Several widely-used first-order saddle-point optimization methods yield an identical continuous-time ordinary differential equation (ODE) that is identical to that of the Gradient Descent Ascent (GDA) method when derived naively. However, the convergence properties of these methods are qualitatively different, even on simple bilinear games. Thus the ODE perspective, which has proved powerful in analyzing single-objective optimization methods, has not played a similar role in saddle-point optimization. We adopt a framework studied in fluid dynamics -- known as High-Resolution Differential Equations (HRDEs) -- to design differential equation models for several saddle-point optimization methods. Critically, these HRDEs are distinct for various saddle-point optimization methods. Moreover, in bilinear games, the convergence properties of the HRDEs match the qualitative features of the corresponding discrete methods. Additionally, we show that the HRDE of Optimistic Gradient Descent Ascent (OGDA) exhibits \emph{last-iterate convergence} for general monotone variational inequalities. Finally, we provide rates of convergence for the \emph{best-iterate convergence} of the OGDA method, relying solely on the first-order smoothness of the monotone operator.
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Consider an online convex optimization problem where the loss functions are self-concordant barriers, smooth relative to a convex function $h$, and possibly non-Lipschitz. We analyze the regret of online mirror descent with $h$. Then, based on the result, we prove the following in a unified manner. Denote by $T$ the time horizon and $d$ the parameter dimension. 1. For online portfolio selection, the regret of $\widetilde{\text{EG}}$, a variant of exponentiated gradient due to Helmbold et al., is $\tilde{O} ( T^{2/3} d^{1/3} )$ when $T > 4 d / \log d$. This improves on the original $\tilde{O} ( T^{3/4} d^{1/2} )$ regret bound for $\widetilde{\text{EG}}$. 2. For online portfolio selection, the regret of online mirror descent with the logarithmic barrier is $\tilde{O}(\sqrt{T d})$. The regret bound is the same as that of Soft-Bayes due to Orseau et al. up to logarithmic terms. 3. For online learning quantum states with the logarithmic loss, the regret of online mirror descent with the log-determinant function is also $\tilde{O} ( \sqrt{T d} )$. Its per-iteration time is shorter than all existing algorithms we know.
Computer simulations have become essential for analyzing complex systems, but high-fidelity simulations often come with significant computational costs. To tackle this challenge, multi-fidelity computer experiments have emerged as a promising approach that leverages both low-fidelity and high-fidelity simulations, enhancing both the accuracy and efficiency of the analysis. In this paper, we introduce a new and flexible statistical model, the Recursive Non-Additive (RNA) emulator, that integrates the data from multi-fidelity computer experiments. Unlike conventional multi-fidelity emulation approaches that rely on an additive auto-regressive structure, the proposed RNA emulator recursively captures the relationships between multi-fidelity data using Gaussian process priors without making the additive assumption, allowing the model to accommodate more complex data patterns. Importantly, we derive the posterior predictive mean and variance of the emulator, which can be efficiently computed in a closed-form manner, leading to significant improvements in computational efficiency. Additionally, based on this emulator, we introduce three active learning strategies that optimize the balance between accuracy and simulation costs to guide the selection of the fidelity level and input locations for the next simulation run. We demonstrate the effectiveness of the proposed approach in a suite of synthetic examples and a real-world problem. An R package for the proposed methodology is provided in an open repository.
We demonstrate the first algorithms for the problem of regression for generalized linear models (GLMs) in the presence of additive oblivious noise. We assume we have sample access to examples $(x, y)$ where $y$ is a noisy measurement of $g(w^* \cdot x)$. In particular, \new{the noisy labels are of the form} $y = g(w^* \cdot x) + \xi + \epsilon$, where $\xi$ is the oblivious noise drawn independently of $x$ \new{and satisfies} $\Pr[\xi = 0] \geq o(1)$, and $\epsilon \sim \mathcal N(0, \sigma^2)$. Our goal is to accurately recover a \new{parameter vector $w$ such that the} function $g(w \cdot x)$ \new{has} arbitrarily small error when compared to the true values $g(w^* \cdot x)$, rather than the noisy measurements $y$. We present an algorithm that tackles \new{this} problem in its most general distribution-independent setting, where the solution may not \new{even} be identifiable. \new{Our} algorithm returns \new{an accurate estimate of} the solution if it is identifiable, and otherwise returns a small list of candidates, one of which is close to the true solution. Furthermore, we \new{provide} a necessary and sufficient condition for identifiability, which holds in broad settings. \new{Specifically,} the problem is identifiable when the quantile at which $\xi + \epsilon = 0$ is known, or when the family of hypotheses does not contain candidates that are nearly equal to a translated $g(w^* \cdot x) + A$ for some real number $A$, while also having large error when compared to $g(w^* \cdot x)$. This is the first \new{algorithmic} result for GLM regression \new{with oblivious noise} which can handle more than half the samples being arbitrarily corrupted. Prior work focused largely on the setting of linear regression, and gave algorithms under restrictive assumptions.
Numerically solving partial differential equations typically requires fine discretization to resolve necessary spatiotemporal scales, which can be computationally expensive. Recent advances in deep learning have provided a new approach to solving partial differential equations that involves the use of neural operators. Neural operators are neural network architectures that learn mappings between function spaces and have the capability to solve partial differential equations based on data. This study utilizes a novel neural operator called Hyena, which employs a long convolutional filter that is parameterized by a multilayer perceptron. The Hyena operator is an operation that enjoys sub-quadratic complexity and state space model to parameterize long convolution that enjoys a global receptive field. This mechanism enhances the model's comprehension of the input's context and enables data-dependent weight for different partial differential equations instances. To measure how effective the layers are in solving partial differential equations, we conduct experiments on Diffusion-Reaction equation and Navier Stokes equation. Our findings indicate Hyena Neural operator can serve as an efficient and accurate model for learning partial differential equations solution operator. The data and code used can be found at: //github.com/Saupatil07/Hyena-Neural-Operator
Realistic reservoir simulation is known to be prohibitively expensive in terms of computation time when increasing the accuracy of the simulation or by enlarging the model grid size. One method to address this issue is to parallelize the computation by dividing the model in several partitions and using multiple CPUs to compute the result using techniques such as MPI and multi-threading. Alternatively, GPUs are also a good candidate to accelerate the computation due to their massively parallel architecture that allows many floating point operations per second to be performed. The numerical iterative solver takes thus the most computational time and is challenging to solve efficiently due to the dependencies that exist in the model between cells. In this work, we evaluate the OPM Flow simulator and compare several state-of-the-art GPU solver libraries as well as custom developed solutions for a BiCGStab solver using an ILU0 preconditioner and benchmark their performance against the default DUNE library implementation running on multiple CPU processors using MPI. The evaluated GPU software libraries include a manual linear solver in OpenCL and the integration of several third party sparse linear algebra libraries, such as cuSparse, rocSparse, and amgcl. To perform our bench-marking, we use small, medium, and large use cases, starting with the public test case NORNE that includes approximately 50k active cells and ending with a large model that includes approximately 1 million active cells. We find that a GPU can accelerate a single dual-threaded MPI process up to 5.6 times, and that it can compare with around 8 dual-threaded MPI processes.
Predictive algorithms are often trained by optimizing some loss function, to which regularization functions are added to impose a penalty for violating constraints. As expected, the addition of such regularization functions can change the minimizer of the objective. It is not well-understood which regularizers change the minimizer of the loss, and, when the minimizer does change, how it changes. We use property elicitation to take first steps towards understanding the joint relationship between the loss and regularization functions and the optimal decision for a given problem instance. In particular, we give a necessary and sufficient condition on loss and regularizer pairs for when a property changes with the addition of the regularizer, and examine some regularizers satisfying this condition standard in the fair machine learning literature. We empirically demonstrate how algorithmic decision-making changes as a function of both data distribution changes and hardness of the constraints.
Exhibiting an explicit Boolean function with a large high-order nonlinearity is an important problem in cryptography, coding theory, and computational complexity. We prove lower bounds on the second-order, third-order, and higher-order nonlinearities of some trace monomial Boolean functions. We prove lower bounds on the second-order nonlinearities of functions $\mathrm{tr}_n(x^7)$ and $\mathrm{tr}_n(x^{2^r+3})$ where $n=2r$. Among all trace monomials, our bounds match the best second-order nonlinearity lower bounds by \cite{Car08} and \cite{YT20} for odd and even $n$ respectively. We prove a lower bound on the third-order nonlinearity for functions $\mathrm{tr}_n(x^{15})$, which is the best third-order nonlinearity lower bound. For any $r$, we prove that the $r$-th order nonlinearity of $\mathrm{tr}_n(x^{2^{r+1}-1})$ is at least $2^{n-1}-2^{(1-2^{-r})n+\frac{r}{2^{r-1}}-1}- O(2^{\frac{n}{2}})$. For $r \ll \log_2 n$, this is the best lower bound among all explicit functions.
We propose and analyze the application of statistical functional depth metrics for the selection of extreme scenarios in day-ahead grid planning. Our primary motivation is screening of probabilistic scenarios for realized load and renewable generation, in order to identify scenarios most relevant for operational risk mitigation. To handle the high-dimensionality of the scenarios across asset classes and intra-day periods, we employ functional measures of depth to sub-select outlying scenarios that are most likely to be the riskiest for the grid operation. We investigate a range of functional depth measures, as well as a range of operational risks, including load shedding, operational costs, reserves shortfall and variable renewable energy curtailment. The effectiveness of the proposed screening approach is demonstrated through a case study on the realistic Texas-7k grid.
Ordered sequences of data, specified with a join operation to combine sequences, serve as a foundation for the implementation of parallel functional algorithms. This abstract data type can be elegantly and efficiently implemented using balanced binary trees, where a join operation is provided to combine two trees and rebalance as necessary. In this work, we present a verified implementation and cost analysis of joinable red-black trees in $\textbf{calf}$, a dependent type theory for cost analysis. We implement red-black trees and auxiliary intermediate data structures in such a way that all correctness invariants are intrinsically maintained. Then, we describe and verify precise cost bounds on the operations, making use of the red-black tree invariants. Finally, we implement standard algorithms on sequences using the simple join-based signature and bound their cost in the case that red-black trees are used as the underlying implementation. All proofs are formally mechanized using the embedding of $\textbf{calf}$ in the Agda theorem prover.
Classical gradient-based density topology optimization is adapted for method-of-moments numerical modeling to design a conductor-based system attaining the minimal antenna Q-factor evaluated via an energy stored operator. Standard topology optimization features are discussed, e.g., the interpolation scheme and density and projection filtering. The performance of the proposed technique is demonstrated in a few examples in terms of the realized Q-factor values and necessary computational time to obtain a design. The optimized designs are compared to the fundamental bound and well-known empirical structures. The presented framework can provide a completely novel design, as presented in the second example.
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