Learning Convex Partitions and Computing Game-theoretic Equilibria from Best Response Queries

Suppose that an $m$-simplex is partitioned into $n$ convex regions having disjoint interiors and distinct labels, and we may learn the label of any point by querying it. The learning objective is to know, for any point in the simplex, a label that occurs within some distance $\epsilon$ from that point. We present two algorithms for this task: Constant-Dimension Generalised Binary Search (CD-GBS), which for constant $m$ uses $poly(n, \log \left( \frac{1}{\epsilon} \right))$ queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant $n$ uses $poly(m, \log \left( \frac{1}{\epsilon} \right))$ queries. We show via Kakutani's fixed-point theorem that these algorithms provide bounds on the best-response query complexity of computing approximate well-supported equilibria of bimatrix games in which one of the players has a constant number of pure strategies. We also partially extend our results to games with multiple players, establishing further query complexity bounds for computing approximate well-supported equilibria in this setting.Comment: 38 pages, 7 figures, second version strengthens lower bound in Theorem 6, adds footnotes with additional comments and fixes typo

Quantum Algorithms for Learning and Testing Juntas

In this article we develop quantum algorithms for learning and testing juntas, i.e. Boolean functions which depend only on an unknown set of k out of n input variables. Our aim is to develop efficient algorithms: - whose sample complexity has no dependence on n, the dimension of the domain the Boolean functions are defined over; - with no access to any classical or quantum membership ("black-box") queries. Instead, our algorithms use only classical examples generated uniformly at random and fixed quantum superpositions of such classical examples; - which require only a few quantum examples but possibly many classical random examples (which are considered quite "cheap" relative to quantum examples). Our quantum algorithms are based on a subroutine FS which enables sampling according to the Fourier spectrum of f; the FS subroutine was used in earlier work of Bshouty and Jackson on quantum learning. Our results are as follows: - We give an algorithm for testing k-juntas to accuracy $\epsilon$ that uses $O(k/\epsilon)$ quantum examples. This improves on the number of examples used by the best known classical algorithm. - We establish the following lower bound: any FS-based k-junta testing algorithm requires $\Omega(\sqrt{k})$ queries. - We give an algorithm for learning $k$-juntas to accuracy $\epsilon$ that uses $O(\epsilon^{-1} k\log k)$ quantum examples and $O(2^k \log(1/\epsilon))$ random examples. We show that this learning algorithms is close to optimal by giving a related lower bound.Comment: 15 pages, 1 figure. Uses synttree package. To appear in Quantum Information Processin

Lower Bounds on the Oracle Complexity of Nonsmooth Convex Optimization via Information Theory

We present an information-theoretic approach to lower bound the oracle complexity of nonsmooth black box convex optimization, unifying previous lower bounding techniques by identifying a combinatorial problem, namely string guessing, as a single source of hardness. As a measure of complexity we use distributional oracle complexity, which subsumes randomized oracle complexity as well as worst-case oracle complexity. We obtain strong lower bounds on distributional oracle complexity for the box $[-1,1]^n$, as well as for the $L^p$-ball for $p \geq 1$ (for both low-scale and large-scale regimes), matching worst-case upper bounds, and hence we close the gap between distributional complexity, and in particular, randomized complexity, and worst-case complexity. Furthermore, the bounds remain essentially the same for high-probability and bounded-error oracle complexity, and even for combination of the two, i.e., bounded-error high-probability oracle complexity. This considerably extends the applicability of known bounds

Oracles and query lower bounds in generalised probabilistic theories

We investigate the connection between interference and computational power within the operationally defined framework of generalised probabilistic theories. To compare the computational abilities of different theories within this framework we show that any theory satisfying three natural physical principles possess a well-defined oracle model. Indeed, we prove a subroutine theorem for oracles in such theories which is a necessary condition for the oracle to be well-defined. The three principles are: causality (roughly, no signalling from the future), purification (each mixed state arises as the marginal of a pure state of a larger system), and strong symmetry existence of non-trivial reversible transformations). Sorkin has defined a hierarchy of conceivable interference behaviours, where the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multi-slit experiment. Given our oracle model, we show that if a classical computer requires at least n queries to solve a learning problem, then the corresponding lower bound in theories lying at the kth level of Sorkin's hierarchy is n/k. Hence, lower bounds on the number of queries to a quantum oracle needed to solve certain problems are not optimal in the space of all generalised probabilistic theories, although it is not yet known whether the optimal bounds are achievable in general. Hence searches for higher-order interference are not only foundationally motivated, but constitute a search for a computational resource beyond that offered by quantum computation.Comment: 17+7 pages. Comments Welcome. Published in special issue "Foundational Aspects of Quantum Information" in Foundations of Physic