141,394 research outputs found
The Computational Complexity of Finding Stationary Points in Non-Convex Optimization
Finding approximate stationary points, i.e., points where the gradient is
approximately zero, of non-convex but smooth objective functions over
unrestricted -dimensional domains is one of the most fundamental problems in
classical non-convex optimization. Nevertheless, the computational and query
complexity of this problem are still not well understood when the dimension
of the problem is independent of the approximation error. In this paper, we
show the following computational and query complexity results:
1. The problem of finding approximate stationary points over unrestricted
domains is PLS-complete.
2. For , we provide a zero-order algorithm for finding
-approximate stationary points that requires at most
value queries to the objective function.
3. We show that any algorithm needs at least queries
to the objective function and/or its gradient to find -approximate
stationary points when . Combined with the above, this characterizes the
query complexity of this problem to be .
4. For , we provide a zero-order algorithm for finding
-KKT points in constrained optimization problems that requires at
most value queries to the objective function. This
closes the gap between the works of Bubeck and Mikulincer [2020] and Vavasis
[1993] and characterizes the query complexity of this problem to be
.
5. Combining our results with the recent result of Fearnley et al. [2022], we
show that finding approximate KKT points in constrained optimization is
reducible to finding approximate stationary points in unconstrained
optimization but the converse is impossible.Comment: Full version of COLT 2023 extended abstrac
Finding Approximate Nash Equilibria of Bimatrix Games via Payoff Queries
We study the deterministic and randomized query complexity of finding approximate equilibria in a k × k bimatrix game. We show that the deterministic query complexity of finding an ϵ-Nash equilibrium when ϵ < ½ is Ω(k2), even in zero-one constant-sum games. In combination with previous results [Fearnley et al. 2013], this provides a complete characterization of the deterministic query complexity of approximate Nash equilibria. We also study randomized querying algorithms. We give a randomized algorithm for finding a (3-√5/2 + ϵ)-Nash equilibrium using O(k.log k/ϵ2) payoff queries, which shows that the ½ barrier for deterministic algorithms can be broken by randomization. For well-supported Nash equilibria (WSNE), we first give a randomized algorithm for finding an ϵ-WSNE of a zero-sum bimatrix game using O(k.log k/ϵ4) payoff queries, and we then use this to obtain a randomized algorithm for finding a (⅔ + ϵ)-WSNE in a general bimatrix game using O(k.log k/ϵ4) payoff queries. Finally, we initiate the study of lower bounds against randomized algorithms in the context of bimatrix games, by showing that randomized algorithms require Ω(k2) payoff queries in order to find an ϵ-Nash equilibrium with ϵ < 1/4k, even in zero-one constant-sum games. In particular, this rules out query-efficient randomized algorithms for finding exact Nash equilibria
Zeroth-Order Negative Curvature Finding: Escaping Saddle Points without Gradients
We consider escaping saddle points of nonconvex problems where only the
function evaluations can be accessed. Although a variety of works have been
proposed, the majority of them require either second or first-order
information, and only a few of them have exploited zeroth-order methods,
particularly the technique of negative curvature finding with zeroth-order
methods which has been proven to be the most efficient method for escaping
saddle points. To fill this gap, in this paper, we propose two zeroth-order
negative curvature finding frameworks that can replace Hessian-vector product
computations without increasing the iteration complexity. We apply the proposed
frameworks to ZO-GD, ZO-SGD, ZO-SCSG, ZO-SPIDER and prove that these ZO
algorithms can converge to -approximate second-order
stationary points with less query complexity compared with prior zeroth-order
works for finding local minima
Query-Efficient Algorithms to Find the Unique Nash Equilibrium in a Two-Player Zero-Sum Matrix Game
We study the query complexity of identifying Nash equilibria in two-player
zero-sum matrix games. Grigoriadis and Khachiyan (1995) showed that any
deterministic algorithm needs to query entries in worst case from
an input matrix in order to compute an -approximate
Nash equilibrium, where . Moreover, they designed a
randomized algorithm that queries
entries from the input matrix in expectation and returns an
-approximate Nash equilibrium when the entries of the matrix are
bounded between and . However, these two results do not completely
characterize the query complexity of finding an exact Nash equilibrium in
two-player zero-sum matrix games. In this work, we characterize the query
complexity of finding an exact Nash equilibrium for two-player zero-sum matrix
games that have a unique Nash equilibrium . We first show
that any randomized algorithm needs to query entries of the input
matrix in expectation in order to find the unique
Nash equilibrium where . We complement this lower
bound by presenting a simple randomized algorithm that, with probability
, returns the unique Nash equilibrium by querying at most entries of the input matrix
. In the special case when the unique Nash
Equilibrium is a pure-strategy Nash equilibrium (PSNE), we design a simple
deterministic algorithm that finds the PSNE by querying at most
entries of the input matrix.Comment: 17 page
Extended Learning Graphs for Triangle Finding
We present new quantum algorithms for Triangle Finding improving its best previously known quantum query complexities for both dense and sparse instances. For dense graphs on n vertices, we get a query complexity of O(n^(5/4)) without any of the extra logarithmic factors present in the previous algorithm of Le Gall [FOCS\u2714]. For sparse graphs with m >= n^(5/4) edges, we get a query complexity of O(n^(11/12) m^(1/6) sqrt(log n)), which is better than the one obtained by Le Gall and Nakajima [ISAAC\u2715] when m >= n^(3/2). We also obtain an algorithm with query complexity O(n^(5/6) (m log n)^(1/6) + d_2 sqrt(n)) where d_2 is the variance of the degree distribution.
Our algorithms are designed and analyzed in a new model of learning graphs that we call extended learning graphs. In addition, we present a framework in order to easily combine and analyze them. As a consequence we get much simpler algorithms and analyses than previous algorithms of Le Gall based on the MNRS quantum walk framework [SICOMP\u2711]
Connecting the dots: a multi-pivot approach to data exploration
The purpose of data browsers is to help users identify and query data effectively without being overwhelmed by large complex graphs of data. A proposed solution to identify and query data in graph-based datasets is Pivoting (or set-oriented browsing), a many-to-many graph browsing technique that allows users to navigate the graph by starting from a set of instances followed by navigation through common links. Relying solely on navigation, however, makes it difficult for users to find paths or even see if the element of interest is in the graph when the points of interest may be many vertices apart. Further challenges include finding paths which require combinations of forward and backward links in order to make the necessary connections which further adds to the complexity of pivoting. In order to mitigate the effects of these problems and enhance the strengths of pivoting we present a multi-pivot approach which we embodied in tool called Visor. Visor allows users to explore from multiple points in the graph, helping users connect key points of interest in the graph on the conceptual level, visually occluding the remainder parts of the graph, thus helping create a road-map for navigation. We carried out an user study to demonstrate the viability of our approach
Social welfare and profit maximization from revealed preferences
Consider the seller's problem of finding optimal prices for her
(divisible) goods when faced with a set of consumers, given that she can
only observe their purchased bundles at posted prices, i.e., revealed
preferences. We study both social welfare and profit maximization with revealed
preferences. Although social welfare maximization is a seemingly non-convex
optimization problem in prices, we show that (i) it can be reduced to a dual
convex optimization problem in prices, and (ii) the revealed preferences can be
interpreted as supergradients of the concave conjugate of valuation, with which
subgradients of the dual function can be computed. We thereby obtain a simple
subgradient-based algorithm for strongly concave valuations and convex cost,
with query complexity , where is the additive
difference between the social welfare induced by our algorithm and the optimum
social welfare. We also study social welfare maximization under the online
setting, specifically the random permutation model, where consumers arrive
one-by-one in a random order. For the case where consumer valuations can be
arbitrary continuous functions, we propose a price posting mechanism that
achieves an expected social welfare up to an additive factor of
from the maximum social welfare. Finally, for profit maximization (which may be
non-convex in simple cases), we give nearly matching upper and lower bounds on
the query complexity for separable valuations and cost (i.e., each good can be
treated independently)
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