Towards Characterizing the First-order Query Complexity of Learning (Approximate) Nash Equilibria in Zero-sum Matrix Games

• URL: http://arxiv.org/abs/2304.12768v2
• Date: Thu, 2 Nov 2023 09:02:30 GMT
• Title: Towards Characterizing the First-order Query Complexity of Learning (Approximate) Nash Equilibria in Zero-sum Matrix Games
• Authors: H\'edi Hadiji (L2S), Sarah Sachs (UvA), Tim van Erven (UvA), Wouter M. Koolen (CWI)
• Abstract summary: We show that exact equilibria can be computed efficiently from $O(fracln Kepsilon)$ instead of $O(fracln Kepsilon2)$ queries. We introduce a new technique for lower bounds, which allows us to obtain lower bounds of order $tildeOmega(frac1Kepsilon)$ for any $epsilon leq 1 / (cK4)$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In the first-order query model for zero-sum $K\times K$ matrix games, players observe the expected pay-offs for all their possible actions under the randomized action played by their opponent. This classical model has received renewed interest after the discovery by Rakhlin and Sridharan that $\epsilon$-approximate Nash equilibria can be computed efficiently from $O(\frac{\ln K}{\epsilon})$ instead of $O(\frac{\ln K}{\epsilon^2})$ queries. Surprisingly, the optimal number of such queries, as a function of both $\epsilon$ and $K$, is not known. We make progress on this question on two fronts. First, we fully characterise the query complexity of learning exact equilibria ($\epsilon=0$), by showing that they require a number of queries that is linear in $K$, which means that it is essentially as hard as querying the whole matrix, which can also be done with $K$ queries. Second, for $\epsilon > 0$, the current query complexity upper bound stands at $O(\min(\frac{\ln(K)}{\epsilon} , K))$. We argue that, unfortunately, obtaining a matching lower bound is not possible with existing techniques: we prove that no lower bound can be derived by constructing hard matrices whose entries take values in a known countable set, because such matrices can be fully identified by a single query. This rules out, for instance, reducing to an optimization problem over the hypercube by encoding it as a binary payoff matrix. We then introduce a new technique for lower bounds, which allows us to obtain lower bounds of order $\tilde\Omega(\log(\frac{1}{K\epsilon})$ for any $\epsilon \leq 1 / (cK^4)$, where $c$ is a constant independent of $K$. We further discuss possible future directions to improve on our techniques in order to close the gap with the upper bounds.

Related papers

• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms [50.15964512954274]
  We study the problem of residual error estimation for matrix and vector norms using a linear sketch. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work. We also show an $Omega(k2/pn1-2/p)$ lower bound for the sparse recovery problem, which is tight up to a $mathrmpoly(log n)$ factor.
  arXiv  Detail & Related papers  (2024-08-16T02:33:07Z)
• Quantum and classical query complexities of functions of matrices [0.0]
  We show that for any continuous function $f(x):[-1,1]rightarrow [-1,1]$, the quantum query complexity of computing $brai f(A) ketjpm varepsilon/4$ is lower bounded by $Omega(widetildedeg_varepsilon(f))$.
  arXiv  Detail & Related papers  (2023-11-13T00:45:41Z)
• Krylov Methods are (nearly) Optimal for Low-Rank Approximation [8.017116107657206]
  We show that any algorithm requires $Omegaleft(log(n)/varepsilon1/2right)$ matrix-vector products, exactly matching the upper bound obtained by Krylov methods. Our lower bound addresses Open Question 1WooWoo14, providing evidence for the lack of progress on algorithms for Spectral LRA.
  arXiv  Detail & Related papers  (2023-04-06T16:15:19Z)
• Optimal Query Complexities for Dynamic Trace Estimation [59.032228008383484]
  We consider the problem of minimizing the number of matrix-vector queries needed for accurate trace estimation in the dynamic setting where our underlying matrix is changing slowly. We provide a novel binary tree summation procedure that simultaneously estimates all $m$ traces up to $epsilon$ error with $delta$ failure probability. Our lower bounds (1) give the first tight bounds for Hutchinson's estimator in the matrix-vector product model with Frobenius norm error even in the static setting, and (2) are the first unconditional lower bounds for dynamic trace estimation.
  arXiv  Detail & Related papers  (2022-09-30T04:15:44Z)
• On the Complexity of Dynamic Submodular Maximization [15.406670088500087]
  We show that any algorithm that maintains a $(0.5+epsilon)$-approximate solution under a cardinality constraint, for any constant $epsilon>0$, must have an amortized query complexity that is $mathitpolynomial$ in $n$. This is in sharp contrast with recent dynamic algorithms of [LMNF+20, Mon20] that achieve $(0.5-epsilon)$-approximation with a $mathsfpolylog(n)$ amortized query complexity.
  arXiv  Detail & Related papers  (2021-11-05T00:04:29Z)
• Semi-supervised Active Regression [21.51757844385258]
  This paper studies the use of partially labelled, potentially biased data for learning tasks. The learner has access to a dataset $X in mathbbRd | X min_beta in mathbbRd | X beta - Y |2 end2 equation while making few additional label queries.
  arXiv  Detail & Related papers  (2021-06-12T03:28:43Z)
• Learning a Latent Simplex in Input-Sparsity Time [58.30321592603066]
  We consider the problem of learning a latent $k$-vertex simplex $KsubsetmathbbRdtimes n$, given access to $AinmathbbRdtimes n$. We show that the dependence on $k$ in the running time is unnecessary given a natural assumption about the mass of the top $k$ singular values of $A$.
  arXiv  Detail & Related papers  (2021-05-17T16:40:48Z)
• Second-Order Information in Non-Convex Stochastic Optimization: Power and Limitations [54.42518331209581]
  We find an algorithm which finds. epsilon$-approximate stationary point (with $|nabla F(x)|le epsilon$) using. $(epsilon,gamma)$surimate random random points. Our lower bounds here are novel even in the noiseless case.
  arXiv  Detail & Related papers  (2020-06-24T04:41:43Z)
• Average Case Column Subset Selection for Entrywise $\ell_1$-Norm Loss [76.02734481158458]
  It is known that in the worst case, to obtain a good rank-$k$ approximation to a matrix, one needs an arbitrarily large $nOmega(1)$ number of columns. We show that under certain minimal and realistic distributional settings, it is possible to obtain a $(k/epsilon)$-approximation with a nearly linear running time and poly$(k/epsilon)+O(klog n)$ columns. This is the first algorithm of any kind for achieving a $(k/epsilon)$-approximation for entrywise
  arXiv  Detail & Related papers  (2020-04-16T22:57:06Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.