Computational complexity of real functions

AbstractRecursive analysis, the theory of computation of functions on real numbers, has been studied from various aspects. We investigate the computational complexity of real functions using the methods of recursive function theory. Partial recursive real functions are defined and their domains are characterized as the recursively open sets. We define the time complexity of recursive real continuous functions and show that the time complexity and the modulus of uniform continuity of a function are closely related. We study the complexity of the roots and the differentiability of polynomial time computable real functions. In particular, a polynomial time computable real function may have a root of arbitrarily high complexity and may be nowhere differentiable. The concepts of the space complexity and nondeterministic computation are used to study the complexity of the integrals and the maximum values of real functions. These problems are shown to be related to the “P=?NP” and the “P=?PSPACE” questions

MERMAIDE: Learning to Align Learners using Model-Based Meta-Learning

We study how a principal can efficiently and effectively intervene on the rewards of a previously unseen learning agent in order to induce desirable outcomes. This is relevant to many real-world settings like auctions or taxation, where the principal may not know the learning behavior nor the rewards of real people. Moreover, the principal should be few-shot adaptable and minimize the number of interventions, because interventions are often costly. We introduce MERMAIDE, a model-based meta-learning framework to train a principal that can quickly adapt to out-of-distribution agents with different learning strategies and reward functions. We validate this approach step-by-step. First, in a Stackelberg setting with a best-response agent, we show that meta-learning enables quick convergence to the theoretically known Stackelberg equilibrium at test time, although noisy observations severely increase the sample complexity. We then show that our model-based meta-learning approach is cost-effective in intervening on bandit agents with unseen explore-exploit strategies. Finally, we outperform baselines that use either meta-learning or agent behavior modeling, in both $0$-shot and $K=1$-shot settings with partial agent information

Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime

The synthesis of classical Computational Complexity Theory with Recursive Analysis provides a quantitative foundation to reliable numerics. Here the operators of maximization, integration, and solving ordinary differential equations are known to map (even high-order differentiable) polynomial-time computable functions to instances which are `hard' for classical complexity classes NP, #P, and CH; but, restricted to analytic functions, map polynomial-time computable ones to polynomial-time computable ones -- non-uniformly! We investigate the uniform parameterized complexity of the above operators in the setting of Weihrauch's TTE and its second-order extension due to Kawamura&Cook (2010). That is, we explore which (both continuous and discrete, first and second order) information and parameters on some given f is sufficient to obtain similar data on Max(f) and int(f); and within what running time, in terms of these parameters and the guaranteed output precision 2^(-n). It turns out that Gevrey's hierarchy of functions climbing from analytic to smooth corresponds to the computational complexity of maximization growing from polytime to NP-hard. Proof techniques involve mainly the Theory of (discrete) Computation, Hard Analysis, and Information-Based Complexity