We consider the query complexity of finding a local minimum of a function defined on a graph, where at most $k$ rounds of interaction with the oracle are allowed. Rounds model parallel settings, where each query takes resources to complete and is executed on a separate processor. Thus the query complexity in $k$ rounds informs how many processors are needed to achieve a parallel time of $k$. We focus on the d-dimensional grid $[n]^d$, where the dimension $d$ is a constant, and consider two regimes for the number of rounds: constant and polynomial in n. We give algorithms and lower bounds that characterize the trade-off between the number of rounds of adaptivity and the query complexity of local search. When the number of rounds $k$ is constant, we show that the query complexity of local search in $k$ rounds is $\Theta\bigl(n^{\frac{d^{k+1} - d^k}{d^k - 1}}\bigl)$, for both deterministic and randomized algorithms. When the number of rounds is polynomial, i.e. $k = n^{\alpha}$ for $0 < \alpha < d/2$, the randomized query complexity is $\Theta\left(n^{d-1 - \frac{d-2}{d}\alpha}\right)$ for all $d \geq 5$. For $d=3$ and $d=4$, we show the same upper bound expression holds and give almost matching lower bounds. The local search analysis also enables us to characterize the query complexity of computing a Brouwer fixed point in rounds. Our proof technique for lower bounding the query complexity in rounds may be of independent interest as an alternative to the classical relational adversary method of Aaronson from the fully adaptive setting.
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We show the $O(\log n)$ time extract minimum function of efficient priority queues can be generalized to the extraction of the $k$ smallest elements in $O(k \log(n/k))$ time (we define $\log(x)$ as $\max(\log_2(x), 1)$.), which we prove optimal for comparison-based priority queues with $o(\log n)$ time insertion. We show heap-ordered tree selection (Kaplan et al., SOSA '19) can be applied on the heap-ordered trees of the classic Fibonacci heap and Brodal queue, in $O(k \log(n/k))$ amortized and worst-case time, respectively. We additionally show the deletion of $k$ elements or selection without extraction can be performed on both heaps, also in $O(k \log(n/k))$ time. Surprisingly, all operations are possible with no modifications to the original Fibonacci heap and Brodal queue data structures. We then apply the result to lazy search trees (Sandlund & Wild, FOCS '20), creating a new interval data structure based on selectable heaps. This gives optimal $O(B+n)$ time lazy search tree performance, lowering insertion complexity into a gap $\Delta_i$ from $O(\log(n/|\Delta_i|) + \log \log n)$ to $O(\log(n/|\Delta_i|))$ time. An $O(1)$ time merge operation is also made possible when used as a priority queue, among other situations. If Brodal queues are used, all runtimes of the lazy search tree can be made worst-case.
We present new scalar and matrix Chernoff-style concentration bounds for a broad class of probability distributions over the binary hypercube $\{0,1\}^n$. Motivated by recent tools developed for the study of mixing times of Markov chains on discrete distributions, we say that a distribution is $\ell_\infty$-independent when the infinity norm of its influence matrix $\mathcal{I}$ is bounded by a constant. We show that any distribution which is $\ell_\infty$-independent satisfies a matrix Chernoff bound that matches the matrix Chernoff bound for independent random variables due to Tropp. Our matrix Chernoff bound is a broad generalization and strengthening of the matrix Chernoff bound of Kyng and Song (FOCS'18). Using our bound, we can conclude as a corollary that a union of $O(\log|V|)$ random spanning trees gives a spectral graph sparsifier of a graph with $|V|$ vertices with high probability, matching results for independent edge sampling, and matching lower bounds from Kyng and Song.
We consider zero-sum games on infinite graphs, with objectives specified as sets of infinite words over some alphabet of colors. A well-studied class of objectives is the one of $\omega$-regular objectives, due to its relation to many natural problems in theoretical computer science. We focus on the strategy complexity question: given an objective, how much memory does each player require to play as well as possible? A classical result is that finite-memory strategies suffice for both players when the objective is $\omega$-regular. We show a reciprocal of that statement: when both players can play optimally with a chromatic finite-memory structure (i.e., whose updates can only observe colors) in all infinite game graphs, then the objective must be $\omega$-regular. This provides a game-theoretic characterization of $\omega$-regular objectives, and this characterization can help in obtaining memory bounds. Moreover, a by-product of our characterization is a new one-to-two-player lift: to show that chromatic finite-memory structures suffice to play optimally in two-player games on infinite graphs, it suffices to show it in the simpler case of one-player games on infinite graphs. We illustrate our results with the family of discounted-sum objectives, for which $\omega$-regularity depends on the value of some parameters.
A graph is called a sum graph if its vertices can be labelled by distinct positive integers such that there is an edge between two vertices if and only if the sum of their labels is the label of another vertex of the graph. Most papers on sum graphs consider combinatorial questions like the minimum number of isolated vertices that need to be added to a given graph to make it a sum graph. In this paper, we initiate the study of sum graphs from the viewpoint of computational complexity. Notice that every $n$-vertex sum graph can be represented by a sorted list of $n$ positive integers where edge queries can be answered in $O(\log n)$ time. Therefore, limiting the size of the vertex labels also upper-bounds the space complexity of storing the graph in the database. We show that every $n$-vertex, $m$-edge, $d$-degenerate graph can be made a sum graph by adding at most $m$ isolated vertices to it, such that the size of each vertex label is at most $O(n^2d)$. This enables us to store the graph using $O(m\log n)$ bits of memory. For sparse graphs (graphs with $O(n)$ edges), this matches the trivial lower bound of $\Omega(n\log n)$. Since planar graphs and forests have constant degeneracy, our result implies an upper bound of $O(n^2)$ on their label size. The previously best known upper bound on the label size of general graphs with the minimum number of isolated vertices was $O(4^n)$, due to Kratochv\'il, Miller & Nguyen. Furthermore, their proof was existential, whereas our labelling can be constructed in polynomial time.
$\newcommand{\Emph}[1]{{\it{#1}}} \newcommand{\FF}{\mathcal{F}}\newcommand{\region}{\mathsf{r}}\newcommand{\restrictY}[2]{#1 \cap {#2}}$For a set of points $P \subseteq \mathbb{R}^2$, and a family of regions $\FF$, a $\Emph{local~t-spanner}$ of $P$, is a sparse graph $G$ over $P$, such that, for any region $\region \in \FF$, the subgraph restricted to $\region$, denoted by $\restrictY{G}{\region} = G_{P \cap \region}$, is a $t$-spanner for all the points of $\region \cap P$. We present algorithms for the construction of local spanners with respect to several families of regions, such as homothets of a convex region. Unfortunately, the number of edges in the resulting graph depends logarithmically on the spread of the input point set. We prove that this dependency can not be removed, thus settling an open problem raised by Abam and Borouny. We also show improved constructions (with no dependency on the spread) of local spanners for fat triangles, and regular $k$-gons. In particular, this improves over the known construction for axis parallel squares. We also study a somewhat weaker notion of local spanner where one allows to shrink the region a "bit". Any spanner is a weak local spanner if the shrinking is proportional to the diameter. Surprisingly, we show a near linear size construction of a weak spanner for axis-parallel rectangles, where the shrinkage is $\Emph{multiplicative}$.
The suffix array $SA[1..n]$ of a text $T$ of length $n$ is a permutation of $\{1,\ldots,n\}$ describing the lexicographical ordering of suffixes of $T$, and it is considered to be among of the most important data structures in string algorithms, with dozens of applications in data compression, bioinformatics, and information retrieval. One of the biggest drawbacks of the suffix array is that it is very difficult to maintain under text updates: even a single character substitution can completely change the contents of the suffix array. Thus, the suffix array of a dynamic text is modelled using suffix array queries, which return the value $SA[i]$ given any $i\in[1..n]$. Prior to this work, the fastest dynamic suffix array implementations were by Amir and Boneh. At ISAAC 2020, they showed how to answer suffix array queries in $\tilde{O}(k)$ time, where $k\in[1..n]$ is a trade-off parameter, with $\tilde{O}(\frac{n}{k})$-time text updates. In a very recent preprint [2021], they also provided a solution with $O(\log^5 n)$-time queries and $\tilde{O}(n^{2/3})$-time updates. We propose the first data structure that supports both suffix array queries and text updates in $O({\rm polylog}\,n)$ time (achieving $O(\log^4 n)$ and $O(\log^{3+o(1)} n)$ time, respectively). Our data structure is deterministic and the running times for all operations are worst-case. In addition to the standard single-character edits (character insertions, deletions, and substitutions), we support (also in $O(\log^{3+o(1)} n)$ time) the "cut-paste" operation that moves any (arbitrarily long) substring of $T$ to any place in $T$. We complement our structure by a hardness result: unless the Online Matrix-Vector Multiplication (OMv) Conjecture fails, no data structure with $O({\rm polylog}\,n)$-time suffix array queries can support the "copy-paste" operation in $O(n^{1-\epsilon})$ time for any $\epsilon>0$.
We consider interpolation learning in high-dimensional linear regression with Gaussian data, and prove a generic uniform convergence guarantee on the generalization error of interpolators in an arbitrary hypothesis class in terms of the class's Gaussian width. Applying the generic bound to Euclidean norm balls recovers the consistency result of Bartlett et al. (2020) for minimum-norm interpolators, and confirms a prediction of Zhou et al. (2020) for near-minimal-norm interpolators in the special case of Gaussian data. We demonstrate the generality of the bound by applying it to the simplex, obtaining a novel consistency result for minimum l1-norm interpolators (basis pursuit). Our results show how norm-based generalization bounds can explain and be used to analyze benign overfitting, at least in some settings.
We consider the exploration-exploitation trade-off in reinforcement learning and we show that an agent imbued with a risk-seeking utility function is able to explore efficiently, as measured by regret. The parameter that controls how risk-seeking the agent is can be optimized exactly, or annealed according to a schedule. We call the resulting algorithm K-learning and show that the corresponding K-values are optimistic for the expected Q-values at each state-action pair. The K-values induce a natural Boltzmann exploration policy for which the `temperature' parameter is equal to the risk-seeking parameter. This policy achieves an expected regret bound of $\tilde O(L^{3/2} \sqrt{S A T})$, where $L$ is the time horizon, $S$ is the number of states, $A$ is the number of actions, and $T$ is the total number of elapsed time-steps. This bound is only a factor of $L$ larger than the established lower bound. K-learning can be interpreted as mirror descent in the policy space, and it is similar to other well-known methods in the literature, including Q-learning, soft-Q-learning, and maximum entropy policy gradient, and is closely related to optimism and count based exploration methods. K-learning is simple to implement, as it only requires adding a bonus to the reward at each state-action and then solving a Bellman equation. We conclude with a numerical example demonstrating that K-learning is competitive with other state-of-the-art algorithms in practice.
This work considers the problem of provably optimal reinforcement learning for episodic finite horizon MDPs, i.e. how an agent learns to maximize his/her long term reward in an uncertain environment. The main contribution is in providing a novel algorithm --- Variance-reduced Upper Confidence Q-learning (vUCQ) --- which enjoys a regret bound of $\widetilde{O}(\sqrt{HSAT} + H^5SA)$, where the $T$ is the number of time steps the agent acts in the MDP, $S$ is the number of states, $A$ is the number of actions, and $H$ is the (episodic) horizon time. This is the first regret bound that is both sub-linear in the model size and asymptotically optimal. The algorithm is sub-linear in that the time to achieve $\epsilon$-average regret for any constant $\epsilon$ is $O(SA)$, which is a number of samples that is far less than that required to learn any non-trivial estimate of the transition model (the transition model is specified by $O(S^2A)$ parameters). The importance of sub-linear algorithms is largely the motivation for algorithms such as $Q$-learning and other "model free" approaches. vUCQ algorithm also enjoys minimax optimal regret in the long run, matching the $\Omega(\sqrt{HSAT})$ lower bound. Variance-reduced Upper Confidence Q-learning (vUCQ) is a successive refinement method in which the algorithm reduces the variance in $Q$-value estimates and couples this estimation scheme with an upper confidence based algorithm. Technically, the coupling of both of these techniques is what leads to the algorithm enjoying both the sub-linear regret property and the asymptotically optimal regret.
We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.
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