Worst-case Problem Research Articles

Towards well-generalizing meta-learning via adversarial task augmentation

Meta-learning aims to use the knowledge from previous tasks to facilitate the learning of novel tasks. Many meta-learning models elaborately design various task-shared inductive bias, and learn it from a large number of tasks, so the generalization capability of the learned inductive bias depends on the diversity of the training tasks. A common assumption in meta-learning is that the training tasks and the test tasks come from the same or similar task distributions. However, this is usually not strictly satisfied in practice, so meta-learning models need to cope with various novel in-domain or cross-domain tasks. To this end, we propose to use task augmentation to increase the diversity of training tasks, thereby improving the generalization capability of meta-learning models. Concretely, we consider the worst-case problem around the base task distribution, and derive the adversarial task augmentation method which can generate inductive bias-adaptive ‘challenging’ tasks. Our method can be used as a simple plug-and-play module for various meta-learning models, and improve their generalization capability. We conduct extensive experiments under in-domain and cross-domain few-shot learning and unsupervised few-shot learning settings, and evaluate our method on different types of data (images and text). Experimental results show that our method can effectively improve the generalization capability of various meta-learning models under different settings.

Model-based design and optimization of GSSR chromatography for peptide purification

Gradient with Steady State Recycle (GSSR) is a recently developed process for center-cut separation by solvent-gradient chromatography. The process comprises a multicolumn, open-loop system with cyclic steady-state operation that simulates a solvent gradient moving countercurrently with respect to the solid phase. However, the feed is always injected into the same column and the product always collected from the same column as in single-column batch chromatography. Here, three-column GSSR chromatography for peptide purification is optimized using state-of-the-art mathematical programming tools. The optimization problem is formulated using a full-discretization approach for steady periodic dynamics. The resulting nonlinear programming problem is solved by an efficient open-source interior-point solver coupled to a high-performance parallel linear solver for sparse symmetric indefinite matrices. The procedure is successfully employed to find optimal solutions for a series of process design problems with increasing number of decision variables. In addition to productivity and recovery, process performance is analyzed in terms of two key performance indicators: dilution ratio and solvent consumption ratio. Finally, the problem of robust process design under uncertainty in the solvent gradient manipulation is examined. The best solution is chosen only among candidate solutions that are robust feasible, i.e., remain feasible for all modifier gradient perturbations within the accuracy range of the gradient pump. This gives rise to a robust approach to optimal design in which the nominal problem is replaced by a worst case problem. Overall, our work illustrates the advantages of using advanced mathematical programming tools in designing and optimizing a GSSR process for which it is difficult to deduce sufficiently general heuristic design rules.

Outer Approximation for Mixed-Integer Nonlinear Robust Optimization

Currently, few approaches are available for mixed-integer nonlinear robust optimization. Those that do exist typically either require restrictive assumptions on the problem structure or do not guarantee robust protection. In this work, we develop an algorithm for convex mixed-integer nonlinear robust optimization problems where a key feature is that the method does not rely on a specific structure of the inner worst-case (adversarial) problem and allows the latter to be non-convex. A major challenge of such a general nonlinear setting is ensuring robust protection, as this calls for a global solution of the non-convex adversarial problem. Our method is able to achieve this up to a tolerance, by requiring worst-case evaluations only up to a certain precision. For example, the necessary assumptions can be met by approximating a non-convex adversarial via piecewise relaxations and solving the resulting problem up to any requested error as a mixed-integer linear problem.In our approach, we model a robust optimization problem as a nonsmooth mixed-integer nonlinear problem and tackle it by an outer approximation method that requires only inexact function values and subgradients. To deal with the arising nonlinear subproblems, we render an adaptive bundle method applicable to this setting and extend it to generate cutting planes, which are valid up to a known precision. Relying on its convergence to approximate critical points, we prove, as a consequence, finite convergence of the outer approximation algorithm.As an application, we study the gas transport problem under uncertainties in demand and physical parameters on realistic instances and provide computational results demonstrating the efficiency of our method.

Hard-instance learning for quantum adiabatic prime factorization

Prime factorization is a difficult problem with classical computing, whose exponential hardness is the foundation of Rivest-Shamir-Adleman cryptography. With programable quantum devices, adiabatic quantum computing has been proposed as a plausible approach to solve prime factorization, having promising advantage over classical computing. Here, we find there are certain hard instances that are consistently intractable for both classical simulated annealing and unconfigured adiabatic quantum computing (AQC). Aiming at an automated architecture for optimal configuration of quantum adiabatic factorization, we apply a deep reinforcement learning (RL) method to configure the AQC algorithm. By setting the success probability of the worst-case problem instances as the reward to RL, we show the AQC performance on the hard instances is dramatically improved by RL configuration. The success probability also becomes more evenly distributed over different problem instances, meaning the configured AQC is more stable as compared to the unconfigured case. Through a technique of transfer learning, we find prominent evidence that the framework of AQC configuration is scalable---the configured AQC as trained on five qubits remains working efficiently on nine qubits with a minimal amount of additional training cost.

Optimal portfolios in the presence of stress scenarios A worst-case approach

Insurance companies and banks regularly have to face stress tests performed by regulatory instances. To model their investment decision problems that includes stress scenarios, we propose the worst-case portfolio approach. Thus, the resulting optimal portfolios are already stress test prone by construction. A central issue of the worst-case portfolio approach is that neither the time nor the order of occurrence of the stress scenarios are known. Even more, there are no probabilistic assumptions regarding the occurrence of the stresses. By defining the relative worst-case loss and introducing the concept of minimum constant portfolio processes, we generalize the traditional concepts of the indifference frontier and the indifference-optimality principle. We prove the existence of a minimum constant portfolio process that is optimal for the multi-stress worst-case problem. As a main result we derive a verification theorem that provides conditions on Lagrange multipliers and nonlinear ordinary differential equations that support the construction of optimal worst-case portfolio strategies. The practical applicability of the verification theorem is demonstrated via numerical solution of various worst-case problems with stresses. There, it is in particular shown that an investor who chooses the worst-case optimal portfolio process may have a preference regarding the order of stresses, but there may also be stress scenarios where he/she is indifferent regarding the order and time of occurrence.

A worst-case approach for interest rate stresses and stock crashes

Abstract In the current low interest rate environment even sovereign bonds cannot be considered as risk-free investments. To care for this aspect we introduce a worst-case continuous-time portfolio problem with combined stresses, that is, both stocks and the money market account can experience shocks in the form of unpredictable downward jumps in their values. We characterize the worst-case optimal portfolio strategy as an indifference strategy that is the solution of a constrained optimization problem. Our results generalize existing results in a multi-asset setting. Numerical examples demonstrate new effects in the presence of interest rate shocks. These insights can be used for risk management in the presence of crash risks.

Mixed-case community detection problem in social networks: Algorithms and analysis

The problem of detecting communities is one of the essential problems in the study of social networks. To devise the algorithms of community detection, one should first define high-quality communities. In fact, there are no agreed methods to measure the quality of the community. In this paper, we consider a novel objective function of this problem. Our goal is to maximize not only the average of the sum of edge weights within communities (i.e., average-case) but also the sum of edge weights within the minimum community (i.e., worst-case). To balance both the average-case and worst-case problems, we introduce a parameter into our objective function and call it the mixed-cased community detection problem. For the worst-case, we devise several approximation algorithms, such as the Greedy, Semi-Sandwich Approximation, and Local Search algorithms. For the average-case, an efficient Terminal-based algorithm is proposed. We prove that the best solution between the average-case and worst-case problems still can provide an approximate guarantee for any mixed-case community detection problem. Moreover, we devise a heuristic algorithm for our problem. Finally, we conduct the experiments in three networks. The experimental results indicate that our proposed methods can form a high-quality community partition.

Smoothed analysis for tensor methods in unsupervised learning

Smoothed analysis is a powerful paradigm in overcoming worst-case intractability in high-dimensional data analysis and unsupervised learning. While polynomial time smoothed analysis guarantees have been obtained for worst-case intractable problems like tensor decomposition and learning mixtures of Gaussians, such guarantees have been hard to obtain for several other important problems in data analysis. A core technical challenge in analyzing algorithms is obtaining lower bounds on the least singular value for random matrix ensembles with dependent entries, that are given by low-degree polynomials of a few base underlying random variables. In this work, we address this challenge by obtaining high-confidence lower bounds on the least singular value of new classes of structured random matrix ensembles of the above kind. We then use these bounds to design algorithms with polynomial time smoothed analysis guarantees for the following three important problems in high-dimensional data analysis:

On Basing One-way Permutations on NP-hard Problems under Quantum Reductions

A fundamental pursuit in complexity theory concerns reducing worst-case problems to average-case problems. There exist complexity classes such as PSPACE that admit worst-case to average-case reductions. However, for many other classes such as NP, the evidence so far is typically negative, in the sense that the existence of such reductions would cause collapses of the polynomial hierarchy(PH). Basing cryptographic primitives, e.g., the average-case hardness of inverting one-way permutations, on NP-completeness is a particularly intriguing instance. As there is evidence showing that classical reductions from NP-hard problems to breaking these primitives result in PH collapses, it seems unlikely to base cryptographic primitives on NP-hard problems. Nevertheless, these results do not rule out the possibilities of the existence of quantum reductions. In this work, we initiate a study of the quantum analogues of these questions. Aside from formalizing basic notions of quantum reductions and demonstrating powers of quantum reductions by examples of separations, our main result shows that if NP-complete problems reduce to inverting one-way permutations using certain types of quantum reductions, thencoNP⊆QIP(2).

A game-theoretic method for resilient control design in industrial multi-agent CPSs with Markovian and coupled dynamics

Multi-agent cyber-physical systems (CPSs) are ubiquitous in various modern critical infrastructures, especially in industrial automation control involved in the petrochemical, metallurgy, textile, and pharmaceutical, which are vulnerable for their hierarchically and horizontally interconnected architecture. In this paper, we investigate a resilient control design problem for industrial multi-agent CPSs of centralised information structure, with Markovian and physically coupled dynamics from a game-theoretic perspective. First, we investigate a worst-case design problem and solve for worst-case disturbance strategy and optimal disturbance attenuation level parameter through an equivalent two-person zero-sum differential game in both finite- and infinite-horizon. Second, we study a corresponding N-person worst-case nonzero-sum differential game and derive state-feedback Nash equilibrium solution of robustness property as resilient control strategy for each agent. The experimental case study is given to demonstrate that using resilient control strategy, each agent could maintain its physical plant operationally normal under cyber attack, malicious disturbance input and physical state interdependency.

Provably Secure Identity-Based Encryption and Signature over Cyclotomic Fields

Identity-based cryptography is a type of public key cryptography with simple key management procedures. To our knowledge, till now, the existing identity-based cryptography based on NTRU is all over power-of-2 cyclotomic rings. Whether there is provably secure identity-based cryptography over more general fields is still open. In this paper, with the help of the results of collision resistance preimage sampleable functions (CRPSF) over cyclotomic fields, we give concrete constructions of provably secure identity-based encryption schemes (IBE) and identity-based signature schemes (IBS) based on NTRU over any cyclotomic field. Our IBE schemes are provably secure under adaptive chosen-plaintext and adaptive chosen-identity attacks, meanwhile, our IBS schemes are existentially unforgeable against adaptively chosen message and adaptively chosen identity attacks for any probabilistic polynomial time (PPT) adversary in the random oracle model. The securities of both schemes are based on the worst-case approximate shortest independent vectors problem (SIVPγ) over corresponding ideal lattices. The secret key size of our IBE (IBS) scheme is short—only one (two) ring element(s). The ciphertext (signature) is also short—only two (three) ring elements. Meanwhile, as the case of NTRUEncrypt, our IBE scheme could encrypt n bits in each encryption process. These properties may make our schemes have more advantages for some IoT applications over postquantum world in theory.

Optimal best-case and worst-case coverage for straight paths in ad hoc networks

Given a set of network nodes and their locations, the best support path problem is to find a path that has the min-max distance to network nodes and the maximal breach path problem is to find a path that has the max-min distance to network nodes. This paper considers straight paths and proposes optimal algorithms identifying both the best support and the maximal breach straight paths. Our study provides a coverage measurement method for straight paths in an ad hoc network. Based on computational geometry and graph theory, we propose plane-sweep algorithms to find the optimal straight paths for both the best-case and worst-case coverage problems in polynomial time. Mathematical analysis and simulations are used to prove the optimality of proposed algorithms.

Efficient Approximations for the Online Dispersion Problem

The dispersion problem has been widely studied in computational geometry and facility location and is closely related to the packing problem. The goal is to locate $n$ points (e.g., facilities or persons) in a $k$-dimensional polytope, so that they are far away from each other and from the boundary of the polytope. In many real-world scenarios, however, the points arrive and depart at different times, and decisions must be made without knowing future events. Therefore, we study, for the first time in the literature, the online dispersion problem in Euclidean space. There are two natural objectives when time is involved: the all-time worst-case (ATWC) problem tries to maximize the minimum distance that ever appears at any time; and the cumulative distance (CD) problem tries to maximize the integral of the minimum distance throughout the whole time interval. Interestingly, the online problems are highly nontrivial even on a segment. For cumulative distance, this remains the case even when the problem is time-dependent but offline, with all the arriving and departure times given in advance. For the online ATWC problem on a segment, we construct a deterministic polynomial-time algorithm which is $(2\ln 2+\epsilon)$-competitive, where $\epsilon>0$ can be arbitrarily small and the algorithm's running time is polynomial in $\frac{1}{\epsilon}$. We show that this algorithm is actually optimal. For the same problem in a square, we provide a $1.591$-competitive algorithm and a $1.183$ lower bound. Furthermore, for arbitrary $k$-dimensional polytopes with $k\geq 2$, we provide a $\frac{2}{1-\epsilon}$-competitive algorithm and a $\frac{7}{6}$ lower bound. All our lower bounds come from the structure of the online problems and hold even when computational complexity is not a concern. Interestingly, for the offline CD problem in arbitrary $k$-dimensional polytopes, we provide a polynomial-time black-box reduction to the online ATWC problem, and the resulting competitive ratio increases by a factor of at most 2. Our techniques also apply to online dispersion problems with different boundary conditions.

A new approach for worst-case regret portfolio optimization problem

This paper considers the worst-case regret portfolio optimization problem when the distributions of the asset returns are uncertain. In general, the solution to this problem is NP hard and approximation methods that minimise the difference between the maximum return and the sum of each portfolio return are often proposed. Applying the duality of semi-infinite programming, the worst-case regret portfolio optimization problem with uncertain distributions can be equivalently reformulated to a linear optimization problem, and the established solution approaches for linear optimization can then be applied. An example of a portfolio optimization problem is provided to show the efficiency of our method and the results demonstrate that our method can satisfy the portfolio risk diversification property under the uncertain distributions of the returns.

A verifiable threshold secret sharing scheme based on lattices

The generalized compact knapsack function is defined as fa(x)=∑iai.xi, where a=(a1,…,am)∈R for some ring R and x=(x1,…,xm)∈S for a specified subset S ⊂ R. It is known that, for appropriate choice of R and S, inverting this function is at least as hard as solving certain worst-case problems on cyclic lattices. In this paper, we exploit collision-resistance as well as homomorphic properties of this function to propose a threshold verifiable secret sharing scheme with two distinguished features: first, the security of the scheme is completely based on lattice problems and second, upon receiving shares, participants can verify consistency of their shares with the secret without any communication.

On ideal lattices, Gröbner bases and generalized hash functions

In this paper, we draw connections between ideal lattices and multivariate polynomial rings over integers using Gröbner bases. Univariate ideal lattices are ideals in the residue class ring, [Formula: see text] (here [Formula: see text] is a monic polynomial) and cryptographic primitives have been built based on these objects. Ideal lattices in the univariate case are generalizations of cyclic lattices. We introduce the notion of multivariate cyclic lattices and show that ideal lattices are a generalization of them in the multivariate case too. Based on multivariate ideal lattices, we construct hash functions using Gröbner basis techniques. We define a worst case problem, shortest substitution problem with respect to an ideal in [Formula: see text], and use its computational hardness to establish the collision resistance of the hash functions.

Reliable capacitated facility location problem with service levels

This study is concerned with a generalized version of the reliable capacitated facility location problem that guarantees a minimum level of service level in terms of satisfied demands under correlated facility disruptions with uncertain joint distribution. A linear mixed integer formulation for the generalized problem is presented and shown to correspond with the worst-case (distributionally robust) problem when the service-level function is submodular on the disruption scenarios. Managerial insights, including the effect of service-level parameters on optimal solutions, are presented based on experimental results. It is concluded that the proposed model may be used to design supply chain networks with the same expected total cost than the traditional reliable capacitated facility location problem, but with an improved total expected service level.

Robust Secure Beamforming for 5G Cellular Networks Coexisting With Satellite Networks

This paper studies the robust secure beamforming (BF) issue of fifth generation (5G) cellular system operating at millimeter wave frequency and coexisting with a satellite network. By employing an uniform planar array at the base station (BS) and assuming known imperfect angle-of-arrival-based channel state informations of multiple eavesdroppers (Eves), a constrained optimization problem is first formulated to maximize the worst-case achievable secrecy rate of the cellular user under the constraints of the transmit power of BS and the interference threshold of satellite earth station. Then, we propose two robust BF methods to solve the complex optimization problem for both coordinated and uncoordinated Eves. For the case of coordinated Eves, we propose a heuristic BF scheme, which transfers the worst-case problem into a min-max one such that the BF weight vectors can be obtained analytically. For uncoordinated Eves, we convert the non-convex problem into a convex one, and further propose an iterative penalty function-based algorithm to obtain the optimal BF weight vectors. Finally, simulation results are provided to confirm the effectiveness and superiority of the proposed robust BF schemes.

Worst-Case Investment Strategy with Delay

AbstractThis paper considers a worst-case investment optimization problem with delay for a fund manager who is in a crash-threatened financial market. Driven by existing of capital inflow/outflow related to history performance, we investigate the optimal investment strategies under the worst-case scenario and the stochastic control framework with delay. The financial market is assumed to be either in a normal state (crash-free) or in a crash state. In the normal state the prices of risky assets behave as geometric Brownian motion, and in the crash state the prices of risky assets suddenly drop by a certain relative amount, which induces to a dropping of the total wealth relative to that of crash-free state. We obtain the ordinary differential equations satisfied by the optimal investment strategies and the optimal value functions under the power and exponential utilities, respectively. Finally, a numerical simulation is provided to illustrate the sensitivity of the optimal strategies with respective to the model parameters.

Identity Based Proxy Re-encryption Scheme under LWE

The proxy re-encryption allows an intermediate proxy to convert a ciphertext for Alice into a ciphertext for Bob without seeing the original message and leaking out relevant information. Unlike many prior identity based proxy re-encryption schemes which are based on the number theoretic assumptions such as large integer factorization and discrete logarithm problem. In this paper, we first propose a novel identity based proxy re-encryption scheme which is based on the hardness of standard Learning With Error(LWE) problem and is CPA secure in the standard model. This scheme can be reduced to the worst-case lattice hard problem that is able to resist attacks from quantum algorithm. The key step in our construction is that the challenger how to answer the private query under a known trapdoor matrix. Our scheme enjoys properties of the non-interactivity, unidirectionality, anonymous and so on. In this paper, we utilize primitives include G-trapdoor for lattice and sample algorithms to realize simple and efficient re-encryption.