The Shapley value (SV) is a prevalent approach of allocating credit to machine learning (ML) entities to understand black box ML models. Enriching such interpretations with higher-order interactions is inevitable for complex systems, where the Shapley Interaction Index (SII) is a direct axiomatic extension of the SV. While it is well-known that the SV yields an optimal approximation of any game via a weighted least square (WLS) objective, an extension of this result to SII has been a long-standing open problem, which even led to the proposal of an alternative index. In this work, we characterize higher-order SII as a solution to a WLS problem, which constructs an optimal approximation via SII and $k$-Shapley values ($k$-SII). We prove this representation for the SV and pairwise SII and give empirically validated conjectures for higher orders. As a result, we propose KernelSHAP-IQ, a direct extension of KernelSHAP for SII, and demonstrate state-of-the-art performance for feature interactions.

Accurate control of robots at high speeds requires a control system that can take into account the kinodynamic interactions of the robot with the environment. Prior works on learning inverse kinodynamic (IKD) models of robots have shown success in capturing the complex kinodynamic effects. However, the types of control problems these approaches can be applied to are limited only to that of following pre-computed kinodynamically feasible trajectories. In this paper we present Optim-FKD, a new formulation for accurate, high-speed robot control that makes use of a learned forward kinodynamic (FKD) model and non-linear least squares optimization. Optim-FKD can be used for accurate, high speed control on any control task specifiable by a non-linear least squares objective. Optim-FKD can solve for control objectives such as path following and time-optimal control in real time, without needing access to pre-computed kinodynamically feasible trajectories. We empirically demonstrate these abilities of our approach through experiments on a scale one-tenth autonomous car. Our results show that Optim-FKD can follow desired trajectories more accurately and can find better solutions to optimal control problems than baseline approaches.

A Bayesian approach termed BAyesian Least Squares Optimization with Nonnegative L1-norm constraint (BALSON) is proposed. The error distribution of data fitting is described by Gaussian likelihood. The parameter distribution is assumed to be a Dirichlet distribution. With the Bayes rule, searching for the optimal parameters is equivalent to finding the mode of the posterior distribution. In order to explicitly characterize the nonnegative L1-norm constraint of the parameters, we further approximate the true posterior distribution by a Dirichlet distribution. We estimate the statistics of the approximating Dirichlet posterior distribution by sampling methods. Four sampling methods have been introduced. With the estimated posterior distributions, the original parameters can be effectively reconstructed in polynomial fitting problems, and the BALSON framework is found to perform better than conventional methods.

Vector quantization aims to form new vectors/matrices with shared values close to the original. It could compress data with acceptable information loss, and could be of great usefulness in areas like Image Processing, Pattern Recognition and Machine Learning. In recent years, the importance of quantization has been soaring as it has been discovered huge potentials in deploying practical neural networks, which is among one of the most popular research topics. Conventional vector quantization methods usually suffer from their own flaws: hand-coding domain rules quantization could produce poor results when encountering complex data, and clustering-based algorithms have the problem of inexact solution and high time consumption. In this paper, we explored vector quantization problem from a new perspective of sparse least square optimization and designed multiple algorithms with their program implementations. Specifically, deriving from a sparse form of coefficient matrix, three types of sparse least squares, with $l_0$, $l_1$, and generalized $l_1 + l_2$ penalizations, are designed and implemented respectively. In addition, to produce quantization results with given amount of quantized values(instead of penalization coefficient $\lambda$), this paper proposed a cluster-based least square quantization method, which could also be regarded as an improvement of information preservation of conventional clustering algorithm. The algorithms were tested on various data and tasks and their computational properties were analyzed. The paper offers a new perspective to probe the area of vector quantization, while the algorithms proposed could provide more appropriate options for quantization tasks under different circumstances.