Minimax Research Articles

On Minimax Detection of Gaussian Stochastic Sequences and Gaussian Stationary Signals

We consider the detection problem for Gaussian stochastic sequences (signals) with unknown covariance matrices in white Gaussian noise. For a given false alarm probability (1st-kind error probability), the quality of minimax detection is given by the best miss probability (2nd-kind error probability) exponent over a growing observation interval. The goal is finding the largest set of covariance matrices (composite hypothesis) such that its minimax testing can be replaced with testing a single particular covariance matrix (simple hypothesis) with no degradation of the detection exponent. We completely describe this maximal set of covariance matrices. We also consider some consequences concerning minimax detection of Gaussian stochastic signals against Gaussian white noise and detection of Gaussian stationary signals.

Path-Dependent Hamilton–Jacobi Equations: The Minimax Solutions Revised

Motivated by optimal control problems and differential games for functional differential equations of retarded type, the paper deals with a Cauchy problem for a path-dependent Hamilton--Jacobi equation with a right-end boundary condition. Minimax solutions of this problem are studied. The existence and uniqueness result is obtained under assumptions that are weaker than those considered earlier. In contrast to previous works, on the one hand, we do not require any properties concerning positive homogeneity of the Hamiltonian in the impulse variable, and on the other hand, we suppose that the Hamiltonian satisfies a Lipshitz continuity condition with respect to the path variable in the uniform (supremum) norm. The progress is related to the fact that a suitable Lyapunov--Krasovskii functional is built that allows to prove a comparison principle. This functional is in some sense equivalent to the square of the uniform norm of the path variable and, at the same time, it possesses appropriate smoothness properties. In addition, the paper provides non-local and infinitesimal criteria of minimax solutions, their stability with respect to perturbations of the Hamiltonian and the boundary functional, as well as consistency of the approach with the non-path-dependent case. Connection of the problem statement under consideration with some other possible statements (regarding the choice of path spaces and derivatives used) known in the theory of path-dependent Hamilton--Jacobi equations is discussed. Some remarks concerning viscosity solutions of the studied Cauchy problem are given.

Individualized Decision Making Under Partial Identification: ThreePerspectives, Two Optimality Results, and One Paradox

Unmeasured confounding is a threat to causal inference and gives rise to biased estimates. In this article, we consider the problem of individualized decision-making under partial identification. Firstly, we argue that when faced with unmeasured confounding, one should pursue individualized decision-making using partial identification in a comprehensive manner. We establish a formal link between individualized decision-making under partial identification and classical decision theory by considering a lower bound perspective of value/utility function. Secondly, building on this unified framework, we provide a novel minimax solution (i.e., a rule that minimizes the maximum regret for so-called opportunists) for individualized decision-making/policy assignment. Lastly, we provide an interesting paradox drawing on novel connections between two challenging domains, that is, individualized decision-making and unmeasured confounding. Although motivated by instrumental variable bounds, we emphasize that the general framework proposed in this article would in principle apply for a rich set of bounds that might be available under partial identification.

Generalized quantum process discrimination problems

We study a broad class of quantum process discrimination problems that can handle many optimization strategies such as the Bayes, Neyman-Pearson, and unambiguous strategies, where each process can consist of multiple time steps and can have an internal memory. Given a collection of candidate processes, our task is to find a discrimination strategy, which may be adaptive and/or entanglement-assisted, that maximizes a given objective function subject to given constraints. Our problem can be formulated as a convex problem. Its Lagrange dual problem with no duality gap and necessary and sufficient conditions for an optimal solution are derived. We also show that if a problem has a certain symmetry and at least one optimal solution exists, then there also exists an optimal solution with the same type of symmetry. A minimax strategy for a process discrimination problem is also discussed. As applications of our results, we provide some problems in which an adaptive strategy is not necessary for optimal discrimination. We also present an example of single-shot channel discrimination for which an analytical solution can be obtained.

THE HARSANYI-RAWLS DEBATE: POLITICAL PHILOSOPHY AS DECISION THEORY UNDER UNCERTAINTY

Social decisions are often made under great uncertainty - in situations where political principles, and even standard subjective expected utility, do not apply smoothly. In the first section, we argue that the core of this problem lies in decision theory itself - it is about how to act when we do not have an adequate representation of the context of the action and of its possible consequences. Thus, we distinguish two criteria to complement decision theory under ignorance - Laplace’s principle of insufficient reason and Wald’s maximin criterion. After that, we apply this analysis to political philosophy, by contrasting Harsanyi’s and Rawls’s theories of justice, respectively based on Laplace’s principle of insufficient reason and Wald’s maximin rule - and we end up highlighting the virtues of Rawls’s principle on practical grounds (it is intuitively attractive because of its computational simplicity, so providing a salient point for convergence) - and connect this argument to our moral intuitions and social norms requiring prudence in the case of decisions made for the sake of others.

Existence and limiting behavior of min-max solutions of the Ginzburg–Landau equations on compact manifolds

We use a natural two-parameter min-max construction to produce critical points of the Ginzburg–Landau functionals on a compact Riemannian manifold of dimension $\geq 2$. We investigate the limiting behavior of these critical points as $\varepsilon \to 0$, and show in particular that some of the energy concentrates on a nontrivial stationary, rectifiable $(n-2)$-varifold as $\varepsilon \to 0$, suggesting connections to the min-max construction of minimal $(n-2)$-submanifolds.

Fairness in Influence Maximization through Randomization

The influence maximization paradigm has been used by researchers in various fields in order to study how information spreads in social networks. While previously the attention was mostly on efficiency, more recently fairness issues have been taken into account in this scope. In the present paper, we propose to use randomization as a mean for achieving fairness. While this general idea is not new, it has not been applied in the area of information spread in networks. Similar to previous works like Fish et al. (WWW '19) and Tsang et al. (IJCAI '19), we study the maximin criterion for (group) fairness. By allowing randomized solutions, we introduce two different variants of this problem. While the original deterministic maximin problem has been shown to be inapproximable, interestingly, we show that both probabilistic variants permit approximation algorithms with a constant multiplicative factor of 1-1/e plus an additive arbitrarily small error that is due to the simulation of the information spread. For an experimental study, we provide implementations of our methods and compare the achieved fairness values to existing methods. Non-surprisingly, the ex-ante values, i.e., minimum expected value of an individual (or group) to obtain the information, of the computed probabilistic strategies are significantly larger than the (ex-post) fairness values of previous methods. This confirms that studying fairness via randomization is a worthwhile direction. More surprisingly, we observe that even the ex-post fairness values, i.e., fairness values of sets sampled according to the probabilistic strategies, computed by our routines dominate over the fairness achieved by previous methods on most of the instances tested.

Autonomous path planning through application of rotated two-parameter overrelaxation 9-point Laplacian iteration technique

Autonomous path navigation is one of the important studies in robotics since a robot’s ability to navigate through an environment brings about many advancements with it. This paper suggests the iteration technique called half-sweep two parameter overrelaxation 9-point laplacian (HSTOR-9P) to be applied on autonomous path navigation and aims to investigate its effectiveness in performing computation for path planning in an indoor static environment. This iteration technique is a harmonic function that solves the Laplace’s equation where the modelling of the environment is based on. The harmonic functions are an appropriate method to be used on autonomous path planning because it satisfies the min-max principle, therefore avoiding the occurrence of local minima which traps robot’s movements, and that it offers complete path planning algorithm. Its performance is tested against its predecessor iteration technique. Results shown that HSTOR-9P iteration technique enables path construction in a lower number of iterations, thus, performs better than its predecessors.

The annuity puzzle and consumption hump under ambiguous life expectancy

We establish a continuous-time model to solve the optimal consumption-saving problem with life annuity. The decision-maker has ambiguous beliefs in the force of mortality and is averse to this model uncertainty. The ambiguity aversion is represented by the max-min principle under the framework of robust control. We show that the optimal annuity demand decreases with the extent of ambiguity aversion, bequest motives, and extent of relative risk aversion. For an individual with non-zero bequest motives, the optimal annuity demand becomes zero when the extent of ambiguity aversion exceeds a certain threshold. We also show that, even if the individual has access to actuarially fair annuities, the optimal consumption would have a hump-shaped path when the individual is ambiguity-averse.

Min-max solutions for super sinh-Gordon equations on compact surfaces

In the present paper we initiate the variational analysis of a super sinh-Gordon system on compact surfaces, yielding the first example of non-trivial solution of min-max type. The proof is based on a linking argument jointly with a suitably defined Nehari manifold and a careful analysis of Palais-Smale sequences. We complement this study with a multiplicity result exploiting the symmetry of the problem.

An Analytic and Numerical Investigation of a Differential Game

In this paper we present an appropriate singular, zero-sum, linear-quadratic differential game. One of the main features of this game is that the weight matrix of the minimizer’s control cost in the cost functional is singular. Due to this singularity, the game cannot be solved either by applying the Isaacs MinMax principle, or the Bellman–Isaacs equation approach. As an application, we introduced an interception differential game with an appropriate regularized cost functional and developed an appropriate dual representation. By developing the variational derivatives of this regularized cost functional, we apply Popov’s approximation method and show how the numerical results coincide with the dual representation.

On Universality and Training in Binary Hypothesis Testing

The classical binary hypothesis testing problem is revisited. We notice that when one of the hypotheses is composite, there is an inherent difficulty in defining an optimality criterion that is both informative and well-justified. For testing in the simple normal location problem (that is, testing for the mean of multivariate Gaussians), we overcome the difficulty as follows. In this problem there exists a natural “hardness” order between parameters as for different parameters the error-probabilities curves (when the parameter is known) are either identical, or one dominates the other. We can thus define minimax performance as the worst-case among parameters which are below some hardness level. Fortunately, there exists a universal minimax test, in the sense that it is minimax for all hardness levels simultaneously. Under this criterion we also find the optimal test for composite hypothesis testing with training data. THIS criterion extends to the wide class of local asymptotic normal models, in an asymptotic sense where the approximation of the error probabilities is additive. Since we have the asymptotically optimal tests for composite hypothesis testing with and without training data, we quantify the loss of universality and gain of training data for these models.

Minimax rates in sparse, high-dimensional change point detection

We study the detection of a sparse change in a high-dimensional mean vector as a minimax testing problem. Our first main contribution is to derive the exact minimax testing rate across all parameter regimes for n independent, p-variate Gaussian observations. This rate exhibits a phase transition when the sparsity level is of order ploglog(8n) and has a very delicate dependence on the sample size: in a certain sparsity regime, it involves a triple iterated logarithmic factor in n. Further, in a dense asymptotic regime, we identify the sharp leading constant, while in the corresponding sparse asymptotic regime, this constant is determined to within a factor of 2. Extensions that cover spatial and temporal dependence, primarily in the dense case, are also provided.

Guaranteed lower eigenvalue bounds for two spectral problems arising in fluid mechanics

In this paper, we obtain guaranteed lower bounds for eigenvalues of two spectral problems arising in fluid mechanics by using the min–max principles of weak form that can be derived by the principles of operator forms. These two problems are the Laplace model for fluid–solid vibrations and the sloshing problem. We deal with the positive semi-definiteness of the associated bilinear forms by adding some constraints to the solution and finite element spaces. Numerical experiments are reported finally to validate our theoretical results.

Immunization Strategies for Funding Multiple Inflation-Linked Retirement Income Benefits

Protecting against unexpected yield curve, inflation, and longevity shifts are some of the most critical issues institutional and private investors must solve when managing post-retirement income benefits. This paper empirically investigates the performance of alternative immunization strategies for funding targeted multiple liabilities that are fixed in timing but random in size (inflation-linked), i.e., that change stochastically according to consumer price or wage level indexes. The immunization procedure is based on a targeted minimax strategy considering the M-Absolute as the interest rate risk measure. We investigate to what extent the inflation-hedging properties of ILBs in asset liability management strategies targeted to immunize multiple liabilities of random size are superior to that of nominal bonds. We use two alternative datasets comprising daily closing prices for U.S. Treasuries and U.S. inflation-linked bonds from 2000 to 2018. The immunization performance is tested over 3-year and 5-year investment horizons, uses real and not simulated bond data and takes into consideration the impact of transaction costs in the performance of immunization strategies and in the selection of optimal investment strategies. The results show that the multiple liability immunization strategy using inflation-linked bonds outperforms the equivalent strategy using nominal bonds and is robust even in a nearly zero interest rate scenario. These results have important implications in the design and structuring of ALM liability-driven investment strategies, particularly for retirement income providers such as pension schemes or life insurance companies.

Collective decision under ignorance.

This paper studies aggregation of preferences under ignorance, in which everybody knows that the true probability distribution over outcomes lies in some objective set but knows nothing about which one in it is true or which one in it is more likely to be true. We consider two decision models which express the precautionary principle under ignorance, the maximin criterion and the alpha -maximin criterion. We show that the Pareto axiom implies dictatorship, in each decision model. The impossibility results force us to choose between two options, one is to give up the precautionary principle as modelled at the social level, the other is to weaken the Pareto axiom. We provide possibility results for each of the options.

Differential Graphical Game With Distributed Global Nash Solution

This article studies multiagent differential graphical games in linear dynamical networks. We point out that in existing graphical game formulations, being global Nash and being distributed are two contradicting properties. In particular, we prove that the best response strategy, most widely used in solving graphical games, leads to Nash but does not provide distributed solutions. On the other hand, the minmax strategy, recently developed in solving graphical games, provides distributed solutions but prevents the agents from reaching a global Nash equilibrium. In this article, we propose a novel differential graphical game formulation, which promises the existence of solutions that 1) lead to global Nash equilibrium and 2) are distributed in the sense that each agent only uses the state information of its own and its direct neighbors. Stability and Nash equilibrium properties of the proposed graphical game are proven, respectively. Simulation studies are conducted to illustrate the theoretical results.

Subgame Maxmin Strategies in Zero-Sum Stochastic Games with Tolerance Levels

We study subgame phi -maxmin strategies in two-player zero-sum stochastic games with a countable state space, finite action spaces, and a bounded and universally measurable payoff function. Here, phi denotes the tolerance function that assigns a nonnegative tolerated error level to every subgame. Subgame phi -maxmin strategies are strategies of the maximizing player that guarantee the lower value in every subgame within the subgame-dependent tolerance level as given by phi . First, we provide necessary and sufficient conditions for a strategy to be a subgame phi -maxmin strategy. As a special case, we obtain a characterization for subgame maxmin strategies, i.e., strategies that exactly guarantee the lower value at every subgame. Secondly, we present sufficient conditions for the existence of a subgame phi -maxmin strategy. Finally, we show the possibly surprising result that each game admits a strictly positive tolerance function phi ^* with the following property: if a player has a subgame phi ^*-maxmin strategy, then he has a subgame maxmin strategy too. As a consequence, the existence of a subgame phi -maxmin strategy for every positive tolerance function phi is equivalent to the existence of a subgame maxmin strategy.

Rawls’s difference principle and maximin rule of allocation: a new analysis

If Rawls’s A Theory of Justice has achieved fame among economists, this is due to his Difference Principle, which says that inequalities of resources should be to the benefit of the less fortunate, or more operationally, that allocations of resources should be ranked by the maximin criterion. We extend the Rawlsian maximin in two ways: first, by resorting to the more general min-of-means formula of decision theory, second, by addressing the case where the resources accruing to each individual are uncertain to society. For the latter purpose, we resort to the ex ante versus ex post distinction of welfare economics. The paper axiomatically characterizes the ex ante and ex post forms of the Rawlsian maximin and compares them in terms of egalitarian criteria. It finally recommends and axiomatizes a compromise egalitarian theory that mixes the two forms.

A novel interpretation of min-max theorem and principle in relativistic quantum chemistry

The development of min–max theorem and min–max principle for solving wave equations in relativistic quantum chemistry is briefly discussed. A new interpretation of the min–max treatment of Dirac equation is suggested. The min–max restores covariance in a one-electron formalism and tries to find a pure state.