Monotonicity Assumption Research Articles

On the Impossibility of Convergence of Mixed Strategies with Optimal No-Regret Learning

We study the limiting behavior of the mixed strategies that result from optimal no-regret learning in a repeated game setting where the stage game is any 2 × 2 competitive game. We consider optimal no-regret algorithms that are mean-based and monotonic in their argument. We show that for any such algorithm, the limiting mixed strategies of the players cannot converge almost surely to any Nash equilibrium. This negative result is also shown to hold under a broad relaxation of these assumptions, including popular variants of Follow-the-Regularized Leader with optimism or adaptive step sizes. Finally, we provide partial evidence that the monotonicity and mean-based assumptions can be removed or relaxed. Our results identify the inherent stochasticity in players’ realizations as a critical factor underlying this divergence, and demonstrate a crucial difference in outcomes between using the opponent’s mixtures and realizations to make updates. Funding: V. Muthukumar was supported by a Simons-Berkeley Research Fellowship, NSF awards IIS-2212182 and CCF-2239151, and generous gifts from Amazon, Adobe, and Google. S. Phade acknowledges the support of the NSF [Grants CNS-1527846 and CCF-1618145] and the NSF Science & Technology Center [Grant CCF-0939370 (Science of Information)]. A. Sahai acknowledges the support of the ML4Wireless center member companies and the NSF [Grants AST-144078 and ECCS-1343398].

Statistical and computational efficiency for smooth tensor estimation with unknown permutations

We consider the problem of structured tensor denoising in the presence of unknown permutations. Such data problems arise commonly in recommendation systems, neuroimaging, community detection, and multiway comparison applications. Here, we develop a general family of smooth tensor models up to arbitrary index permutations; the model incorporates the popular tensor block models and Lipschitz hypergraphon models as special cases. We show that a constrained least-squares estimator in the block-wise polynomial family achieves the minimax error bound. A phase transition phenomenon is revealed with respect to the smoothness threshold needed for optimal recovery. In particular, we find that a polynomial of degree up to ( m − 2 ) ( m + 1 ) / 2 is sufficient for accurate recovery of order-m tensors, whereas higher degrees exhibit no further benefits. This phenomenon reveals the intrinsic distinction for smooth tensor estimation problems with and without unknown permutations. Furthermore, we provide an efficient polynomial-time Borda count algorithm that provably achieves the optimal rate under monotonicity assumptions. The efficacy of our procedure is demonstrated through both simulations and Chicago crime data analysis.

A nested primal–dual iterated Tikhonov method for regularized convex optimization

AbstractProximal–gradient methods are widely employed tools in imaging that can be accelerated by adopting variable metrics and/or extrapolation steps. One crucial issue is the inexact computation of the proximal operator, often implemented through a nested primal–dual solver, which represents the main computational bottleneck whenever an increasing accuracy in the computation is required. In this paper, we propose a nested primal–dual method for the efficient solution of regularized convex optimization problems. Our proposed method approximates a variable metric proximal–gradient step with extrapolation by performing a prefixed number of primal–dual iterates, while adjusting the steplength parameter through an appropriate backtracking procedure. Choosing a prefixed number of inner iterations allows the algorithm to keep the computational cost per iteration low. We prove the convergence of the iterates sequence towards a solution of the problem, under a relaxed monotonicity assumption on the scaling matrices and a shrinking condition on the extrapolation parameters. Furthermore, we investigate the numerical performance of our proposed method by equipping it with a scaling matrix inspired by the Iterated Tikhonov method. The numerical results show that the combination of such scaling matrices and Nesterov-like extrapolation parameters yields an effective acceleration towards the solution of the problem.

Measuring psychache as a suicide risk variable: A Mokken analysis of the Holden's Psychache Scale

BackgroundThe Psychache Scale (PAS) is a questionnaire measuring trait-level psychological pain with satisfactory internal consistency and a strong correlation with suicidal ideation severity. However, inconsistent results have been reported for the PAS dimensionality. In the present study we used a non-parametric item response theory model, called Mokken Scale Analysis (MSA), to refine an unidimensional version of the PAS. MethodsThe sample was composed of 400 Italian adults (312 females and 88 males) nonrandomly recruited from the general population. ResultsA final set of 10 items satisfied the unidimensionality, local independence, and monotonicity assumptions, although it did not satisfy the double monotonicity assumption. The internal consistency of the PAS-10 was satisfactory (ordinal alpha = 0.98, ω = 0.97, and AVE = 0.82), and ROC curves analysis indicated good discriminant validity when differentiating participants with higher suicide risk from those with lower suicide risk. LimitationsStructural invariance between nonclinical and clinical samples was not investigated, and the presence of suicide ideation and behaviors was assessed with self-report measures with potential under-reporting of the phenomenon. ConclusionThe PAS-10 resulted to be a potentially valid and unidimensional measure of psychological pain (i.e., psychache) that could be used to screen adults at higher risk for suicide. Future studies are needed to investigate psychometric properties of the PAS-10 in clinical samples and to replicate results in independent samples.

Optimizing Oxford Shoulder Scores with computerized adaptive testing reduces redundancy while maintaining precision.

The Oxford Shoulder Score (OSS) is a 12-item measure commonly used for the assessment of shoulder surgeries. This study explores whether computerized adaptive testing (CAT) provides a shortened, individually tailored questionnaire while maintaining test accuracy. A total of 16,238 preoperative OSS were available in the National Joint Registry (NJR) for England, Wales, Northern Ireland, the Isle of Man, and the States of Guernsey dataset (April 2012 to April 2022). Prior to CAT, the foundational item response theory (IRT) assumptions of unidimensionality, monotonicity, and local independence were established. CAT compared sequential item selection with stopping criteria set at standard error (SE) < 0.32 and SE < 0.45 (equivalent to reliability coefficients of 0.90 and 0.80) to full-length patient-reported outcome measure (PROM) precision. Confirmatory factor analysis (CFA) for unidimensionality exhibited satisfactory fit with root mean square standardized residual (RSMSR) of 0.06 (cut-off ≤ 0.08) but not with comparative fit index (CFI) of 0.85 or Tucker-Lewis index (TLI) of 0.82 (cut-off > 0.90). Monotonicity, measured by H value, yielded 0.482, signifying good monotonic trends. Local independence was generally met, with Yen's Q3 statistic > 0.2 for most items. The median item count for completing the CAT simulation with a SE of 0.32 was 3 (IQR 3 to 12), while for a SE of 0.45 it was 2 (IQR 2 to 6). This constituted only 25% and 16%, respectively, when compared to the 12-item full-length questionnaire. Calibrating IRT for the OSS has resulted in the development of an efficient and shortened CAT while maintaining accuracy and reliability. Through the reduction of redundant items and implementation of a standardized measurement scale, our study highlights a promising approach to alleviate time burden and potentially enhance compliance with these widely used outcome measures.

Caballero–Engel meet Lasry–Lions: A uniqueness result

In a Mean Field Game (MFG) each decision maker cares about the cross sectional distribution of the state and the dynamics of the distribution is generated by the agents’ optimal decisions. We prove the uniqueness of the equilibrium in a class of MFG where the decision maker controls the state at optimally chosen times. This setup accommodates several problems featuring non-convex adjustment costs, and complements the well known drift-control case studied by Lasry–Lions. Examples of such problems are described by Caballero and Engel in several papers, which introduce the concept of the generalized hazard function of adjustment. We extend the analysis to a general “impulse control problem” by introducing the concept of the “Impulse Hamiltonian”. Under the monotonicity assumption (a form of strategic substitutability), we establish the uniqueness of equilibrium. In this context, the Impulse Hamiltonian and its derivative play a similar role to the classical Hamiltonian that arises in the drift-control case.

Solving split equality equilibrium and fixed point problems in Banach spaces

In this paper, we introduce a new algorithm for approximating a common solution of Split Equality Generalised Mixed Equilibrium Problem (SEGMEP) and Split Equality Fixed Point Problem (SEFPP) for two infinite families of closed uniformly Li-Lipschitz continuous and Ki-Lipschitz continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings (i ∈ N and ϕ is the Lyapunov functional) in Banach spaces. Under standard and mild assumption of monotonicity and lower semicontinuity of the SEGMEP associated mappings, we establish the strong convergence of the scheme without imposing any compactness type conditions on either the operators or the spaces considered. We apply our result to approximate the solution of Split Equality Convex Minimization Problem (SECMP) and Split Equality Variational Inclusion Problem (SEVIP). A numerical example is presented to illustrate the performance and implementability of our method. Our results extend, generalize and complement several related works in the literature.

A flexible time-varying coefficient rate model for panel count data.

Panel count regression is often required in recurrent event studies, where the interest is to model the event rate. Existing rate models are unable to handle time-varying covariate effects due to theoretical and computational difficulties. Mean models provide a viable alternative but are subject to the constraints of the monotonicity assumption, which tends to be violated when covariates fluctuate over time. In this paper, we present a new semiparametric rate model for panel count data along with related theoretical results. For model fitting, we present an efficient EM algorithm with three different methods for variance estimation. The algorithm allows us to sidestep the challenges of numerical integration and difficulties with the iterative convex minorant algorithm. We showed that the estimators are consistent and asymptotically normally distributed. Simulation studies confirmed an excellent finite sample performance. To illustrate, we analyzed data from a real clinical study of behavioral risk factors for sexually transmitted infections.

Adaptive, Doubly Optimal No-Regret Learning in Strongly Monotone and Exp-Concave Games with Gradient Feedback

Feasible Online Learning with Gradient Feedback Online gradient descent (OGD) is well-known to be doubly optimal under strong convexity or monotonicity assumptions: (1) in the single-agent setting, it achieves an optimal regret of [Formula: see text] for strongly convex cost functions, and (2) in the multiagent setting of strongly monotone games with each agent employing OGD we obtain last-iterate convergence of the joint action to a unique Nash equilibrium at an optimal rate of [Formula: see text]. Whereas these finite-time guarantees highlight its merits, OGD has the drawback that it requires knowing the strong convexity/monotonicity parameters. In “Adaptive, Doubly Optimal No-Regret Learning in Strongly Monotone and Exp-Concave Games with Gradient Feedback,” M. Jordan, T. Lin, and Z. Zhou design a fully adaptive OGD algorithm, AdaOGD, that does not require a priori knowledge of these parameters. In the single-agent setting, the algorithm achieves [Formula: see text] regret under strong convexity, which is optimal up to a log factor. Further, if each agent employs AdaOGD in strongly monotone games, the joint action converges in a last-iterate sense to a unique Nash equilibrium at a rate of [Formula: see text], again optimal up to log factors. The algorithms are illustrated in a learning version of the classic newsvendor problem, in which, because of lost sales, only (noisy) gradient feedback can be observed. The results immediately yield the first feasible and near-optimal algorithm for both the single-retailer and multiretailer settings.

An Instrumental Variable Approach to Learning the Causal Exposure–Response Relationship in a Randomized Dose Comparison Trial

We consider the problem of estimating the causal effect of drug exposure on clinical response in a randomized dose comparison trial where each group receives a different dose. Because exposure is not randomized, the exposure–response relationship is subject to confounding in this setting. Conventional statistical methods for confounding adjustment with a continuous exposure typically assume that there are no unmeasured confounders. This article provides an instrumental variable approach that does not require the assumption of no unmeasured confounders. Specifically, we use randomized dose assignment as an instrumental variable and characterize the causal exposure–response relationship using a control variable under a mild monotonicity assumption on individual dose–exposure profiles. Based on this characterization, we derive partial identification bounds for the causal exposure–response relationship, which are model-free but may be too wide to be useful in practice. For practicality, we further develop a simple estimation method based on a regression model for the mean response conditional on exposure and the control variable. Simulation results show that, in the presence of unmeasured confounding, the model-based estimation method reduces estimation bias effectively at the expense of increased variability, as compared to existing methods. The method is illustrated with real data from a study of chimeric antigen receptor T cell therapy for treating chronic lymphocytic leukemia.

CIDR: A Cooperative Integrated Dynamic Refining Method for Minimal Feature Removal Problem

The minimal feature removal problem in the post-hoc explanation area aims to identify the minimal feature set (MFS). Prior studies using the greedy algorithm to calculate the minimal feature set lack the exploration of feature interactions under a monotonic assumption which cannot be satisfied in general scenarios. In order to address the above limitations, we propose a Cooperative Integrated Dynamic Refining method (CIDR) to efficiently discover minimal feature sets. Specifically, we design Cooperative Integrated Gradients (CIG) to detect interactions between features. By incorporating CIG and characteristics of the minimal feature set, we transform the minimal feature removal problem into a knapsack problem. Additionally, we devise an auxiliary Minimal Feature Refinement algorithm to determine the minimal feature set from numerous candidate sets. To the best of our knowledge, our work is the first to address the minimal feature removal problem in the field of natural language processing. Extensive experiments demonstrate that CIDR is capable of tracing representative minimal feature sets with improved interpretability across various models and datasets.

On the Convergence of an Adaptive Momentum Method for Adversarial Attacks

Adversarial examples are commonly created by solving a constrained optimization problem, typically using sign-based methods like Fast Gradient Sign Method (FGSM). These attacks can benefit from momentum with a constant parameter, such as Momentum Iterative FGSM (MI-FGSM), to enhance black-box transferability. However, the monotonic time-varying momentum parameter is required to guarantee convergence in theory, creating a theory-practice gap. Additionally, recent work shows that sign-based methods fail to converge to the optimum in several convex settings, exacerbating the issue. To address these concerns, we propose a novel method which incorporates both an innovative adaptive momentum parameter without monotonicity assumptions and an adaptive step-size scheme that replaces the sign operation. Furthermore, we derive a regret upper bound for general convex functions. Experiments on multiple models demonstrate the efficacy of our method in generating adversarial examples with human-imperceptible noise while achieving high attack success rates, indicating its superiority over previous adversarial example generation methods.

Optimal Decision Making Under Strategic Behavior

We are witnessing an increasing use of data-driven predictive models to inform decisions in high-stakes situations, from lending and hiring to university admissions. As decisions have implications for individuals and society, there is increasing pressure on decision makers to be transparent about their decision policies. At the same time, individuals may use knowledge, gained by transparency, to invest effort strategically in order to maximize their chances of receiving a beneficial decision. Our goal is to find decision policies that are optimal in terms of utility in such a strategic setting. First, we characterize how strategic investment of effort by individuals leads to a change in the feature distribution. Using this characterization, we show that, in general, optimal decision policies are hard to find in polynomial time, and there are cases in which deterministic policies are suboptimal. Then, we demonstrate that, if the cost individuals pay to change their features satisfies a natural monotonicity assumption, we can narrow down the search for the optimal policy to a particular family of decision policies with a set of desirable properties, which allow for a highly effective polynomial time-heuristic search algorithm using dynamic programming. Finally, under no assumptions on the cost, we develop an iterative search algorithm that is guaranteed to converge to locally optimal decision policies. Experiments on synthetic and real credit card data illustrate our theoretical findings and show that the decision policies found by our algorithms achieve higher utility than those that do not account for strategic behavior. This paper was accepted by Vivek Farias, data science. Funding: This work was supported by Machine Learning Cluster of Excellence [EXC #2064/1, Project #390727645], H2020 European Research Council [Grant 945719], and Bundesministerium für Bildung und Forschung [Grant 01IS18039B]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2021.02567

Implementing the time-to-event continual reassessment method in the presence of partial orders in a phase I head and neck cancer trial

BackgroundIn this article we describe the methodology of the time-to-event continual reassessment method in the presence of partial orders (PO-TITE-CRM) and the process of implementing this trial design into a phase I trial in head and neck cancer called ADePT-DDR. The ADePT-DDR trial aims to find the maximum tolerated dose of an ATR inhibitor given in conjunction with radiotherapy in patients with head and neck squamous cell carcinoma.MethodsThe PO-TITE-CRM is a phase I trial design that builds upon the time-to-event continual reassessment method (TITE-CRM) to allow for the presence of partial ordering of doses. Partial orders occur in the case where the monotonicity assumption does not hold and the ordering of doses in terms of toxicity is not fully known.ResultsWe arrived at a parameterisation of the design which performed well over a range of scenarios. Results from simulations were used iteratively to determine the best parameterisation of the design and we present the final set of simulations. We provide details on the methodology as well as insight into how it is applied to the trial.ConclusionsWhilst being a very efficient design we highlight some of the difficulties and challenges that come with implementing such a design. As the issue of partial ordering may become more frequent due to the increasing investigations of combination therapies we believe this account will be beneficial to those wishing to implement a design with partial orders.Trial registrationADePT-DDR was added to the European Clinical Trials Database (EudraCT number: 2020-001034-35) on 2020-08-07.

Periodic solutions for second-order even and noneven Hamiltonian systems

AbstractIn this paper, we consider the second-order Hamiltonian system $$ \ddot{x}+V^{\prime}(x)=0,\quad x\in \mathbb{R}^{N}. $$ x ¨ + V ′ ( x ) = 0 , x ∈ R N . We use the monotonicity assumption introduced by Bartsch and Mederski (Arch. Ration. Mech. Anal. 215:283–306, 2015). When V is even, we can release the strict convexity hypothesis, which is used by Bartsch and Mederski combined with the monotonicity assumption. When V is noneven, we weaken the strict convexity assumption and introduce another hypothesis (see (V10)). Then in both cases, we can build the homomorphism between the Nehari manifold and the unit sphere of some suitable space. Using the Nehari manifold method introduced by Szulkin (J. Funct. Anal. 257:3802–3822 2009), we prove the existence of T-periodic solutions with minimal period T.

Uniqueness for a class p-Laplacian problems when a parameter is large

We prove uniqueness of positive solutions for the problem \[-\Delta_{p}u=\lambda f(u)\text{ in }\Omega,\ u=0\text{ on }\partial \Omega,\] where \(1\lt p\lt 2\) and \(p\) is close to 2, \(\Omega\) is bounded domain in \(\mathbb{R}^{n}\) with smooth boundary \(\partial \Omega\), \(f:[0,\infty)\rightarrow [0,\infty )\) with \(f(z)\sim z^{\beta }\) at \(\infty\) for some \(\beta \in (0,1)\), and \(\lambda\) is a large parameter. The monotonicity assumption on \(f\) is not required even for \(u\) large.

Dual-Reinforcement-Learning-Based Attack Path Prediction for 5G Industrial Cyber–Physical Systems

5G industrial cyber-physical systems have attracted substantial research interests due to their capability in the interconnection of everything. However, integrating the 5G network may expose systems to more potential risks. To reveal attack propagation, an attack path prediction approach based on dual reinforcement learning is proposed. First, a dual-network model is established, incorporating the security constraints for attacks against the 5G network into the attack graph. Second, employing reinforcement learning, Q-value updating functions and reward mechanisms based on topology and vulnerability are designed. Finally, an optimal attack path prediction algorithm is developed. Unlike traditional methods, the proposed approach does not rely on the monotonicity assumption that a system component has only one vulnerability, enabling it to accurately predict the optimal attack paths. Our simulation results demonstrate that the proposed approach can identify possible attack sources and paths from a 5G industrial cyber-physical system.

A New Inertial-Projection Method for Solving Split Generalized Mixed Equilibrium and Hierarchical Fixed Point Problems

In this paper, we introduce a new iterative algorithm of inertial form for approximating the common solution of Split Generalized Mixed Equilibrium Problem (SGMEP) and Hierarchical Fixed Point Problem (HFPP) in real Hilbert spaces. Motivated by the subgradient extragradient method, we incorporate the inertial technique to accelerate the convergence of the proposed method. Under standard and mild assumption of monotonicity and lower semicontinuity of the SGMEP and HFPP associated mappings, we establish the strong convergence of the iterative algorithm. Some numerical experiments are presented to illustrate the performance and behaviour of our method as well as comparing it with some related methods in the literature.

A new double inertial subgradient extragradient method for solving a non-monotone variational inequality problem in Hilbert space

In this paper, we introduced a new double inertial subgradient extragradient method for solving a variational inequality problem in Hilbert space. In our method, the mapping needed not to satisfy any assumption of monotonicity and two different self-adaptive step sizes were used for avoiding the need of Lipschitz constant of the mapping. The strong convergence of the proposed method was proved under some new conditions. Finally, some numerical examples were presented to illustrate the convergence of our method and compare with some related methods in the literature.

Some LMIs for the design of dynamic output feedback contractive controllers for a class of nonlinear systems

This paper deals with the problem of designing an output feedback controller for the class of linear systems interconnected with a nonlinearity satisfying a monotonicity assumption. The control design objective is to ensure the incremental exponential stability of the closed loop. First, a set of sufficient conditions by relying on a design based on Linear Matrix Inequalities (LMIs) are proposed. The results are then expanded to include in the design a filter to cancel additional (unwanted) harmonics of the steady-state solution. The proposed results are validated via a simplified example of application.