Ariel Neufeld

Ariel Neufeld, Tuan Anh Nguyen, Philipp Schmocker

We introduce multilevel Picard (MLP) approximations for McKean--Vlasov stochastic differential equations (SDEs) with nonconstant diffusion coefficient. Under standard Lipschitz assumptions on the coefficients, we show that the MLP algorithm approximates the solution of the SDE in the $L^2$-sense without the curse of dimensionality. The latter means that its computational cost grows at most polynomially in both the dimension and the reciprocal of the prescribed error tolerance. In two numerical experiments, we demonstrate its applicability by approximating McKean--Vlasov SDEs in dimensions up to 1000.

Arnulf Jentzen, Benno Kuckuck, Ariel Neufeld et al.

Stochastic gradient descent (SGD) optimization algorithms are key ingredients in a series of machine learning applications. In this article we perform a rigorous strong error analysis for SGD optimization algorithms. In particular, we prove for every arbitrarily small $\varepsilon \in (0,\infty)$ and every arbitrarily large $p\in (0,\infty)$ that the considered SGD optimization algorithm converges in the strong $L^p$-sense with order $\frac{1}{2}-\varepsilon$ to the global minimum of the objective function of the considered stochastic approximation problem under standard convexity-type assumptions on the objective function and relaxed assumptions on the moments of the stochastic errors appearing in the employed SGD optimization algorithm. The key ideas in our convergence proof are, first, to employ techniques from the theory of Lyapunov-type functions for dynamical systems to develop a general convergence machinery for SGD optimization algorithms based on such functions, then, to apply this general machinery to concrete Lyapunov-type functions with polynomial structures, and, thereafter, to perform an induction argument along the powers appearing in the Lyapunov-type functions in order to achieve for every arbitrarily large $ p \in (0,\infty) $ strong $ L^p $-convergence rates. This article also contains an extensive review of results on SGD optimization algorithms in the scientific literature.

Dong-Young Lim, Ariel Neufeld, Sotirios Sabanis et al.

We introduce a new Langevin dynamics based algorithm, called e-TH$\varepsilon$O POULA, to solve optimization problems with discontinuous stochastic gradients which naturally appear in real-world applications such as quantile estimation, vector quantization, CVaR minimization, and regularized optimization problems involving ReLU neural networks. We demonstrate both theoretically and numerically the applicability of the e-TH$\varepsilon$O POULA algorithm. More precisely, under the conditions that the stochastic gradient is locally Lipschitz in average and satisfies a certain convexity at infinity condition, we establish non-asymptotic error bounds for e-TH$\varepsilon$O POULA in Wasserstein distances and provide a non-asymptotic estimate for the expected excess risk, which can be controlled to be arbitrarily small. Three key applications in finance and insurance are provided, namely, multi-period portfolio optimization, transfer learning in multi-period portfolio optimization, and insurance claim prediction, which involve neural networks with (Leaky)-ReLU activation functions. Numerical experiments conducted using real-world datasets illustrate the superior empirical performance of e-TH$\varepsilon$O POULA compared to SGLD, TUSLA, ADAM, and AMSGrad in terms of model accuracy.

Ariel Neufeld, Philipp Schmocker

In this paper, we derive an $L^p$-chaos expansion based on iterated Stratonovich integrals with respect to a given exponentially integrable continuous semimartingale. By omitting the orthogonality of the expansion, we show that every $p$-integrable functional, $p \in [1,\infty)$, can be approximated by a finite sum of iterated Stratonovich integrals. Using (possibly random) neural networks as integrands, we therefere obtain universal approximation results for $p$-integrable financial derivatives in the $L^p$-sense. Moreover, we can approximately solve the $L^p$-hedging problem (coinciding for $p = 2$ with the quadratic hedging problem), where the approximating hedging strategy can be computed in closed form within short runtime.

Ariel Neufeld, Julian Sester, Mario Šikić

We introduce a general framework for Markov decision problems under model uncertainty in a discrete-time infinite horizon setting. By providing a dynamic programming principle we obtain a local-to-global paradigm, namely solving a local, i.e., a one time-step robust optimization problem leads to an optimizer of the global (i.e. infinite time-steps) robust stochastic optimal control problem, as well as to a corresponding worst-case measure. Moreover, we apply this framework to portfolio optimization involving data of the S&P 500. We present two different types of ambiguity sets; one is fully data-driven given by a Wasserstein-ball around the empirical measure, the second one is described by a parametric set of multivariate normal distributions, where the corresponding uncertainty sets of the parameters are estimated from the data. It turns out that in scenarios where the market is volatile or bearish, the optimal portfolio strategies from the corresponding robust optimization problem outperforms the ones without model uncertainty, showcasing the importance of taking model uncertainty into account.

Jianjun Chen, Yongming Li, Ariel Neufeld

In this paper we provide a quantum Monte Carlo algorithm to solve multidimensional Black-Scholes PDEs with correlation for option pricing. The payoff function of the option is of general form and is only required to be continuous and piecewise affine, which covers most of the relevant payoff functions used in finance. We provide a rigorous error analysis and complexity analysis of our algorithm. In particular, we prove that the computational complexity of our algorithm is bounded polynomially in the space dimension $d$ of the PDE and the reciprocal of the prescribed accuracy $\varepsilon$. Moreover, we show that for payoff functions which are bounded, our algorithm indeed has a speed-up compared to classical Monte Carlo methods. Furthermore, we provide numerical simulations in two dimensions using our developed package within the Qiskit framework tailored to price continuous piecewise affine options with respect to the Black-Scholes model, as well as discuss the potential extension of the numerical simulations to arbitrary space dimension.

Ariel Neufeld, Julian Sester, Daiying Yin

We present an approach, based on deep neural networks, that allows identifying robust statistical arbitrage strategies in financial markets. Robust statistical arbitrage strategies refer to trading strategies that enable profitable trading under model ambiguity. The presented novel methodology allows to consider a large amount of underlying securities simultaneously and does not depend on the identification of cointegrated pairs of assets, hence it is applicable on high-dimensional financial markets or in markets where classical pairs trading approaches fail. Moreover, we provide a method to build an ambiguity set of admissible probability measures that can be derived from observed market data. Thus, the approach can be considered as being model-free and entirely data-driven. We showcase the applicability of our method by providing empirical investigations with highly profitable trading performances even in 50 dimensions, during financial crises, and when the cointegration relationship between asset pairs stops to persist.

Ariel Neufeld, Julian Sester

In this paper we demonstrate both theoretically as well as numerically that neural networks can detect model-free static arbitrage opportunities whenever the market admits some. Due to the use of neural networks, our method can be applied to financial markets with a high number of traded securities and ensures almost immediate execution of the corresponding trading strategies. To demonstrate its tractability, effectiveness, and robustness we provide examples using real financial data. From a technical point of view, we prove that a single neural network can approximately solve a class of convex semi-infinite programs, which is the key result in order to derive our theoretical results that neural networks can detect model-free static arbitrage strategies whenever the financial market admits such opportunities.

Zeyi Chen, Ariel Neufeld, Qikun Xiang

We develop an estimator-based stochastic fixed-point framework for approximately computing the 2-Wasserstein barycenter of continuous, non-parametric probability measures. Notably, we provide the first rigorous convergence analysis for implementable estimator-based stochastic extensions of the fixed-point iterative scheme proposed by Álvarez-Esteban, del Barrio, Cuesta-Albertos, and Matrán (2016). In particular, we establish almost sure convergence, and identify sufficient conditions for geometric rates of convergence under controlled errors in optimal transport (OT) map estimation. We subsequently propose a concrete, provably convergent, and computationally tractable stochastic algorithm that accommodates input measures satisfying Caffarelli-type regularity conditions, which form a dense subset of the Wasserstein space. This algorithm leverages a modified entropic OT map estimator to enable efficient and scalable implementation. To facilitate quantitative evaluation, we further propose a novel and efficient procedure for synthetically generating benchmark instances, in which the input measures exhibit non-trivial features and the corresponding barycenters are approximately known. Numerical experiments on both synthetic and real-world datasets demonstrate the strong computational efficiency, estimation accuracy, and sampling flexibility of our approach.

Daniel Bartl, Ariel Neufeld, Kyunghyun Park

We examine nonlinear Kolmogorov partial differential equations (PDEs). Here the nonlinear part of the PDE comes from its Hamiltonian where one maximizes over all possible drift and diffusion coefficients which fall within a $\varepsilon$-neighborhood of pre-specified baseline coefficients. Our goal is to quantify and compute how sensitive those PDEs are to such a small nonlinearity, and then use the results to develop an efficient numerical method for their approximation. We show that as $\varepsilon\downarrow 0$, the nonlinear Kolmogorov PDE equals the linear Kolmogorov PDE defined with respect to the corresponding baseline coefficients plus $\varepsilon$ times a correction term which can be also characterized by the solution of another linear Kolmogorov PDE involving the baseline coefficients. As these linear Kolmogorov PDEs can be efficiently solved in high-dimensions by exploiting their Feynman-Kac representation, our derived sensitivity analysis then provides a Monte Carlo based numerical method which can efficiently solve these nonlinear Kolmogorov equations. We establish an error and complexity analysis for our numerical method. Moreover, we provide numerical examples in up to 100 dimensions to empirically demonstrate the applicability of our numerical method.

Ariel Neufeld, Tuan Anh Nguyen

We prove that multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation are capable of approximating solutions of semilinear Kolmogorov PDEs in $L^\mathfrak{p}$-sense, $\mathfrak{p}\in [2,\infty)$, in the case of gradient-independent, Lipschitz-continuous nonlinearities, while the computational effort of the multilevel Picard approximations and the required number of parameters in the neural networks grow at most polynomially in both dimension $d\in \mathbb{N}$ and reciprocal of the prescribed accuracy $ε$.

Luxu Liang, Ariel Neufeld, Ying Zhang

In this paper, we provide a non-asymptotic analysis of the convergence of the stochastic gradient Hamiltonian Monte Carlo (SGHMC) algorithm to a target measure in Wasserstein-1 and Wasserstein-2 distance. Crucially, compared to the existing literature on SGHMC, we allow its stochastic gradient to be discontinuous. This allows us to provide explicit upper bounds, which can be controlled to be arbitrarily small, for the expected excess risk of non-convex stochastic optimization problems with discontinuous stochastic gradients, including, among others, the training of neural networks with ReLU activation function. To illustrate the applicability of our main results, we consider numerical experiments on quantile estimation and on several optimization problems involving ReLU neural networks relevant in finance and artificial intelligence.

Ariel Neufeld, Philipp Schmocker, Sizhou Wu

In this paper, we present a randomized extension of the deep splitting algorithm introduced in [Beck, Becker, Cheridito, Jentzen, and Neufeld (2021)] using random neural networks suitable to approximately solve both high-dimensional nonlinear parabolic PDEs and PIDEs with jumps having (possibly) infinite activity. We provide a full error analysis of our so-called random deep splitting method. In particular, we prove that our random deep splitting method converges to the (unique viscosity) solution of the nonlinear PDE or PIDE under consideration. Moreover, we empirically analyze our random deep splitting method by considering several numerical examples including both nonlinear PDEs and nonlinear PIDEs relevant in the context of pricing of financial derivatives under default risk. In particular, we empirically demonstrate in all examples that our random deep splitting method can approximately solve nonlinear PDEs and PIDEs in 10'000 dimensions within seconds.

Ariel Neufeld, Philipp Schmocker

We introduce a Banach space-valued extension of random feature learning, a data-driven supervised machine learning technique for large-scale kernel approximation. By randomly initializing the feature maps, only the linear readout needs to be trained, which reduces the computational complexity substantially. Viewing random feature models as Banach space-valued random variables, we prove a universal approximation result in the corresponding Bochner space. Moreover, we derive approximation rates and an explicit algorithm to learn an element of the given Banach space by such models. The framework of this paper includes random trigonometric/Fourier regression and in particular random neural networks which are single-hidden-layer feedforward neural networks whose weights and biases are randomly initialized, whence only the linear readout needs to be trained. For the latter, we can then lift the universal approximation property of deterministic neural networks to random neural networks, even within function spaces over non-compact domains, e.g., weighted spaces, $L^p$-spaces, and (weighted) Sobolev spaces, where the latter includes the approximation of the (weak) derivatives. In addition, we analyze when the training costs for approximating a given function grow polynomially in both the input/output dimension and the reciprocal of a pre-specified tolerated approximation error. Furthermore, we demonstrate in a numerical example the empirical advantages of random feature models over their deterministic counterparts.

Ariel Neufeld, Philipp Schmocker

This paper extends the universal approximation property of single-hidden-layer feedforward neural networks beyond compact domains, which is of particular interest for the approximation within weighted $C^k$-spaces and weighted Sobolev spaces over unbounded domains. More precisely, by assuming that the activation function is non-polynomial, we establish universal approximation results within function spaces defined over non-compact subsets of a Euclidean space, including $L^p$-spaces, weighted $C^k$-spaces, and weighted Sobolev spaces, where the latter two include the approximation of the (weak) derivatives. Moreover, we provide some dimension-independent rates for approximating a function with sufficiently regular and integrable Fourier transform by neural networks with non-polynomial activation function.

Anastasis Kratsios, Ariel Neufeld, Philipp Schmocker · eth-zurich

Neural operators (NOs) are a class of deep learning models designed to simultaneously solve infinitely many related problems by casting them into an infinite-dimensional space, whereon these NOs operate. A significant gap remains between theory and practice: worst-case parameter bounds from universal approximation theorems suggest that NOs may require an unrealistically large number of parameters to solve most operator learning problems, which stands in direct opposition to a slew of experimental evidence. This paper closes that gap for a specific class of {NOs}, generative {equilibrium operators} (GEOs), using (realistic) finite-dimensional deep equilibrium layers, when solving families of convex optimization problems over a separable Hilbert space $X$. Here, the inputs are smooth, convex loss functions on $X$, and outputs are the associated (approximate) solutions to the optimization problem defined by each input loss. We show that when the input losses lie in suitable infinite-dimensional compact sets, our GEO can uniformly approximate the corresponding solutions to arbitrary precision, with rank, depth, and width growing only logarithmically in the reciprocal of the approximation error. We then validate both our theoretical results and the trainability of GEOs on three applications: (1) nonlinear PDEs, (2) stochastic optimal control problems, and (3) hedging problems in mathematical finance under liquidity constraints.

Ariel Neufeld, Philipp Schmocker

In this paper, we solve stochastic partial differential equations (SPDEs) numerically by using (possibly random) neural networks in the truncated Wiener chaos expansion of their corresponding solution. Moreover, we provide some approximation rates for learning the solution of SPDEs with additive and/or multiplicative noise. Finally, we apply our results in numerical examples to approximate the solution of three SPDEs: the stochastic heat equation, the Heath-Jarrow-Morton equation, and the Zakai equation.

Hari Hara Suthan Chittoor, Paul Robert Griffin, Ariel Neufeld et al.

Long-term time series forecasting (LTSF) involves predicting a large number of future values of a time series based on the past values. This is an essential task in a wide range of domains including weather forecasting, stock market analysis and disease outbreak prediction. Over the decades LTSF algorithms have transitioned from statistical models to deep learning models like transformer models. Despite the complex architecture of transformer based LTSF models `Are Transformers Effective for Time Series Forecasting? (Zeng et al., 2023)' showed that simple linear models can outperform the state-of-the-art transformer based LTSF models. Recently, quantum machine learning (QML) is evolving as a domain to enhance the capabilities of classical machine learning models. In this paper we initiate the application of QML to LTSF problems by proposing QuLTSF, a simple hybrid QML model for multivariate LTSF. Through extensive experiments on a widely used weather dataset we show the advantages of QuLTSF over the state-of-the-art classical linear models, in terms of reduced mean squared error and mean absolute error.

Ariel Neufeld, Philipp Schmocker, Viet Khoa Tran

Quantum neural networks (QNNs) are an analog of classical neural networks in the world of quantum computing, which are represented by a unitary matrix with trainable parameters. Inspired by the universal approximation property of classical neural networks, ensuring that every continuous function can be arbitrarily well approximated uniformly on a compact set of a Euclidean space, some recent works have established analogous results for QNNs, ranging from single-qubit to multi-qubit QNNs, and even hybrid classical-quantum models. In this paper, we study the approximation capabilities of QNNs for periodic functions with respect to the supremum norm. We use the Jackson inequality to approximate a given function by implementing its approximating trigonometric polynomial via a suitable QNN. In particular, we see that by restricting to the class of periodic functions, one can achieve a quadratic reduction of the number of parameters, producing better approximation results than in the literature. Moreover, the smoother the function, the fewer parameters are needed to construct a QNN to approximate the function.

Ariel Neufeld, Julian Sester

We present a novel $Q$-learning algorithm tailored to solve distributionally robust Markov decision problems where the corresponding ambiguity set of transition probabilities for the underlying Markov decision process is a Wasserstein ball around a (possibly estimated) reference measure. We prove convergence of the presented algorithm and provide several examples also using real data to illustrate both the tractability of our algorithm as well as the benefits of considering distributional robustness when solving stochastic optimal control problems, in particular when the estimated distributions turn out to be misspecified in practice.

Dong-Young Lim, Ariel Neufeld, Sotirios Sabanis et al.

We consider non-convex stochastic optimization problems where the objective functions have super-linearly growing and discontinuous stochastic gradients. In such a setting, we provide a non-asymptotic analysis for the tamed unadjusted stochastic Langevin algorithm (TUSLA) introduced in Lovas et al. (2020). In particular, we establish non-asymptotic error bounds for the TUSLA algorithm in Wasserstein-1 and Wasserstein-2 distances. The latter result enables us to further derive non-asymptotic estimates for the expected excess risk. To illustrate the applicability of the main results, we consider an example from transfer learning with ReLU neural networks, which represents a key paradigm in machine learning. Numerical experiments are presented for the aforementioned example which support our theoretical findings. Hence, in this setting, we demonstrate both theoretically and numerically that the TUSLA algorithm can solve the optimization problem involving neural networks with ReLU activation function. Besides, we provide simulation results for synthetic examples where popular algorithms, e.g. ADAM, AMSGrad, RMSProp, and (vanilla) stochastic gradient descent (SGD) algorithm, may fail to find the minimizer of the objective functions due to the super-linear growth and the discontinuity of the corresponding stochastic gradient, while the TUSLA algorithm converges rapidly to the optimal solution. Moreover, we provide an empirical comparison of the performance of TUSLA with popular stochastic optimizers on real-world datasets, as well as investigate the effect of the key hyperparameters of TUSLA on its performance.

Ariel Neufeld, Julian Sester

We introduce a novel and highly tractable supervised learning approach based on neural networks that can be applied for the computation of model-free price bounds of, potentially high-dimensional, financial derivatives and for the determination of optimal hedging strategies attaining these bounds. In particular, our methodology allows to train a single neural network offline and then to use it online for the fast determination of model-free price bounds of a whole class of financial derivatives with current market data. We show the applicability of this approach and highlight its accuracy in several examples involving real market data. Further, we show how a neural network can be trained to solve martingale optimal transport problems involving fixed marginal distributions instead of financial market data.

Christian Beck, Sebastian Becker, Patrick Cheridito et al.

In this article, we introduce and analyze a deep learning based approximation algorithm for SPDEs. Our approach employs neural networks to approximate the solutions of SPDEs along given realizations of the driving noise process. If applied to a set of simulated noise trajectories, it yields empirical distributions of SPDE solutions, from which functionals like the mean and variance can be estimated. We test the performance of the method on stochastic heat equations with additive and multiplicative noise as well as stochastic Black-Scholes equations with multiplicative noise and Zakai equations from nonlinear filtering theory. In all cases, the proposed algorithm yields accurate results with short runtimes in up to 100 space dimensions.

Pushpendu Ghosh, Ariel Neufeld, Jajati Keshari Sahoo

We employ both random forests and LSTM networks (more precisely CuDNNLSTM) as training methodologies to analyze their effectiveness in forecasting out-of-sample directional movements of constituent stocks of the S&P 500 from January 1993 till December 2018 for intraday trading. We introduce a multi-feature setting consisting not only of the returns with respect to the closing prices, but also with respect to the opening prices and intraday returns. As trading strategy, we use Krauss et al. (2017) and Fischer & Krauss (2018) as benchmark. On each trading day, we buy the 10 stocks with the highest probability and sell short the 10 stocks with the lowest probability to outperform the market in terms of intraday returns -- all with equal monetary weight. Our empirical results show that the multi-feature setting provides a daily return, prior to transaction costs, of 0.64% using LSTM networks, and 0.54% using random forests. Hence we outperform the single-feature setting in Fischer & Krauss (2018) and Krauss et al. (2017) consisting only of the daily returns with respect to the closing prices, having corresponding daily returns of 0.41% and of 0.39% with respect to LSTM and random forests, respectively.

Christian Beck, Sebastian Becker, Patrick Cheridito et al.

In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handle extremely high-dimensional PDEs. We test the method on different examples from physics, stochastic control and mathematical finance. In all cases, it yields very good results in up to 10,000 dimensions with short run times.

Dominik Alfke, Weston Baines, Jan Blechschmidt et al.

We present a novel technique based on deep learning and set theory which yields exceptional classification and prediction results. Having access to a sufficiently large amount of labelled training data, our methodology is capable of predicting the labels of the test data almost always even if the training data is entirely unrelated to the test data. In other words, we prove in a specific setting that as long as one has access to enough data points, the quality of the data is irrelevant.