In this paper we investigate the computational complexity of solving ordinary differential equations (ODEs) y ' = p ( y ) over unbounded time domains, where p is a vector of polynomials. Contrarily to the bounded (compact) time case, this problem has not been well-studied, apparently due to the "intuition" that it can always be reduced to the bounded case by using rescaling techniques. However, as we show in this paper, rescaling techniques do not seem to provide meaningful insights on the complexity of this problem, since the use of such techniques introduces a dependence on parameters which are hard to compute.We present algorithms which numerically solve these ODEs over unbounded time domains. These algorithms have guaranteed accuracy, i.e. given some arbitrarily large time t and error bound e as input, they will output a value y ? which satisfies ? y ( t ) - y ? ? ? e . We analyze the complexity of these algorithms and show that they compute y ? in time polynomial in several quantities including the time t, the accuracy of the output e and the length of the curve y from 0 to t, assuming it exists until time t. We consider both algebraic complexity and bit complexity.
The reason to focus on the Lipschitz condition for @math seems to be fundamental when studying the computational complexity of . Indeed it is well-known (see e.g. [Theorem 7.3] Ko91 ) that if @math is not Lipschitz, then the solution of can have arbitrarily high complexity, even if @math is assumed to be polynomial-time (precision-)computable and has a unique solution. The Lipschitz condition plays an instrumental role in the complexity because it is used to derive the number of steps of the algorithm, for example in Picard-Lindel " o f theorem it is used to bound the number of iterations. It was recently shown @cite_10 , following an open problem from Ko [Section 7.2] Ko91 , that if @math is Lipschitz and polynomial-time (precision-)computable, then the solution of over @math can still be PSPACE-complete (but it was already known not to be of higher complexity @cite_8 ).
@cite_10 @cite_8
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We provide sharp worst-case evaluation complexity bounds for nonconvex minimization problems with general inexpensive constraints, i.e. problems where the cost of evaluating enforcing of the (possibly nonconvex or even disconnected) constraints, if any, is negligible compared to that of evaluating the objective function. These bounds unify, extend or improve all known upper and lower complexity bounds for unconstrained and convexly-constrained problems. It is shown that, given an accuracy level @math , a degree of highest available Lipschitz continuous derivatives @math and a desired optimality order @math between one and @math , a conceptual regularization algorithm requires no more than @math evaluations of the objective function and its derivatives to compute a suitably approximate @math -th order minimizer. With an appropriate choice of the regularization, a similar result also holds if the @math -th derivative is merely H "older rather than Lipschitz continuous. We provide an example that shows that the above complexity bound is sharp for unconstrained and a wide class of constrained problems, we also give reasons for the optimality of regularization methods from a worst-case complexity point of view, within a large class of algorithms that use the same derivative information. In answer to Ko's question raised in 1983, we show that an initial value problem given by a polynomial-time computable, Lipschitz continuous function can have a polynomial-space complete solution. The key insight is simple: the Lipschitz condition means that the feedback in the differential equation is weak. We define a class of polynomial-space computation tableaux with equally restricted feedback, and show that they are still polynomial-space complete. The same technique also settles Ko's two later questions on Volterra integral equations. We introduce the general and powerful scheme of predicting information re-use in optimization algorithms. This allows us to devise a computationally efficient algorithm for bandit convex optimization with new state-of-the-art guarantees for both Lipschitz loss functions and loss functions with Lipschitz gradients. This is the first algorithm admitting both a polynomial time complexity and a regret that is polynomial in the dimension of the action space that improves upon the original regret bound for Lipschitz loss functions, achieving a regret of @math . Our algorithm further improves upon the best existing polynomial-in-dimension bound (both computationally and in terms of regret) for loss functions with Lipschitz gradients, achieving a regret of @math . In this paper, we study games with continuous action spaces and non-linear payoff functions. Our key insight is that Lipschitz continuity of the payoff function allows us to provide algorithms for finding approximate equilibria in these games. We begin by studying Lipschitz games, which encompass, for example, all concave games with Lipschitz continuous payoff functions. We provide an efficient algorithm for computing approximate equilibria in these games. Then we turn our attention to penalty games, which encompass biased games and games in which players take risk into account. Here we show that if the penalty function is Lipschitz continuous, then we can provide a quasi-polynomial time approximation scheme. Finally, we study distance biased games, where we present simple strongly polynomial time algorithms for finding best responses in @math , @math , and @math biased games, and then use these algorithms to provide strongly polynomial algorithms that find @math , @math , and @math approximations for these norms, respectively. Sampling logconcave functions arising in statistics and machine learning has been a subject of intensive study. Recent developments include analyses for Langevin dynamics and Hamiltonian Monte Carlo (HMC). While both approaches have dimension-independent bounds for the underlying @math processes under sufficiently strong smoothness conditions, the resulting discrete algorithms have complexity and number of function evaluations growing with the dimension. Motivated by this problem, in this paper, we give a general algorithm for solving multivariate ordinary differential equations whose solution is close to the span of a known basis of functions (e.g., polynomials or piecewise polynomials). The resulting algorithm has polylogarithmic depth and essentially tight runtime - it is nearly linear in the size of the representation of the solution. We apply this to the sampling problem to obtain a nearly linear implementation of HMC for a broad class of smooth, strongly logconcave densities, with the number of iterations (parallel depth) and gradient evaluations being @math in the dimension (rather than polynomial as in previous work). This class includes the widely-used loss function for logistic regression with incoherent weight matrices and has been subject of much study recently. We also give a faster algorithm with @math for the more general and standard class of strongly convex functions with Lipschitz gradient. These results are based on (1) an improved contraction bound for the exact HMC process and (2) logarithmic bounds on the degree of polynomials that approximate solutions of the differential equations arising in implementing HMC. Section 1 of the paper contains a general discussion of robustness. In Section 2 the influence function of the Hampel-Rousseeuw least median of squares estimator is derived. Linearly invariant weak metrics are constructed in Section 3. It is shown in Section 4 that @math -estimators satisfy an exact Holder condition of order 1 2 at models with normal errors. In Section 5 the breakdown points of the Hampel-Krasker dispersion and regression functionals are shown to be 0. The exact breakdown point of the Krasker-Welsch dispersion functional is obtained as well as bounds for the corresponding regression functional. Section 6 contains the construction of a linearly equivariant, high breakdown and locally Lipschitz dispersion functional for any design distribution. In Section 7 it is shown that there is no inherent contradiction between efficiency and a high breakdown point. Section 8 contains a linearly equivariant, high breakdown regression functional which is Lipschitz continuous at models with normal errors. In this paper we prove that computing the solution of an initial-value problem y = p(y) with initial condition y(t0) = y0 ∈ Rd at time t0 + T with precision 2−μ where p is a vector of polynomials can be done in time polynomial in the value of T, μ and Y = [equation]. Contrary to existing results, our algorithm works over any bounded or unbounded domain. Furthermore, we do not assume any Lipschitz condition on the initial-value problem. In a multi-armed bandit problem, an online algorithm chooses from a set of strategies in a sequence of @math trials so as to maximize the total payoff of the chosen strategies. While the performance of bandit algorithms with a small finite strategy set is quite well understood, bandit problems with large strategy sets are still a topic of very active investigation, motivated by practical applications such as online auctions and web advertisement. The goal of such research is to identify broad and natural classes of strategy sets and payoff functions which enable the design of efficient solutions. In this work we study a very general setting for the multi-armed bandit problem in which the strategies form a metric space, and the payoff function satisfies a Lipschitz condition with respect to the metric. We refer to this problem as the "Lipschitz MAB problem". We present a complete solution for the multi-armed problem in this setting. That is, for every metric space (L,X) we define an isometry invariant Max Min COV(X) which bounds from below the performance of Lipschitz MAB algorithms for @math , and we present an algorithm which comes arbitrarily close to meeting this bound. Furthermore, our technique gives even better results for benign payoff functions. We propose a new distributed algorithm based on alternating direction method of multipliers (ADMM) to minimize sum of locally known convex functions using communication over a network. This optimization problem emerges in many applications in distributed machine learning and statistical estimation. Our algorithm allows for a general choice of the communication weight matrix, which is used to combine the iterates at different nodes. We show that when functions are convex, both the objective function values and the feasibility violation converge with rate @math , where @math is the number of iterations. We then show that when functions are strongly convex and have Lipschitz continuous gradients, the sequence generated by our algorithm converges linearly to the optimal solution. In particular, an @math -optimal solution can be computed with @math iterations, where @math is the condition number of the problem. Our analysis highlights the effect of network and communication weights on the convergence rate through degrees of the nodes, the smallest nonzero eigenvalue, and operator norm of the communication matrix. We present an accelerated gradient method for nonconvex optimization problems with Lipschitz continuous first and second derivatives. In a time @math , the method finds an @math -stationary point, meaning a point @math such that @math . The method improves upon the @math complexity of gradient descent and provides the additional second-order guarantee that @math for the computed @math . Furthermore, our method is Hessian free, i.e., it only requires gradient computations, and is therefore suitable for large-scale applications. We present an accelerated gradient method for non-convex optimization problems with Lipschitz continuous first and second derivatives. The method requires time @math to find an @math -stationary point, meaning a point @math such that @math . The method improves upon the @math complexity of gradient descent and provides the additional second-order guarantee that @math for the computed @math . Furthermore, our method is Hessian free, i.e. it only requires gradient computations, and is therefore suitable for large scale applications. In this work we compute lower Lipschitz bounds of @math pooling operators for @math as well as @math pooling operators preceded by half-rectification layers. These give sufficient conditions for the design of invertible neural network layers. Numerical experiments on MNIST and image patches confirm that pooling layers can be inverted with phase recovery algorithms. Moreover, the regularity of the inverse pooling, controlled by the lower Lipschitz constant, is empirically verified with a nearest neighbor regression. For certain subclasses of NP, @math P, or #P characterized by local constraints, it is known that if there exist any problems within that subclass that are not polynomial time computable, then all the problems in the subclass are NP-complete, @math P-complete, or #P-complete. Such dichotomy results have been proved for characterizations such as constraint satisfaction problems and directed and undirected graph homomorphism problems, often with additional restrictions. Here we give a dichotomy result for the more expressive framework of Holant problems. For example, these additionally allow for the expression of matching problems, which have had pivotal roles in the development of complexity theory. As our main result we prove the dichotomy theorem that, for the class @math P, every set of symmetric Holant signatures of any arities that is not polynomial time computable is @math P-complete. The result exploits some special properties of the class @math P and characterizes four distinct tractable ... In this work we analyze the sample complexity of classification by differentially private algorithms. Differential privacy is a strong and well-studied notion of privacy introduced by (2006) that ensures that the output of an algorithm leaks little information about the data point provided by any of the participating individuals. Sample complexity of private PAC and agnostic learning was studied in a number of prior works starting with (, 2008) but a number of basic questions still remain open, most notably whether learning with privacy requires more samples than learning without privacy. We show that the sample complexity of learning with (pure) differential privacy can be arbitrarily higher than the sample complexity of learning without the privacy constraint or the sample complexity of learning with approximate differential privacy. Our second contribution and the main tool is an equivalence between the sample complexity of (pure) differentially private learning of a concept class @math (or @math ) and the randomized one-way communication complexity of the evaluation problem for concepts from @math . Using this equivalence we prove the following bounds: 1. @math , where @math is the Littlestone's (1987) dimension characterizing the number of mistakes in the online-mistake-bound learning model. Known bounds on @math then imply that @math can be much higher than the VC-dimension of @math . 2. For any @math , there exists a class @math such that @math but @math . 3. For any @math , there exists a class @math such that the sample complexity of (pure) @math -differentially private PAC learning is @math but the sample complexity of the relaxed @math -differentially private PAC learning is @math . This resolves an open problem of (2013b). Let @math be an arbitrary polynomial of degree @math with @math non-zero integer coefficients of absolute value less than @math . In this paper, we answer the open question whether the real roots of @math can be computed with a number of arithmetic operations over the rational numbers that is polynomial in the input size of the sparse representation of @math . More precisely, we give a deterministic, complete, and certified algorithm that determines isolating intervals for all real roots of @math with @math many exact arithmetic operations over the rational numbers. When using approximate but certified arithmetic, the bit complexity of our algorithm is bounded by @math , where @math means that we ignore logarithmic. Hence, for sufficiently sparse polynomials (i.e. @math for a positive constant @math ), the bit complexity is @math . We also prove that the latter bound is optimal up to logarithmic factors. Verifying the robustness property of a general Rectified Linear Unit (ReLU) network is an NP-complete problem [Katz, Barrett, Dill, Julian and Kochenderfer CAV17]. Although finding the exact minimum adversarial distortion is hard, giving a certified lower bound of the minimum distortion is possible. Current available methods of computing such a bound are either time-consuming or delivering low quality bounds that are too loose to be useful. In this paper, we exploit the special structure of ReLU networks and provide two computationally efficient algorithms Fast-Lin and Fast-Lip that are able to certify non-trivial lower bounds of minimum distortions, by bounding the ReLU units with appropriate linear functions Fast-Lin, or by bounding the local Lipschitz constant Fast-Lip. Experiments show that (1) our proposed methods deliver bounds close to (the gap is 2-3X) exact minimum distortion found by Reluplex in small MNIST networks while our algorithms are more than 10,000 times faster; (2) our methods deliver similar quality of bounds (the gap is within 35 and usually around 10 ; sometimes our bounds are even better) for larger networks compared to the methods based on solving linear programming problems but our algorithms are 33-14,000 times faster; (3) our method is capable of solving large MNIST and CIFAR networks up to 7 layers with more than 10,000 neurons within tens of seconds on a single CPU core. In addition, we show that, in fact, there is no polynomial time algorithm that can approximately find the minimum @math adversarial distortion of a ReLU network with a @math approximation ratio unless @math = @math , where @math is the number of neurons in the network. We introduce a new approach to isolate the real roots of a square-free polynomial @math with real coefficients. It is assumed that each coefficient of @math can be approximated to any specified error bound. The presented method is exact, complete and deterministic. Due to its similarities to the Descartes method, we also consider it practical and easy to implement. Compared to previous approaches, our new method achieves a significantly better bit complexity. It is further shown that the hardness of isolating the real roots of @math is exclusively determined by the geometry of the roots and not by the complexity or the size of the coefficients. For the special case where @math has integer coefficients of maximal bitsize @math , our bound on the bit complexity writes as @math which improves the best bounds known for existing practical algorithms by a factor of @math . The crucial idea underlying the new approach is to run an approximate version of the Descartes method, where, in each subdivision step, we only consider approximations of the intermediate results to a certain precision. We give an upper bound on the maximal precision that is needed for isolating the roots of @math . For integer polynomials, this bound is by a factor @math lower than that of the precision needed when using exact arithmetic explaining the improved bound on the bit complexity. We give the first dimension-efficient algorithms for learning Rectified Linear Units (ReLUs), which are functions of the form @math with @math . Our algorithm works in the challenging Reliable Agnostic learning model of Kalai, Kanade, and Mansour (2009) where the learner is given access to a distribution @math on labeled examples but the labeling may be arbitrary. We construct a hypothesis that simultaneously minimizes the false-positive rate and the loss on inputs given positive labels by @math , for any convex, bounded, and Lipschitz loss function. The algorithm runs in polynomial-time (in @math ) with respect to any distribution on @math (the unit sphere in @math dimensions) and for any error parameter @math (this yields a PTAS for a question raised by F. Bach on the complexity of maximizing ReLUs). These results are in contrast to known efficient algorithms for reliably learning linear threshold functions, where @math must be @math and strong assumptions are required on the marginal distribution. We can compose our results to obtain the first set of efficient algorithms for learning constant-depth networks of ReLUs. Our techniques combine kernel methods and polynomial approximations with a "dual-loss" approach to convex programming. As a byproduct we obtain a number of applications including the first set of efficient algorithms for "convex piecewise-linear fitting" and the first efficient algorithms for noisy polynomial reconstruction of low-weight polynomials on the unit sphere. Given a simplicial complex with weights on its simplices, and a nontrivial cycle on it, we are interested in finding the cycle with minimal weight which is homologous to the given one. Assuming that the homology is defined with integer ( @math ) coefficients, we show the following (Theorem 5.2): For a finite simplicial complex @math of dimension greater than @math , the boundary matrix @math is totally unimodular if and only if @math is torsion-free for all pure subcomplexes @math in @math of dimensions @math and @math , respectively, where @math . Because of the total unimodularity of the boundary matrix, we can solve the optimization problem, which is inherently an integer programming problem, as a linear program and obtain an integer solution. Thus, the problem of finding optimal cycles in a given homology class can be solved in polynomial time. This result is surprising in the backdrop of a recent result which says that the problem is NP-hard under @math coefficients which, being a field, is in general easier to deal with. Our result implies, among other things, that one can compute in polynomial time an optimal @math -cycle in a given homology class for any triangulation of an orientable compact @math -manifold or for any finite simplicial complex embedded in @math . Our optimization approach can also be used for various related problems, such as finding an optimal chain homologous to a given one when these are not cycles. Our result can also be viewed as providing a topological characterization of total unimodularity. We study the problem of approximating the commuting-operator value of a two-player non-local game. It is well-known that it is @math -complete to decide whether the classical value of a non-local game is 1 or @math . Furthermore, as long as @math is small enough, this result does not depend on the gap @math . In contrast, a recent result of Fitzsimons, Ji, Vidick, and Yuen shows that the complexity of computing the quantum value grows without bound as the gap @math decreases. In this paper, we show that this also holds for the commuting-operator value of a game. Specifically, in the language of multi-prover interactive proofs, we show that the power of @math (proofs with two provers, one round, completeness probability @math , soundness probability @math , and commuting-operator strategies) can increase without bound as the gap @math gets arbitrarily small. Our results also extend naturally in two ways, to perfect zero-knowledge protocols, and to lower bounds on the complexity of computing the approximately-commuting value of a game. Thus we get lower bounds on the complexity class @math - @math of perfect zero-knowledge multi-prover proofs with approximately-commuting operator strategies, as the gap @math gets arbitrarily small. While we do not know any computable time upper bound on the class @math , a result of the first author and Vidick shows that for @math and @math , the class @math , with constant communication from the provers, is contained in @math . We give a lower bound of @math (ignoring constants inside the function) for this class, which is tight up to polynomial factors assuming the exponential time hypothesis.