Regularizing continual learning techniques is important for anticipating algorithmic behavior under new realizations of data. We introduce a new approach to continual learning by imposing the properties of a parabolic partial differential equation (PDE) to regularize the expected behavior of the loss over time. This class of parabolic PDEs has a number of favorable properties that allow us to analyze the error incurred through forgetting and the error induced through generalization. Specifically, we do this through imposing boundary conditions where the boundary is given by a memory buffer. By using the memory buffer as a boundary, we can enforce long term dependencies by bounding the expected error by the boundary loss. Finally, we illustrate the empirical performance of the method on a series of continual learning tasks.

Regularizing neural networks is important for anticipating model behavior in regions of the data space that are not well represented. In this work, we propose a regularization technique for enforcing a level of smoothness in the mapping between the input space and the loss. We specify the level of regularity by requiring that the loss of the network satisfies an elliptic operator over the data domain. To do this, we modify the usual empirical risk minimization objective such that we instead minimize a new objective that satisfies an elliptic operator over points within the domain. This allows us to use existing theory on elliptic operators to anticipate the behavior of the error for points outside the training set. We propose a tractable computational method that approximates the behavior of the elliptic operator while being computationally efficient. Finally, we analyze the properties of the proposed regularization to understand the performance on common problems of distribution shift and group imbalance. Numerical experiments empirically confirm the promise of the proposed regularization technique.

One of the major challenges in estimating conditional potential outcomes and conditional average treatment effects (CATE) is the presence of hidden confounders. Since testing for hidden confounders cannot be accomplished only with observational data, conditional unconfoundedness is commonly assumed in the literature of CATE estimation. Nevertheless, under this assumption, CATE estimation can be significantly biased due to the effects of unobserved confounders. In this work, we consider the case where in addition to a potentially large observational dataset, a small dataset from a randomized controlled trial (RCT) is available. Notably, we make no assumptions on the existence of any covariate information for the RCT dataset, we only require the outcomes to be observed. We propose a CATE estimation method based on a pseudo-confounder generator and a CATE model that aligns the learned potential outcomes from the observational data with those observed from the RCT. Our method is applicable to many practical scenarios of interest, particularly those where privacy is a concern (e.g., medical applications). Extensive numerical experiments are provided demonstrating the effectiveness of our approach for both synthetic and real-world datasets.

Predictive models are often deployed for decision-making tasks for which they were not explicitly trained. When only partial observations of the relevant state are available, as in most real-world applications, there is a strong possibility of hidden confounding. Therefore, partial observability often makes the outcome of an action unidentifiable, and could render a model’s predictions unreliable for action planning. We present an identification bound and propose an algorithm to account for hidden confounding during model-predictive control. To that end, we introduce a generalized causal sensitivity model for action-state dynamics. We place a constraint on the hidden confounding between trajectories of future actions and states, enabling sharp bounds on interventional outcomes. Unlike previous sensitivity models, ours accommodates hidden confounding with memory, while maintaining computational and statistical tractability. We benchmark on a wide variety of multivariate stochastic differential equations with arbitrary confounding. The results suggest that a calibrated sensitivity model helps controllers achieve higher rewards.

McKean-Vlasov stochastic differential equations (MV-SDEs) provide a mathematical description of the behavior of an infinite number of interacting particles by imposing a dependence on the particle density. We study the influence of explicitly including distributional information in the parameterization of the SDE. We propose a series of semi-parametric methods for representing MV-SDEs, and corresponding estimators for inferring parameters from data based on the properties of the MV-SDE. We analyze the characteristics of the different architectures and estimators, and consider their applicability in relevant machine learning problems. We empirically compare the performance of the different architectures and estimators on real and synthetic datasets for time series and probabilistic modeling. The results suggest that explicitly including distributional dependence in the parameterization of the SDE is effective in modeling temporal data with interaction under an exchangeability assumption while maintaining strong performance for standard Itô-SDEs due to the richer class of probability flows associated with MV-SDEs.

The goal of this paper is to develop distributionally robust optimization (DRO) estimators, specifically for multidimensional Extreme Value Theory (EVT) statistics. EVT supports using semi-parametric models called max-stable distributions built from spatial Poisson point processes. While powerful, these models are only asymptotically valid for large samples. However, since extreme data is by definition scarce, the potential for model misspecification error is inherent to these applications, thus DRO estimators are natural. In order to mitigate over-conservative estimates while enhancing out-of-sample performance, we study DRO estimators informed by semi-parametric max-stable constraints in the space of point processes. We study both tractable convex formulations for some problems of interest (e.g. CVaR) and more general neural network based estimators. Both approaches are validated using synthetically generated data, recovering prescribed characteristics, and verifying the efficacy of the proposed techniques. Additionally, the proposed method is applied to a real data set of financial returns for comparison to a previous analysis. We established the proposed model as a novel formulation in the multivariate EVT domain, and innovative with respect to performance when compared to relevant alternate proposals.

Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often necessary to compute the solutions or a function of the solutions to a parametric PDE in multiple scenarios corresponding to different parameters of this PDE. This process often requires resolving the PDEs from scratch, which is time-consuming. To better employ existing simulations for the PDEs, we propose a framework for finding solutions to parabolic PDEs across different scenarios by meta-learning an underlying base distribution.%tasks. We build upon this base distribution to propose a method for computing solutions to parametric PDEs under different parameter settings. Finally, we illustrate the application of the proposed methods through extensive experiments in generative modeling, stochastic control, and finance. The empirical results suggest that the proposed approach improves generalization to solving new PDEs.

Point processes often have a natural interpretation with respect to a continuous process. We propose a point process construction that describes arrival time observations in terms of the state of a latent diffusion process. In this framework, we relate the return times of a diffusion in a continuous path space to new arrivals of the point process. This leads to a continuous sample path that is used to describe the underlying mechanism generating the arrival distribution. These models arise in many disciplines, such as financial settings where actions in a market are determined by a hidden continuous price or in neuroscience where a latent stimulus generates spike trains. Based on the developments in Itô’s excursion theory, we propose methods for inferring and sampling from the point process derived from the latent diffusion process. We illustrate the approach with numerical examples using both simulated and real data. The proposed methods and framework provide a basis for interpreting point processes through the lens of diffusions.

Understanding multivariate dependencies in both the bulk and the tails of a distribution is an important problem for many applications, such as ensuring algorithms are robust to observations that are infrequent but have devastating effects. Archimax copulas are a family of distributions endowed with a precise representation that allows simultaneous modeling of the bulk and the tails of a distribution. Rather than separating the two as is typically done in practice, incorporating additional information from the bulk may improve inference of the tails, where observations are limited. Building on the stochastic representation of Archimax copulas, we develop a non-parametric inference method and sampling algorithm. Our proposed methods, to the best of our knowledge, are the first that allow for highly flexible and scalable inference and sampling algorithms, enabling the increased use of Archimax copulas in practical settings. We experimentally compare to state-of-the-art density modeling techniques, and the results suggest that the proposed method effectively extrapolates to the tails while scaling to higher dimensional data. Our findings suggest that the proposed algorithms can be used in a variety of applications where understanding the interplay between the bulk and the tails of a distribution is necessary, such as healthcare and safety.

We propose a neural network architecture that enables non-parametric calibration and generation of multivariate extreme value distributions (MEVs). MEVs arise from Extreme Value Theory (EVT) as the necessary class of models when extrapolating a distributional fit over large spatial and temporal scales based on data observed in intermediate scales. In turn, EVT dictates that $d$-max-decreasing, a stronger form of convexity, is an essential shape constraint in the characterization of MEVs. As far as we know, our proposed architecture provides the first class of non-parametric estimators for MEVs that preserve these essential shape constraints. We show that the architecture approximates the dependence structure encoded by MEVs at parametric rate. Moreover, we present a new method for sampling high-dimensional MEVs using a generative model. We demonstrate our methodology on a wide range of experimental settings, ranging from environmental sciences to financial mathematics and verify that the structural properties of MEVs are retained compared to existing methods.

Renewable energy or alternative energy is extracted through renewable resources. These are considered as an alternative from conventional fossil fuel-based sources because conventional energy sources are depleting rapidly and raised concerns over increasing environmental impacts. Among many renewable sources, solar energy has a substantial part to meet the increased energy demand with reduced environmental effects. Solar irradiance and temperature are key factors upon which photovoltaic (PV) power generation depends but its optimum operating point gets affected by variation in the above-mentioned environmental factors. Finding the optimum operating point is a challenge due to the nonlinear solar behavior and varying nature of environmental conditions. To overcome these challenges, maximum power point (MPP) searching algorithms are exploited to get optimum power from the PV energy system. Maximum power point tracking (MPPT) behavior is different for various weather conditions, for instance, partial shading (PS), and uniform irradiance (UI) conditions. Numerous MPPT methods came to be used to find the optimum power. This work deals with the development of a novel technique for MPP finding of a PV system on the basis of the Water Cycle Algorithm (WCA) under PS conditions. It turns out to be good in terms of exploration and exploitation. Thus, it has the capability to avoid getting stuck in local minima (LM) and to find the global maxima (GM). The performance of the WCA technique is examined on four different types of P-V patterns for UI, PS and fast changing environmental conditions through MATLAB simulation and experimental setup. The findings of WCA are compared with the previous well-known soft computing methods such as PSO, ACS, DFO, and conventional method P&O to evaluate performance. The outcomes reveal that the WCA algorithm overtakes P&O from the perspective of robustness, accuracy, efficiency, and stability, as well as PSO in respect of converging speed and efficiency.

It has been conjectured that the Fisher divergence is more robust to model uncertainty than the conventional Kullback-Leibler (KL) divergence. This motivates the design of a new class of robust generative auto-encoders (AE) referred to as Fisher auto-encoders. Our approach is to design Fisher AEs by minimizing the Fisher divergence between the intractable joint distribution of observed data and latent variables, with that of the postulated/modeled joint distribution. In contrast to KL-based variational AEs (VAEs), the Fisher AE can exactly quantify the distance between the true and the model-based posterior distributions. Qualitative and quantitative results are provided on both MNIST and celebA datasets demonstrating the competitive performance of Fisher AEs in terms of robustness compared to other AEs such as VAEs and Wasserstein AEs.

We propose a new generative modeling technique for learning multidimensional cumulative distribution functions (CDFs) in the form of copulas. Specifically, we consider certain classes of copulas known as Archimedean and hierarchical Archimedean copulas, popular for their parsimonious representation and ability to model different tail dependencies. We consider their representation as mixture models with Laplace transforms of latent random variables from generative neural networks. This alternative representation allows for computational efficiencies and easy sampling, especially in high dimensions. We describe multiple methods for optimizing the network parameters. Finally, we present empirical results that demonstrate the efficacy of our proposed method in learning multidimensional CDFs and its computational efficiency compared to existing methods.

We develop a framework for estimating unknown partial differential equations (PDEs) from noisy data, using a deep learning approach. Given noisy samples of a solution to an unknown PDE, our method interpolates the samples using a neural network, and extracts the PDE by equating derivatives of the neural network approximation. Our method applies to PDEs which are linear combinations of user-defined dictionary functions, and generalizes previous methods that only consider parabolic PDEs. We introduce a regularization scheme that prevents the function approximation from overfitting the data and forces it to be a solution of the underlying PDE. We validate the model on simulated data generated by the known PDEs and added Gaussian noise, and we study our method under different levels of noise. We also compare the error of our method with a Cramer-Rao lower bound for an ordinary differential equation (ODE). Our results indicate that our method outperforms other methods in estimating PDEs, especially in the low signal-to-noise (SNR) regime.