arXiv subjects

Panos Parpas

Publications and source records attributed to Panos Parpas.

At least 19 recordsLinked to original sources

arXiv2026

Simulation-Based Neural Policies for Portfolio Choice: Architecture, Training, and Interpretability

Many economic decision problems, lifecycle consumption-saving and dynamic portfolio choice, are finite-horizon stochastic control problems with continuous states and actions. When the state is low-dimensional these problems are solved by dynamic programming on a grid. The grid cost grows exponentially in the state dimension, known as the curse of dimensionality, which motivates replacing the value-function grid with a neural policy optimized directly through simulation. Such policies are usually studied in the high-dimensional settings that motivate them, precisely where no reference solution exists. So the contribution of any single architectural or training choice cannot be isolated and diagnosed. We therefore take a step back and treat both the architecture and the solution method as the objects of study. To this end, we consider a lifecycle problem with a sufficiently low-dimensional normalized state space to admit an accurate dynamic programming solution, which is used for evaluation. We compare four architectures. The simplest consists of a single time-conditioned network. We then consider two networks concatenated across the regime switch, followed by one network per date trained backward against frozen downstream policies. Finally, we evaluate a constrained variant of the per-date architecture. Decoupling the policy across time gives each date a short, well-posed objective, which we pair with direction-dominant optimization that normalizes away gradient magnitude. Architectures that lead to similar realized utility objective can nevertheless differ in whether they respect the underlying problem's economics. We therefore evaluate each design jointly based on welfare, a solution-free Bellman residual, shape restrictions, and the resulting policy functions.

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arXiv2025

Adaptive Multilevel Newton: A Quadratically Convergent Optimization Method

Newton's method may exhibit slower convergence than vanilla Gradient Descent in its initial phase on strongly convex problems. Classical Newton-type multilevel methods mitigate this but, like Gradient Descent, achieve only linear convergence near the minimizer. We introduce an adaptive multilevel Newton-type method with a principled automatic switch to full Newton once its quadratic phase is reached. The local quadratic convergence for strongly convex functions with Lipschitz continuous Hessians and for self-concordant functions is established and confirmed empirically. Although per-iteration cost can exceed that of classical multilevel schemes, the method is efficient and consistently outperforms Newton's method, Gradient Descent, and the multilevel Newton method, indicating that second-order methods can outperform first-order methods even when Newton's method is initially slow. The promising empirical results open new avenues for designing reduced-cost second- and high-order methods with extremely fast convergence rates.

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arXiv2025

Convergence rates of Newton's method for strongly self-concordant minimization

Newton's method has been thoroughly studied for the class of self-concordant functions. However, a local analysis specific to strongly self-concordant functions (a subclass of the former) is missing from the literature. The local quadratic rate of strongly self-concordant functions follows, of course, from the known results for self-concordant functions. However, it is not known whether strongly self-concordant functions enjoy better theoretical properties. In this paper, we study the local convergence of Newton's method for this subclass. We show that its quadratic convergence rate differs from that of general self-concordant functions. In particular, it is provably faster for a wide range of objective functions and benefits from a larger region of local convergence. Thus, the results of this paper close the gap in the theoretical understanding of Newton's method applied to strongly self-concordant functions.

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arXiv2024

Kernel Sum of Squares for Data Adapted Kernel Learning of Dynamical Systems from Data: A global optimization approach

This paper examines the application of the Kernel Sum of Squares (KSOS) method for enhancing kernel learning from data, particularly in the context of dynamical systems. Traditional kernel-based methods, despite their theoretical soundness and numerical efficiency, frequently struggle with selecting optimal base kernels and parameter tuning, especially with gradient-based methods prone to local optima. KSOS mitigates these issues by leveraging a global optimization framework with kernel-based surrogate functions, thereby achieving more reliable and precise learning of dynamical systems. Through comprehensive numerical experiments on the Logistic Map, Henon Map, and Lorentz System, KSOS is shown to consistently outperform gradient descent in minimizing the relative-$\rho$ metric and improving kernel accuracy. These results highlight KSOS's effectiveness in predicting the behavior of chaotic dynamical systems, demonstrating its capability to adapt kernels to underlying dynamics and enhance the robustness and predictive power of kernel-based approaches, making it a valuable asset for time series analysis in various scientific fields.

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arXiv2024

Multilevel Regularized Newton Methods with Fast Convergence Rates

We introduce new multilevel methods for solving large-scale unconstrained optimization problems. Specifically, the philosophy of multilevel methods is applied to Newton-type methods that regularize the Newton sub-problem using second order information from a coarse (low dimensional) sub-problem. The new \emph{regularized multilevel methods} provably converge from any initialization point and enjoy faster convergence rates than Gradient Descent. In particular, for arbitrary functions with Lipschitz continuous Hessians, we show that their convergence rate interpolates between the rate of Gradient Descent and that of the cubic Newton method. If, additionally, the objective function is assumed to be convex, then the proposed method converges with the fast $\mathcal{O}(k^{-2})$ rate. Hence, since the updates are generated using a \emph{coarse} model in low dimensions, the theoretical results of this paper significantly speed-up the convergence of Newton-type or preconditioned gradient methods in practical applications. Preliminary numerical results suggest that the proposed multilevel algorithms are significantly faster than current state-of-the-art methods.

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arXiv2023

Curvature Aligned Simplex Gradient: Principled Sample Set Construction For Numerical Differentiation

The simplex gradient, a popular numerical differentiation method due to its flexibility, lacks a principled method by which to construct the sample set, specifically the location of function evaluations. Such evaluations, especially from real-world systems, are often noisy and expensive to obtain, making it essential that each evaluation is carefully chosen to reduce cost and increase accuracy. This paper introduces the curvature aligned simplex gradient (CASG), which provably selects the optimal sample set under a mean squared error objective. As CASG requires function-dependent information often not available in practice, we additionally introduce a framework which exploits a history of function evaluations often present in practical applications. Our numerical results, focusing on applications in sensitivity analysis and derivative free optimization, show that our methodology significantly outperforms or matches the performance of the benchmark gradient estimator given by forward differences (FD) which is given exact function-dependent information that is not available in practice. Furthermore, our methodology is comparable to the performance of central differences (CD) that requires twice the number of function evaluations.

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arXiv2023

Simba: A Scalable Bilevel Preconditioned Gradient Method for Fast Evasion of Flat Areas and Saddle Points

The convergence behaviour of first-order methods can be severely slowed down when applied to high-dimensional non-convex functions due to the presence of saddle points. If, additionally, the saddles are surrounded by large plateaus, it is highly likely that the first-order methods will converge to sub-optimal solutions. In machine learning applications, sub-optimal solutions mean poor generalization performance. They are also related to the issue of hyper-parameter tuning, since, in the pursuit of solutions that yield lower errors, a tremendous amount of time is required on selecting the hyper-parameters appropriately. A natural way to tackle the limitations of first-order methods is to employ the Hessian information. However, methods that incorporate the Hessian do not scale or, if they do, they are very slow for modern applications. Here, we propose Simba, a scalable preconditioned gradient method, to address the main limitations of the first-order methods. The method is very simple to implement. It maintains a single precondition matrix that it is constructed as the outer product of the moving average of the gradients. To significantly reduce the computational cost of forming and inverting the preconditioner, we draw links with the multilevel optimization methods. These links enables us to construct preconditioners in a randomized manner. Our numerical experiments verify the scalability of Simba as well as its efficacy near saddles and flat areas. Further, we demonstrate that Simba offers a satisfactory generalization performance on standard benchmark residual networks. We also analyze Simba and show its linear convergence rate for strongly convex functions.

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arXiv2023

Using Witten Laplacians to locate index-1 saddle points

We introduce a new stochastic algorithm to locate the index-1 saddle points of a function $V:\mathbb R^d \to \mathbb R$, with $d$ possibly large. This algorithm can be seen as an equivalent of the stochastic gradient descent which is a natural stochastic process to locate local minima. It relies on two ingredients: (i) the concentration properties on index-1 saddle points of the first eigenmodes of the Witten Laplacian (associated with $V$) on $1$-forms and (ii) a probabilistic representation of a partial differential equation involving this differential operator. Numerical examples on simple molecular systems illustrate the efficacy of the proposed approach.

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arXiv2023

Privacy Risk for anisotropic Langevin dynamics using relative entropy bounds

The privacy preserving properties of Langevin dynamics with additive isotropic noise have been extensively studied. However, the isotropic noise assumption is very restrictive: (a) when adding noise to existing learning algorithms to preserve privacy and maintain the best possible accuracy one should take into account the relative magnitude of the outputs and their correlations; (b) popular algorithms such as stochastic gradient descent (and their continuous time limits) appear to possess anisotropic covariance properties. To study the privacy risks for the anisotropic noise case, one requires general results on the relative entropy between the laws of two Stochastic Differential Equations with different drifts and diffusion coefficients. Our main contribution is to establish such a bound using stability estimates for solutions to the Fokker-Planck equations via functional inequalities. With additional assumptions, the relative entropy bound implies an $(ε,δ)$-differential privacy bound or translates to bounds on the membership inference attack success and we show how anisotropic noise can lead to better privacy-accuracy trade-offs. Finally, the benefits of anisotropic noise are illustrated using numerical results in quadratic loss and neural network setups.

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arXiv2023

Adaptive Multilevel Newton: A Quadratically Convergent Optimization Method

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arXiv2022

Parameter Estimation for the McKean-Vlasov Stochastic Differential Equation

We consider the problem of parameter estimation for a stochastic McKean-Vlasov equation, and the associated system of weakly interacting particles. We study two cases: one in which we observe multiple independent trajectories of the McKean-Vlasov SDE, and another in which we observe multiple particles from the interacting particle system. In each case, we begin by establishing consistency and asymptotic normality of the (approximate) offline maximum likelihood estimator, in the limit as the number of observations $N\rightarrow\infty$. We then propose an online maximum likelihood estimator, which is based on a continuous-time stochastic gradient ascent scheme with respect to the asymptotic log-likelihood of the interacting particle system. We characterise the asymptotic behaviour of this estimator in the limit as $t\rightarrow\infty$, and also in the joint limit as $t\rightarrow\infty$ and $N\rightarrow\infty$. In these two cases, we obtain a.s. or $\mathbb{L}_1$ convergence to the stationary points of a limiting contrast function, under suitable conditions which guarantee ergodicity and uniform-in-time propagation of chaos. We also establish, under the additional condition of global strong concavity, $\mathbb{L}_2$ convergence to the unique maximiser of the asymptotic log-likelihood of the McKean-Vlasov SDE, with an asymptotic convergence rate which depends on the learning rate, the number of observations, and the dimension of the non-linear process. Our theoretical results are supported by two numerical examples, a linear mean field model and a stochastic opinion dynamics model.

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arXiv2022

Data-driven initialization of deep learning solvers for Hamilton-Jacobi-Bellman PDEs

A deep learning approach for the approximation of the Hamilton-Jacobi-Bellman partial differential equation (HJB PDE) associated to the Nonlinear Quadratic Regulator (NLQR) problem. A state-dependent Riccati equation control law is first used to generate a gradient-augmented synthetic dataset for supervised learning. The resulting model becomes a warm start for the minimization of a loss function based on the residual of the HJB PDE. The combination of supervised learning and residual minimization avoids spurious solutions and mitigate the data inefficiency of a supervised learning-only approach. Numerical tests validate the different advantages of the proposed methodology.

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arXiv2022

Uncertainty quantification for subgradient descent, with applications to relaxations of discrete problems

We consider the problem of minimizing a convex function that depends on an uncertain parameter $θ$. The uncertainty in the objective function means that the optimum, $x^*(θ)$, is also a function of $θ$. We propose an efficient method to compute $x^*(θ)$ and its statistics. We use a chaos expansion of $x^*(θ)$ along a truncated basis and study a restarted subgradient method that compute the optimal coefficients. We establish the convergence rate of the method as the number of basis functions increases, and hence the dimensionality of the optimization problem is increased. We give a non-asymptotic convergence rate for subgradient descent, building on earlier work that looked at gradient and accelerated gradient descent. Additionally, this work explicitly deals with the issue of projections, and suggests a method to deal with non-trivial projections. We show how this algorithm can be used to quantify uncertainty in discrete problems by utilising the (convex) Lovasz Extension for the min s,t-cut graph problem.

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arXiv2022

Fast Multilevel Algorithms for Compressive Principle Component Pursuit

Recovering a low-rank matrix from highly corrupted measurements arises in compressed sensing of structured high-dimensional signals (e.g., videos and hyperspectral images among others). Robust principal component analysis (RPCA), solved via principal component pursuit (PCP), recovers a low-rank matrix from sparse corruptions that are of unknown value and support by decomposing the observation matrix into two terms: a low-rank matrix and a sparse one, accounting for sparse noise and outliers. In the more general setting, where only a fraction of the data matrix has been observed, low-rank matrix recovery is achieved by solving the compressive principle component pursuit (CPCP). Both PCP and CPCP are well-studied convex programs, and numerous iterative algorithms have been proposed for their optimisation. Nevertheless, these algorithms involve singular value decomposition (SVD) at each iteration, which renders their applicability challenging in the case of massive data. In this paper, we propose a multilevel approach for the solution of PCP and CPCP problems. The core principle behind our algorithm is to apply SVD in models of lower-dimensionality than the original one and then lift its solution to the original problem dimension. We show that the proposed algorithms are easy to implement, converge at the same rate but with much lower iteration cost. Numerical experiments on numerous synthetic and real problems indicate that the proposed multilevel algorithms are several times faster than their original counterparts, namely PCP and CPCP.

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arXiv2022

Stochastic Mirror Descent for Convex Optimization with Consensus Constraints

The mirror descent algorithm is known to be effective in situations where it is beneficial to adapt the mirror map to the underlying geometry of the optimization model. However, the effect of mirror maps on the geometry of distributed optimization problems has not been previously addressed. In this paper we study an exact distributed mirror descent algorithm in continuous-time under additive noise. We establish a linear convergence rate of the proposed dynamics for the setting of convex optimization. Our analysis draws motivation from the Augmented Lagrangian and its relation to gradient tracking. To further explore the benefits of mirror maps in a distributed setting we present a preconditioned variant of our algorithm with an additional mirror map over the Lagrangian dual variables. This allows our method to adapt to both the geometry of the primal variables, as well as to the geometry of the consensus constraint. We also propose a Gauss-Seidel type discretization scheme for the proposed method and establish its linear convergence rate. For certain classes of problems we identify mirror maps that mitigate the effect of the graph's spectral properties on the convergence rate of the algorithm. Using numerical experiments we demonstrate the efficiency of the methodology on convex models, both with and without constraints. Our findings show that the proposed method outperforms other methods, especially in scenarios where the model's geometry is not captured by the standard Euclidean norm

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arXiv2021

Uncertainty Quantification for Gradient and Accelerated Gradient Descent Methods on Strongly Convex Functions

We consider the problem of minimizing a strongly convex function that depends on an uncertain parameter $θ$. The uncertainty in the objective function means that the optimum, $x^*(θ)$, is also a function of $θ$. We propose an efficient method to compute $x^*(θ)$ and its statistics. We use a chaos expansion of $x^*(θ)$ along a truncated basis and study first-order methods that compute the optimal coefficients. We establish the convergence rate of the method as the number of basis functions, and hence the dimensionality of the optimization problem is increased. We give the first non-asymptotic rates for the gradient descent and the accelerated gradient descent methods. Our analysis exploits convexity and does not rely on a diminishing step-size strategy. As a result, it is much faster than the state-of-the-art both in theory and in our preliminary numerical experiments. A surprising side-effect of our analysis is that the proposed method also acts as a variance reduction technique to the problem of estimating $x^*(θ)$.

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arXiv2021

A Multilevel Method for Self-Concordant Minimization

The analysis of second-order optimization methods based either on sub-sampling, randomization or sketching has two serious shortcomings compared to the conventional Newton method. The first shortcoming is that the analysis of the iterates has only been shown to be scale-invariant only under specific assumptions on the problem structure. The second shortfall is that the fast convergence rates of second-order methods have only been established by making assumptions regarding the input data. In this paper, we propose a randomized Newton method for self-concordant functions to address both shortfalls. We propose a Self-concordant Iterative-minimization-Galerkin-based Multilevel Algorithm (SIGMA) and establish its super-linear convergence rate using the theory of self-concordant functions. Our analysis is based on the connections between multigrid optimization methods, and the role of coarse-grained or reduced-order models in the computation of search directions. We take advantage of the insights from theanalysis to significantly improve the performance of second-order methods in machine learning applications. We report encouraging initial experiments that suggest SIGMA outperforms other state-of-the-art sub-sampled/sketched Newton methods for both medium and large-scale problems.

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arXiv2020

On stochastic mirror descent with interacting particles: convergence properties and variance reduction

An open problem in optimization with noisy information is the computation of an exact minimizer that is independent of the amount of noise. A standard practice in stochastic approximation algorithms is to use a decreasing step-size. This however leads to a slower convergence. A second alternative is to use a fixed step-size and run independent replicas of the algorithm and average these. A third option is to run replicas of the algorithm and allow them to interact. It is unclear which of these options works best. To address this question, we reduce the problem of the computation of an exact minimizer with noisy gradient information to the study of stochastic mirror descent with interacting particles. We study the convergence of stochastic mirror descent and make explicit the tradeoffs between communication and variance reduction. We provide theoretical and numerical evidence to suggest that interaction helps to improve convergence and reduce the variance of the estimate.

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