In 1983, Aldous proved that randomization can speedup local search. For example, it reduces the query complexity of local search over grid \left[ {1:n} \right]^d from \Theta (n^{d - 1} ) to {\rm O}(d^{1/2} n^{d/2} ). It remains open whether randomization helps fixed-point computation. Inspired by the recent advances on the complexity of equilibrium computation, we solve this open problem by giving an asymptotically tight bound of (\Omega (n))^{d - 1} on the randomized query complexity for computing a fixed point of a discrete Brouwer function over grid \left[ {1:n} \right]^d Our result can be extended to the black-box query model for Sperner's Lemma in any dimension. It also yields a tight bound for the computation of d-dimensional approximate Brouwer fixed points as defined by Scarf and by Hirsch, Papadimitriou, and Vavasis. Since the randomized query complexity of global optimization over \left[ {1:n} \right]^d is \Theta (n^d ), the randomized query model over \left[ {1:n} \right]^d strictly separates these three important search problems: Global optimization is harder than fixed-point computation, and fixed-point computation is harder than local search. Our result indeed demonstrates that randomization does not help much in fixed-point computation in the black-box query model. Our randomized lower bound matches the deterministic complexity of this problem, which is \Theta (n^{d - 1} ).
In 1983, Aldous proved that randomization can speedup local search. For example, it reduces the query complexity of local search over grid \left[ {1:n} \right]^d from \Theta (n^{d - 1} ) to {\rm O}(d^{1/2} n^{d/2} ). It remains open whether randomization helps fixed-point computation. Inspired by the recent advances on the complexity of equilibrium computation, we solve this open problem by giving an asymptotically tight bound of (\Omega (n))^{d - 1} on the randomized query complexity for computing a fixed point of a discrete Brouwer function over grid \left[ {1:n} \right]^d Our result can be extended to the black-box query model for Sperner's Lemma in any dimension. It also yields a tight bound for the computation of d-dimensional approximate Brouwer fixed points as defined by Scarf and by Hirsch, Papadimitriou, and Vavasis. Since the randomized query complexity of global optimization over \left[ {1:n} \right]^d is \Theta (n^d ), the randomized query model over \left[ {1:n} \right]^d strictly separates these three important search problems: Global optimization is harder than fixed-point computation, and fixed-point computation is harder than local search. Our result indeed demonstrates that randomization does not help much in fixed-point computation in the black-box query model. Our randomized lower bound matches the deterministic complexity of this problem, which is \Theta (n^{d - 1} ).
In 1983, Aldous proved that randomization can speedup local search. For example, it reduces the query complexity of local search over grid \left[ {1:n} \right]^d from \Theta (n^{d - 1} ) to {\rm O}(d^{1/2} n^{d/2} ). It remains open whether randomization helps fixed-point computation. Inspired by the recent advances on the complexity of equilibrium computation, we solve this open problem by giving an asymptotically tight bound of (\Omega (n))^{d - 1} on the randomized query complexity for computing a fixed point of a discrete Brouwer function over grid \left[ {1:n} \right]^d Our result can be extended to the black-box query model for Sperner's Lemma in any dimension. It also yields a tight bound for the computation of d-dimensional approximate Brouwer fixed points as defined by Scarf and by Hirsch, Papadimitriou, and Vavasis. Since the randomized query complexity of global optimization over \left[ {1:n} \right]^d is \Theta (n^d ), the randomized query model over \left[ {1:n} \right]^d strictly separates these three important search problems: Global optimization is harder than fixed-point computation, and fixed-point computation is harder than local search. Our result indeed demonstrates that randomization does not help much in fixed-point computation in the black-box query model. Our randomized lower bound matches the deterministic complexity of this problem, which is \Theta (n^{d - 1} ).
In 1983, Aldous proved that randomization can speedup local search. For example, it reduces the query complexity of local search over [1:n]^d from Theta (n^{d-1}) to O (d^{1/2}n^{d/2}). It remains open whether randomization helps fixed-point computation. Inspired by this open problem and recent advances on equilibrium computation, we have been fascinated by the following question: Is a fixed-point or an equilibrium fundamentally harder to find than a local optimum? In this paper, we give a nearly-tight bound of Omega(n)^{d-1} on the randomized query complexity for computing a fixed point of a discrete Brouwer function over [1:n]^d. Since the randomized query complexity of global optimization over [1:n]^d is Theta (n^{d}), the randomized query model over [1:n]^d strictly separates these three important search problems: Global optimization is harder than fixed-point computation, and fixed-point computation is harder than local search. Our result indeed demonstrates that randomization does not help much in fixed-point computation in the query model; the deterministic complexity of this problem is Theta (n^{d-1}).
The problem Local Search, which finds a local minimum of a black-box function on a given graph, is of both practical and theoretical importance to combinatorial optimization, complexity theory, and many other areas in theoretical computer science. In this paper, we study the problem in both the randomized and the quantum query models and give new lower and upper bound techniques in both models. The lower bound technique works for any graph that contains a product graph as a subgraph. Applying it to the Boolean hypercube {0,1}n and the constant-dimensional grids [n]d, two particular product graphs that recently drew much attention, we get the following tight results: $\text{{\it RLS\/}}(\{0,1\}^n)=\Theta(2^{n/2}n^{1/2})$, $\text{{\it QLS\/}}(\{0,1\}^n)=\Theta(2^{n/3}n^{1/6})$, $\text{{\it RLS\/}}([n]^d)=\Theta(n^{d/2})$ for d≥4, $\text{{\it QLS\/}}([n]^d)=\Theta(n^{d/3})$ for d≥6. Here $\text{{\it RLS\/}}(G)$ and $\text{{\it QLS\/}}(G)$ are the randomized and quantum query complexities of Local Search on G, respectively. These improve the previous results by Aaronson [in Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, 2004, pp. 465–474], Ambainis (unpublished), and Santha and Szegedy [in Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, 2004, pp. 494–501]. Our new algorithms work well when the underlying graph expands slowly. As an application to [n]2, a new quantum algorithm using O(n(loglogn)1.5) queries is given. This improves the previously best known upper bound of O(n2/3) (see Aaronson [in Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, 2004, pp. 465–474]), and implies that Local Search on grids exhibits different properties in low dimensions.
The Local Search problem, which finds a local minimum of a black-box function on a given graph, is of both practical and theoretical importance to many areas in computer science and natural sciences. In this paper, we show that for the Boolean hypercube $\B^n$, the randomized query complexity of Local Search is Θ(2n/2n1/2) and the quantum query complexity is Θ(2n/3n1/6). We also show that for the constant dimensional grid [N1/d]d, the randomized query complexity is Θ(N1/2) for d≥4 and the quantum query complexity is Θ(N1/3) for d≥6. New lower bounds for lower dimensional grids are also given. These improve the previous results by Aaronson [STOC'04], and Santha and Szegedy [STOC'04]. Finally we show for [N1/2]2 a new upper bound of O(N1/4(loglogN)3/2) on the quantum query complexity, which implies that Local Search on grids exhibits different properties at low dimensions.
We consider the query complexity of finding a local minimum of a function defined on a graph. This abstract problem is fundamental to many optimization tasks, such as finding a local minimum of the loss function when training deep neural networks. In such applications, each query is an expensive loss evaluation, making it crucial to parallelize computations. This motivates our study of local search where at most k rounds of interaction (aka adaptivity) with the oracle are allowed. We focus on the d -dimensional grid \{1, 2, \ldots, n \}^{d} , where the dimension d \geq 2 is a constant. Our main contribution is to give algorithms and lower bounds that characterize the query complexity of finding a local minimum in k rounds, when k is constant and polynomial in n , respectively. 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. The local search analysis also enables us to characterize the query complexity of computing a Brouwer fixed point in rounds.
We study the problem of \emph{local search} on a graph. Given a real-valued black-box function f on the graph's vertices, this is the problem of determining a local minimum of f---a vertex v for which f(v) is no more than f evaluated at any of v's neighbors. In 1983, Aldous gave the first strong lower bounds for the problem, showing that any randomized algorithm requires Ω(2n/2−o(n)) queries to determine a local minima on the n-dimensional hypercube. The next major step forward was not until 2004 when Aaronson, introducing a new method for query complexity bounds, both strengthened this lower bound to Ω(2n/2/n2) and gave an analogous lower bound on the quantum query complexity. While these bounds are very strong, they are known only for narrow families of graphs (hypercubes and grids). We show how to generalize Aaronson's techniques in order to give randomized (and quantum) lower bounds on the query complexity of local search for the family of vertex-transitive graphs. In particular, we show that for any vertex-transitive graph G of N vertices and diameter d, the randomized and quantum query complexities for local search on G are Ω(N/dlogN) and Ω(4N/dlogN), respectively.