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 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 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. 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. Relative to the large literature on upper bounds on complexity of convex optimization, lesser attention has been paid to the fundamental hardn4516420ess of these problems. Given the extensive use of convex optimization in machine learning and statistics, gaining an understanding of these complexity-theoretic issues is important. In this paper, we study the complexity of stochastic convex optimization in an oracle model of computation. We introduce a new notion of discrepancy between functions, and use it to reduce problems of stochastic convex optimization to statistical parameter estimation, which can be lower bounded using information-theoretic methods. Using this approach, we improve upon known results and obtain tight minimax complexity estimates for various function classes. In this paper, we present a new conditional gradient type method for convex optimization by calling a linear optimization ( @math ) oracle to minimize a series of linear functions over the feasible set. Different from the classic conditional gradient method, the conditional gradient sliding (CGS) algorithm developed herein can skip the computation of gradients from time to time and, as a result, can achieve the optimal complexity bounds in terms of not only the number of calls to the @math oracle but also the number of gradient evaluations. More specifically, we show that the CGS method requires @math and @math gradient evaluations, respectively, for solving smooth and strongly convex problems, while still maintaining the optimal @math bound on the number of calls to the @math oracle. We also develop variants of the CGS method which can achieve the optimal complexity bounds for solving stochastic optimization problems and an important class... This paper revisits information complexity of black-box convex optimization, first studied in the seminal work of Nemirovski and Yudin, from the perspective of feedback information theory. These days, large-scale convex programming arises in a variety of applications, and it is important to refine our understanding of its fundamental limitations. The goal of black-box convex optimization is to minimize an unknown convex objective function from a given class over a compact, convex domain using an iterative scheme that generates approximate solutions by querying an oracle for local information about the function being optimized. The information complexity of a given problem class is defined as the smallest number of queries needed to minimize every function in the class to some desired accuracy. We present a simple information-theoretic approach that not only recovers many of the results of Nemirovski and Yudin, but also gives some new bounds pertaining to optimal rates at which iterative convex optimization schemes approach the solution. As a bonus, we give a particularly simple derivation of the minimax lower bound for a certain active learning problem on the unit interval. We prove lower bounds on the complexity of finding @math -stationary points (points @math such that @math ) of smooth, high-dimensional, and potentially non-convex functions @math . We consider oracle-based complexity measures, where an algorithm is given access to the value and all derivatives of @math at a query point @math . We show that for any (potentially randomized) algorithm @math , there exists a function @math with Lipschitz @math th order derivatives such that @math requires at least @math queries to find an @math -stationary point. Our lower bounds are sharp to within constants, and they show that gradient descent, cubic-regularized Newton's method, and generalized @math th order regularization are worst-case optimal within their natural function classes. 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 provide tight upper and lower bounds on the complexity of minimizing the average of m convex functions using gradient and prox oracles of the component functions. We show a significant gap between the complexity of deterministic vs randomized optimization. For smooth functions, we show that accelerated gradient descent (AGD) and an accelerated variant of SVRG are optimal in the deterministic and randomized settings respectively, and that a gradient oracle is sufficient for the optimal rate. For non-smooth functions, having access to prox oracles reduces the complexity and we present optimal methods based on smoothing that improve over methods using just gradient accesses. We provide tight upper and lower bounds on the complexity of minimizing the average of @math convex functions using gradient and prox oracles of the component functions. We show a significant gap between the complexity of deterministic vs randomized optimization. For smooth functions, we show that accelerated gradient descent (AGD) and an accelerated variant of SVRG are optimal in the deterministic and randomized settings respectively, and that a gradient oracle is sufficient for the optimal rate. For non-smooth functions, having access to prox oracles reduces the complexity and we present optimal methods based on smoothing that improve over methods using just gradient accesses. Despite a large literature on upper bounds on complexity of convex optimization, relatively less attention has been paid to the fundamental hardness of these problems. Given the extensive use of convex optimization in machine learning and statistics, gaining a understanding of these complexity-theoretic issues is important. In this paper, we study the complexity of stochastic convex optimization in an oracle model of computation. We improve upon known results and obtain tight minimax complexity estimates for various function classes. We also discuss implications of these results for the understanding the inherent complexity of large-scale learning and estimation problems. We propose a new stochastic first-order algorithm for solving sparse regression problems. In each iteration, our algorithm utilizes a stochastic oracle of the subgradient of the objective function. Our algorithm is based on a stochastic version of the estimate sequence technique introduced by Nesterov (Introductory lectures on convex optimization: a basic course, Kluwer, Amsterdam, 2003). The convergence rate of our algorithm depends continuously on the noise level of the gradient. In particular, in the limiting case of noiseless gradient, the convergence rate of our algorithm is the same as that of optimal deterministic gradient algorithms. We also establish some large deviation properties of our algorithm. Unlike existing stochastic gradient methods with optimal convergence rates, our algorithm has the advantage of readily enforcing sparsity at all iterations, which is a critical property for applications of sparse regressions. This paper employs a powerful argument, called an algorithmic argument, to prove lower bounds of the quantum query complexity of a multiple-block ordered search problem, which is a natural generalization of the ordered search problem. Apart from much studied polynomial and adversary methods for quantum query complexity lower bounds, our argument shows that the multiple-block ordered search needs a large number of nonadaptive oracle queries on a black-box model of quantum computation that is also supplemented with advice. Our argument is also applied to the notions of computational complexity theory: quantum truth-table reducibility and quantum truth-table autoreducibility. In this paper, we initiate a systematic investigation of differentially private algorithms for convex empirical risk minimization. Various instantiations of this problem have been studied before. We provide new algorithms and matching lower bounds for private ERM assuming only that each data point's contribution to the loss function is Lipschitz bounded and that the domain of optimization is bounded. We provide a separate set of algorithms and matching lower bounds for the setting in which the loss functions are known to also be strongly convex. Our algorithms run in polynomial time, and in some cases even match the optimal non-private running time (as measured by oracle complexity). We give separate algorithms (and lower bounds) for @math - and @math -differential privacy; perhaps surprisingly, the techniques used for designing optimal algorithms in the two cases are completely different. Our lower bounds apply even to very simple, smooth function families, such as linear and quadratic functions. This implies that algorithms from previous work can be used to obtain optimal error rates, under the additional assumption that the contributions of each data point to the loss function is smooth. We show that simple approaches to smoothing arbitrary loss functions (in order to apply previous techniques) do not yield optimal error rates. In particular, optimal algorithms were not previously known for problems such as training support vector machines and the high-dimensional median. Consider the convex optimization problem min x ƒ (g(x)) where both ƒ and g are unknown but can be estimated through sampling. We consider the stochastic compositional gradient descent method (SCGD) that updates based on random function and subgradient evaluations, which are generated by a conditional sampling oracle. We focus on the case where samples are corrupted with Markov noise. Under certain diminishing stepsize assumptions, we prove that the iterate of SCGD converges almost surely to an optimal solution if such a solution exists. Under specific constant stepsize assumptions, we obtain finite-sample error bounds for the averaged iterates of the algorithm. We illustrate an application to online value evaluation in dynamic programming. Interesting theoretical associations have been established by recent papers between the fields of active learning and stochastic convex optimization due to the common role of feedback in sequential querying mechanisms. In this paper, we continue this thread in two parts by exploiting these relations for the first time to yield novel algorithms in both fields, further motivating the study of their intersection. First, inspired by a recent optimization algorithm that was adaptive to unknown uniform convexity parameters, we present a new active learning algorithm for one-dimensional thresholds that can yield minimax rates by adapting to unknown noise parameters. Next, we show that one can perform @math -dimensional stochastic minimization of smooth uniformly convex functions when only granted oracle access to noisy gradient signs along any coordinate instead of real-valued gradients, by using a simple randomized coordinate descent procedure where each line search can be solved by @math -dimensional active learning, provably achieving the same error convergence rate as having the entire real-valued gradient. Combining these two parts yields an algorithm that solves stochastic convex optimization of uniformly convex and smooth functions using only noisy gradient signs by repeatedly performing active learning, achieves optimal rates and is adaptive to all unknown convexity and smoothness parameters. Convex empirical risk minimization is a basic tool in machine learning and statistics. We provide new algorithms and matching lower bounds for differentially private convex empirical risk minimization assuming only that each data point's contribution to the loss function is Lipschitz and that the domain of optimization is bounded. We provide a separate set of algorithms and matching lower bounds for the setting in which the loss functions are known to also be strongly convex.Our algorithms run in polynomial time, and in some cases even match the optimal non-private running time (as measured by oracle complexity). We give separate algorithms (and lower bounds) for (aepsi;, 0)- and (aepsi;,a#x03B4;)-differential privacy, perhaps surprisingly, the techniques used for designing optimal algorithms in the two cases are completely different. Our lower bounds apply even to very simple, smooth function families, such as linear and quadratic functions. This implies that algorithms from previous work can be used to obtain optimal error rates, under the additional assumption that the contributions of each data point to the loss function is smooth. We show that simple approaches to smoothing arbitrary loss functions (in order to apply previous techniques) do not yield optimal error rates. In particular, optimal algorithms were not previously known for problems such as training support vector machines and the high-dimensional median. The emergence of real life graphs with billions of nodes poses significant challenges for managing and querying these graphs. One of the fundamental queries submitted to graphs is the shortest distance query. Online BFS (breadth-first search) and offline pre-computing pairwise shortest distances are prohibitive in time or space complexity for billion-node graphs. In this paper, we study the feasibility of building distance oracles for billion-node graphs. A distance oracle provides approximate answers to shortest distance queries by using a pre-computed data structure for the graph. Sketch-based distance oracles are good candidates because they assign each vertex a sketch of bounded size, which means they have linear space complexity. However, state-of-the-art sketch-based distance oracles lack efficiency or accuracy when dealing with big graphs. In this paper, we address the scalability and accuracy issues by focusing on optimizing the three key factors that affect the performance of distance oracles: landmark selection, distributed BFS, and answer generation. We conduct extensive experiments on both real networks and synthetic networks to show that we can build distance oracles of affordable cost and efficiently answer shortest distance queries even for billion-node graphs. Consider the stochastic composition optimization problem where the objective is a composition of two expected-value functions. We propose a new stochastic first-order method, namely the accelerated stochastic compositional proximal gradient (ASC-PG) method, which updates based on queries to the sampling oracle using two different timescales. The ASC-PG is the first proximal gradient method for the stochastic composition problem that can deal with nonsmooth regularization penalty. We show that the ASC-PG exhibits faster convergence than the best known algorithms, and that it achieves the optimal sample-error complexity in several important special cases. We further demonstrate the application of ASC-PG to reinforcement learning and conduct numerical experiments. We propose a stochastic variance-reduced cubic regularized Newton method for non-convex optimization. At the core of our algorithm is a novel semi-stochastic gradient along with a semi-stochastic Hessian, which are specifically designed for cubic regularization method. We show that our algorithm is guaranteed to converge to an @math -approximately local minimum within @math second-order oracle calls, which outperforms the state-of-the-art cubic regularization algorithms including subsampled cubic regularization. Our work also sheds light on the application of variance reduction technique to high-order non-convex optimization methods. Thorough experiments on various non-convex optimization problems support our theory.