This paper studies the complexity of solving two classes of non-cooperative games in a distributed manner in which the players communicate with a set of system nodes over noisy communication channels. The complexity of solving each game class is defined as the minimum number of iterations required to find a Nash equilibrium (NE) of any game in that class with @math accuracy. First, we consider the class @math of all @math -player non-cooperative games with a continuous action space that admit at least one NE. Using information-theoretic inequalities, we derive a lower bound on the complexity of solving @math that depends on the Kolmogorov @math -capacity of the constraint set and the total capacity of the communication channels. We also derive a lower bound on the complexity of solving games in @math which depends on the volume and surface area of the constraint set. We next consider the class of all @math -player non-cooperative games with at least one NE such that the players' utility functions satisfy a certain (differential) constraint. We derive lower bounds on the complexity of solving this game class under both Gaussian and non-Gaussian noise models. Our result in the non-Gaussian case is derived by establishing a connection between the Kullback-Leibler distance and Fisher information.
The paper @cite_16 considered a model in which the algorithm observes noisy versions of the oracle's response and established lower bounds on the complexity of convex optimization problems under first order as well as gradient-only oracles. In @cite_13 , complexity lower bounds were obtained for convex optimization problems with a stochastic zero-order oracle. The paper @cite_14 studied the complexity of convex optimization problems under a zero-order stochastic oracle in which the optimization algorithm submits two queries at each iteration and the oracle responds to both queries. These results were extended to the case in which the algorithm makes queries about multiple points at each iteration in @cite_8 . In @cite_3 , the complexity of convex optimization problems was studied under an erroneous oracle model wherein the oracle's responses to queries are subject to absolute relative errors.
@cite_14 @cite_8 @cite_3 @cite_16 @cite_13
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We give novel algorithms for stochastic strongly-convex optimization in the gradient oracle model which return a O(1 T)-approximate solution after T iterations. The first algorithm is deterministic, and achieves this rate via gradient updates and historical averaging. The second algorithm is randomized, and is based on pure gradient steps with a random step size. his rate of convergence is optimal in the gradient oracle model. This improves upon the previously known best rate of O(log(T T), which was obtained by applying an online strongly-convex optimization algorithm with regret O(log(T)) to the batch setting. We complement this result by proving that any algorithm has expected regret of Ω(log(T)) in the online stochastic strongly-convex optimization setting. This shows that any online-to-batch conversion is inherently suboptimal for stochastic strongly-convex optimization. This is the first formal evidence that online convex optimization is strictly more difficult than batch stochastic convex optimization. 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 @math , as well as for the @math -ball for @math (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. We study the intrinsic limitations of sequential convex optimization through the lens of feedback information theory. In the oracle model of optimization, an algorithm queries an oracle for noisy information about the unknown objective function and the goal is to (approximately) minimize every function in a given class using as few queries as possible. We show that, in order for a function to be optimized, the algorithm must be able to accumulate enough information about the objective. This, in turn, puts limits on the speed of optimization under specific assumptions on the oracle and the type of feedback. Our techniques are akin to the ones used in statistical literature to obtain minimax lower bounds on the risks of estimation procedures; the notable difference is that, unlike in the case of i.i.d. data, a sequential optimization algorithm can gather observations in a controlled manner, so that the amount of information at each step is allowed to change in time. In particular, we show that optimization algorithms often obey the law of diminishing returns: the signal-to-noise ratio drops as the optimization algorithm approaches the optimum. To underscore the generality of the tools, we use our approach to derive fundamental lower bounds for a certain active learning problem. Overall, the present work connects the intuitive notions of “information” in optimization, experimental design, estimation, and active learning to the quantitative notion of Shannon information. We consider the problem of optimizing convex and concave functions with access to an erroneous zeroth-order oracle. In particular, for a given function x → f (x) we consider optimization when one is given access to absolute error oracles that return values in [f (x) - ∊, f (x) + ∊] or relative error oracles that return value in [(1 - ∊)f (x), (1 + ∊)f (x)], for some ∊ > 0. We show stark information theoretic impossibility results for minimizing convex functions and maximizing concave functions over polytopes in this model.