results with (almost) tight lower bounds.
The tube method or the volume-of-tube method approximates the tail probability of the maximum of a smooth Gaussian random field with zero mean and unit variance. This method evaluates the volume of a spherical tube about the index set, and then transforms it to the tail probability. In this study, we generalize the tube method to a case in which the variance is not constant. We provide the volume formula for a spherical tube with a non-constant radius in terms of curvature tensors, and the tail probability formula of the maximum of a Gaussian random field with inhomogeneous variance, as well as its Laplace approximation. In particular, the critical radius of the tube is generalized for evaluation of the asymptotic approximation error. As an example, we discuss the approximation of the largest eigenvalue distribution of the Wishart matrix with a non-identity matrix parameter. The Bonferroni method is the tube method when the index set is a finite set. We provide the formula for the asymptotic approximation error for the Bonferroni method when the variance is not constant.
In this work, we study the $k$-means cost function. Given a dataset $X \subseteq \mathbb{R}^d$ and an integer $k$, the goal of the Euclidean $k$-means problem is to find a set of $k$ centers $C \subseteq \mathbb{R}^d$ such that $\Phi(C, X) \equiv \sum_{x \in X} \min_{c \in C} ||x - c||^2$ is minimized. Let $\Delta(X,k) \equiv \min_{C \subseteq \mathbb{R}^d} \Phi(C, X)$ denote the cost of the optimal $k$-means solution. For any dataset $X$, $\Delta(X,k)$ decreases as $k$ increases. In this work, we try to understand this behaviour more precisely. For any dataset $X \subseteq \mathbb{R}^d$, integer $k \geq 1$, and a precision parameter $\varepsilon > 0$, let $L(X, k, \varepsilon)$ denote the smallest integer such that $\Delta(X, L(X, k, \varepsilon)) \leq \varepsilon \cdot \Delta(X,k)$. We show upper and lower bounds on this quantity. Our techniques generalize for the metric $k$-median problem in arbitrary metric spaces and we give bounds in terms of the doubling dimension of the metric. Finally, we observe that for any dataset $X$, we can compute a set $S$ of size $O \left(L(X, k, \varepsilon/c) \right)$ using $D^2$-sampling such that $\Phi(S,X) \leq \varepsilon \cdot \Delta(X,k)$ for some fixed constant $c$. We also discuss some applications of our bounds.
A new format for commutator-free Lie group methods is proposed based on explicit classical Runge-Kutta schemes. In this format exponentials are reused at every stage and the storage is required only for two quantities: the right hand side of the differential equation evaluated at a given Runge-Kutta stage and the function value updated at the same stage. The next stage of the scheme is able to overwrite these values. The result is proven for a 3-stage third order method and a conjecture for higher order methods is formulated. Five numerical examples are provided in support of the conjecture. This new class of structure-preserving integrators has a wide variety of applications for numerically solving differential equations on manifolds.
In the oracle identification problem we have oracle access to bits of an unknown string $x$ of length $n$, with the promise that it belongs to a known set $C\subseteq\{0,1\}^n$. The goal is to identify $x$ using as few queries to the oracle as possible. We develop a quantum query algorithm for this problem with query complexity $O\left(\sqrt{\frac{n\log M }{\log(n/\log M)+1}}\right)$, where $M$ is the size of $C$. This bound is already derived by Kothari in 2014, for which we provide a more elegant simpler proof.
In applications of remote sensing, estimation, and control, timely communication is not always ensured by high-rate communication. This work proposes distributed age-efficient transmission policies for random access channels with $M$ transmitters. In the first part of this work, we analyze the age performance of stationary randomized policies by relating the problem of finding age to the absorption time of a related Markov chain. In the second part of this work, we propose the notion of \emph{age-gain} of a packet to quantify how much the packet will reduce the instantaneous age of information at the receiver side upon successful delivery. We then utilize this notion to propose a transmission policy in which transmitters act in a distributed manner based on the age-gain of their available packets. In particular, each transmitter sends its latest packet only if its corresponding age-gain is beyond a certain threshold which could be computed adaptively using the collision feedback or found as a fixed value analytically in advance. Both methods improve age of information significantly compared to the state of the art. In the limit of large $M$, we prove that when the arrival rate is small (below $\frac{1}{eM}$), slotted ALOHA-type algorithms are asymptotically optimal. As the arrival rate increases beyond $\frac{1}{eM}$, while age increases under slotted ALOHA, it decreases significantly under the proposed age-based policies. For arrival rates $\theta$, $\theta=\frac{1}{o(M)}$, the proposed algorithms provide a multiplicative factor of at least two compared to the minimum age under slotted ALOHA (minimum over all arrival rates). We conclude that, as opposed to the common practice, it is beneficial to increase the sampling rate (and hence the arrival rate) and transmit packets selectively based on their age-gain.
Learning low-dimensional embeddings of knowledge graphs is a powerful approach used to predict unobserved or missing edges between entities. However, an open challenge in this area is developing techniques that can go beyond simple edge prediction and handle more complex logical queries, which might involve multiple unobserved edges, entities, and variables. For instance, given an incomplete biological knowledge graph, we might want to predict "em what drugs are likely to target proteins involved with both diseases X and Y?" -- a query that requires reasoning about all possible proteins that {\em might} interact with diseases X and Y. Here we introduce a framework to efficiently make predictions about conjunctive logical queries -- a flexible but tractable subset of first-order logic -- on incomplete knowledge graphs. In our approach, we embed graph nodes in a low-dimensional space and represent logical operators as learned geometric operations (e.g., translation, rotation) in this embedding space. By performing logical operations within a low-dimensional embedding space, our approach achieves a time complexity that is linear in the number of query variables, compared to the exponential complexity required by a naive enumeration-based approach. We demonstrate the utility of this framework in two application studies on real-world datasets with millions of relations: predicting logical relationships in a network of drug-gene-disease interactions and in a graph-based representation of social interactions derived from a popular web forum.
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