Abstract: A general methodology is presented for optimizing a value at risk (VaR) associated with an allocation of objects (i.e., a strategy) having variable performance and loss characteristics. For purposes of illustration, investment strategies prescribing a portfolio of items from a set of candidates with unknown and generally correlated joint losses are discussed. The framework is based on approximating the VaR using nonparametric estimates of the portfolio loss density and, using mathematical insights, an efficient approach to computing the VaR gradient with respect to the strategy. The approach also allows inclusion of constraints on the strategy (e.g. a maximum fraction per item) and allows the VaR optimization problem to be solved using optimization techniques such as sequential quadratic programming.

Abstract: A computing device including a processor configured to receive an exact objective function over a state space. The processor may receive an approximated objective function that approximates the exact objective function. The processor may compute an estimated optimal state of the exact objective function. Computing the estimated optimal state may include, starting at an initial state, computing a preliminary estimated optimal state by performing a plurality of fast-step iterations of a Monte Carlo algorithm with respective fast-step acceptance probabilities determined based at least in part on the approximated objective function. Computing the estimated optimal state may further include performing a correction iteration that has a correction-step acceptance probability determined based at least in part on respective values of the approximated objective function and the exact objective function computed at the preliminary estimated optimal state. The processor may output the estimated optimal state.

Abstract: A method for use with a computing device. The method may include receiving a data set including a plurality of univariate data points and determining a target kernel bandwidth for a kernel density estimator (KDE). Determining the target kernel bandwidth may include computing a plurality of sample KDEs and selecting the target kernel bandwidth based on the sample KDEs. The method may further include computing the KDE for the data set using the target kernel bandwidth. For one or more tail regions of the data set, the method may further include computing one or more respective tail extensions. The method may further include computing and outputting a renormalized piecewise density estimator that, in each tail region, equals a renormalization of the respective tail extension for that tail region, and, outside the one or more tail regions, equals a renormalization of the KDE.

Abstract: A computing system, including a processor configured to train a machine learning model in a plurality of backpropagation iterations. Each backpropagation iteration may include generating a coordinate pair sequence. Each coordinate pair may be unique within the coordinate pair sequence and may include non-matching coordinates. The backpropagation iteration may further include receiving parametrizing angles respectively associated with the coordinate pairs. The backpropagation iteration may further include computing a unitary matrix parametrized by the parametrizing angles, computing a loss gradient matrix, and computing a Jacobian-vector product (JVP). Computing the JVP may include computing a rotated unitary matrix and a rotated loss gradient matrix for each coordinate pair. The JVP may be computed from the rotated unitary matrix and the rotated loss gradient matrix. The backpropagation iteration may further include updating the parametrizing angles based at least in part on the JVP.

Abstract: The systems, devices, articles, and methods generally relate to sampling from an available probability distribution. The samples may be used to create a desirable probability distribution, for instance for use in computing values used in computational techniques including: Importance Sampling and Markov chain Monte Carlo systems. An analog processor may operate as a sample generator, for example by: programming the analog processor with a configuration of the number of programmable parameters for the analog processor, which corresponds to a probability distribution over qubits of the analog processor, evolving the analog processor, and reading out states for the qubits. The states for the qubits in the plurality of qubits correspond to a sample from the probability distribution. Operation of the sampling device may be summarized as including updating a set of samples to include the sample from the probability distribution, and returning the set of samples.

Abstract: A method for use with a computing device. The method may include receiving a data set including a plurality of univariate data points and determining a target kernel bandwidth for a kernel density estimator (KDE). Determining the target kernel bandwidth may include computing a plurality of sample KDEs and selecting the target kernel bandwidth based on the sample KDEs. The method may further include computing the KDE for the data set using the target kernel bandwidth. For one or more tail regions of the data set, the method may further include computing one or more respective tail extensions. The method may further include computing and outputting a renormalized piecewise density estimator that, in each tail region, equals a renormalization of the respective tail extension for that tail region, and, outside the one or more tail regions, equals a renormalization of the KDE.

Abstract: The systems, devices, articles, and methods generally relate to sampling from an available probability distribution. The samples maybe used to create a desirable probability distribution, for instance for use in computing values used in computational techniques including: Importance Sampling and Markov chain Monte Carlo systems. An analog processor may operate as a sample generator, for example by: programming the analog processor with a configuration of the number of programmable parameters for the analog processor, which corresponds to a probability distribution over qubits of the analog processor, evolving the analog processor, and reading out states for the qubits. The states for the qubits in the plurality of qubits correspond to a sample from the probability distribution. Operation of the sampling device may be summarized as including updating a set of samples to include the sample from the probability distribution, and returning the set of samples.

Abstract: A hybrid computer comprising a quantum processor can be operated to perform a scalable comparison of high-entropy samplers. Performing a scalable comparison of high-entropy samplers can include comparing entropy and KL divergence of post-processed samplers. A hybrid computer comprising a quantum processor generates samples for machine learning. The quantum processor is trained by matching data statistics to statistics of the quantum processor. The quantum processor is tuned to match moments of the data.

Abstract: A hybrid computer comprising a quantum processor can be operated to perform a scalable comparison of high-entropy samplers. Performing a scalable comparison of high-entropy samplers can include comparing entropy and KL divergence of post-processed samplers. A hybrid computer comprising a quantum processor generates samples for machine learning. The quantum processor is trained by matching data statistics to statistics of the quantum processor. The quantum processor is tuned to match moments of the data.

Abstract: Quantum processor based techniques minimize an objective function for example by operating the quantum processor as a sample generator providing low-energy samples from a probability distribution with high probability. The probability distribution is shaped to assign relative probabilities to samples based on their corresponding objective function values until the samples converge on a minimum for the objective function. Problems having a number of variables and/or a connectivity between variables that does not match that of the quantum processor may be solved. Interaction with the quantum processor may be via a digital computer. The digital computer stores a hierarchical stack of software modules to facilitate interacting with the quantum processor via various levels of programming environment, from a machine language level up to an end-use applications level.

Abstract: Computational systems implement problem solving using heuristic solvers or optimizers. Such may iteratively evaluate a result of processing, and modify the problem or representation thereof before repeating processing on the modified problem, until a termination condition is reached. Heuristic solvers or optimizers may execute on one or more digital processors and/or one or more quantum processors. The system may autonomously select between types of hardware devices and/or types of heuristic optimization algorithms. Such may coordinate or at least partially overlap post-processing operations with processing operations, for instance performing post-processing on an ith batch of samples while generating an (i+1)th batch of samples, e.g., so post-processing operation on the ith batch of samples does not extend in time beyond the generation of the (i+1)th batch of samples. Heuristic optimizers selection is based on pre-processing assessment of the problem, e.g.

Abstract: Computational systems implement problem solving using heuristic solvers or optimizers. Such may iteratively evaluate a result of processing, and modify the problem or representation thereof before repeating processing on the modified problem, until a termination condition is reached. Heuristic solvers or optimizers may execute on one or more digital processors and/or one or more quantum processors. The system may autonomously select between types of hardware devices and/or types of heuristic optimization algorithms. Such may coordinate or at least partially overlap post-processing operations with processing operations, for instance performing post-processing on an ith batch of samples while generating an (i+1)th batch of samples, e.g., so post-processing operation on the ith batch of samples does not extend in time beyond the generation of the (i+1)th batch of samples. Heuristic optimizers selection is based on pre-processing assessment of the problem, e.g.

Abstract: A hybrid computer comprising a quantum processor can be operated to perform a scalable comparison of high-entropy samplers. Performing a scalable comparison of high-entropy samplers can include comparing entropy and KL divergence of post-processed samplers. A hybrid computer comprising a quantum processor generates samples for machine learning. The quantum processor is trained by matching data statistics to statistics of the quantum processor. The quantum processor is tuned to match moments of the data.

Abstract: The systems, devices, articles, and methods generally relate to sampling from an available probability distribution. The samples maybe used to create a desirable probability distribution, for instance for use in computing values used in computational techniques including: Importance Sampling and Markov chain Monte Carlo systems. An analog processor may operate as a sample generator, for example by: programming the analog processor with a configuration of the number of programmable parameters for the analog processor, which corresponds to a probability distribution over qubits of the analog processor, evolving the analog processor, and reading out states for the qubits. The states for the qubits in the plurality of qubits correspond to a sample from the probability distribution. Operation of the sampling device may be summarized as including updating a set of samples to include the sample from the probability distribution, and returning the set of samples.

Abstract: The systems, devices, articles, and methods generally relate to sampling from an available probability distribution. The samples maybe used to create a desirable probability distribution, for instance for use in computing values used in computational techniques including: Importance Sampling and Markov chain Monte Carlo systems. An analog processor may operate as a sample generator, for example by: programming the analog processor with a configuration of the number of programmable parameters for the analog processor, which corresponds to a probability distribution over qubits of the analog processor, evolving the analog processor, and reading out states for the qubits. The states for the qubits in the plurality of qubits correspond to a sample from the probability distribution. Operation of the sampling device may be summarized as including updating a set of samples to include the sample from the probability distribution, and returning the set of samples.

Abstract: Quantum processor based techniques minimize an objective function for example by operating the quantum processor as a sample generator providing low-energy samples from a probability distribution with high probability. The probability distribution is shaped to assign relative probabilities to samples based on their corresponding objective function values until the samples converge on a minimum for the objective function. Problems having a number of variables and/or a connectivity between variables that does not match that of the quantum processor may be solved. Interaction with the quantum processor may be via a digital computer. The digital computer stores a hierarchical stack of software modules to facilitate interacting with the quantum processor via various levels of programming environment, from a machine language level up to an end-use applications level.

Abstract: Quantum processor based techniques minimize an objective function for example by operating the quantum processor as a sample generator providing low-energy samples from a probability distribution with high probability. The probability distribution is shaped to assign relative probabilities to samples based on their corresponding objective function values until the samples converge on a minimum for the objective function. Problems having a number of variables and/or a connectivity between variables that does not match that of the quantum processor may be solved. Interaction with the quantum processor may be via a digital computer. The digital computer stores a hierarchical stack of software modules to facilitate interacting with the quantum processor via various levels of programming environment, from a machine language level up to an end-use applications level.

Abstract: The systems, devices, articles, and methods generally relate to sampling from an available probability distribution. The samples maybe used to create a desirable probability distribution, for instance for use in computing values used in computational techniques including: Importance Sampling and Markov chain Monte Carlo systems. An analog processor may operate as a sample generator, for example by: programming the analog processor with a configuration of the number of programmable parameters for the analog processor, which corresponds to a probability distribution over qubits of the analog processor, evolving the analog processor, and reading out states for the qubits. The states for the qubits in the plurality of qubits correspond to a sample from the probability distribution. Operation of the sampling device may be summarized as including updating a set of samples to include the sample from the probability distribution, and returning the set of samples.

Abstract: The techniques and structures described herein generally relate to sampling from an available probability distribution to create a desirable probability distribution. This resultant distribution can be used for computing values used in computational techniques including: Importance Sampling and Markov chain Monte Carlo systems.

Abstract: Quantum processor based techniques minimize an objective function for example by operating the quantum processor as a sample generator providing low-energy samples from a probability distribution with high probability. The probability distribution is shaped to assign relative probabilities to samples based on their corresponding objective function values until the samples converge on a minimum for the objective function. Problems having a number of variables and/or a connectivity between variables that does not match that of the quantum processor may be solved. Interaction with the quantum processor may be via a digital computer. The digital computer stores a hierarchical stack of software modules to facilitate interacting with the quantum processor via various levels of programming environment, from a machine language level up to an end-use applications level.