This paper proposes a generalized version of SO(3)-equivariant architectures including Spherical CNN. By utilizing the algebraic properties of Fourier transform and the tools in non-commutative harmonic analysis, the authors are able to construct (and prove) a most generalized version of SO(3)-equivariant architecture. Specifically, it only requires that, when an input image is rotated, each fragment (i.e., the output, Fourier coefficient vectors) of each layer will be multiplied by a Wigner-D matrix. To include non-linearities without performing inverse Fourier transform, the authors propose to use Clebsch-Gordon transformation. The experiments show that the proposed CG-Net can outperform Spherical CNN in several tasks.

Chung's lemma is a classical tool for establishing asymptotic convergence rates of (stochastic) optimization methods under strong convexity-type assumptions and appropriate polynomial diminishing step sizes. In this work, we develop a generalized version of Chung's lemma, which provides a simple non-asymptotic convergence framework for a more general family of step size rules. We demonstrate broad applicability of the proposed generalized Chung's lemma by deriving tight non-asymptotic convergence rates for a large variety of stochastic methods. In particular, we obtain partially new non-asymptotic complexity results for stochastic optimization methods, such as stochastic gradient descent and random reshuffling, under a general $(\theta,\mu)$-Polyak-Lojasiewicz (PL) condition and for various step sizes strategies, including polynomial, constant, exponential, and cosine step sizes rules. Notably, as a by-product of our analysis, we observe that exponential step sizes can adapt to the objective function's geometry, achieving the optimal convergence rate without requiring exact knowledge of the underlying landscape. Our results demonstrate that the developed variant of Chung's lemma offers a versatile, systematic, and streamlined approach to establish non-asymptotic convergence rates under general step size rules.

A classic setting of the stochastic K-armed bandit problem is considered in this note. In this problem it has been known that KL-UCB policy achieves the asymptotically optimal regret bound and KL-UCB policy empirically performs better than the KL-UCB policy although the regret bound for the original form of the KL-UCB policy has been unknown. This note demonstrates that a simple proof of the asymptotic optimality of the KL-UCB policy can be given by the same technique as those used for analyses of other known policies. In the problem of the stochastic bandit problems, it is known that there exists a (problem-dependent) regret lower bound [1][2]. It can be achieved by, for example, the DMED policy [3] for the model of nonparametric distributions over [0, 1]. One of the conference version [6] of [5] also proposed KL-UCB policy, which empirically performs better than KL-UCB but does not have a theoretical guarantee.

Having the right assortment of shipping boxes in the fulfillment warehouse to pack and ship customer's online orders is an indispensable and integral part of nowadays eCommerce business, as it will not only help maintain a profitable business but also create great experiences for customers. However, it is an extremely challenging operations task to strategically select the best combination of tens of box sizes from thousands of feasible ones to be responsible for hundreds of thousands of orders daily placed on millions of inventory products. In this paper, we present a machine learning approach to tackle the task by formulating the box design problem prescriptively as a generalized version of weighted k-medoids clustering problem, where the parameters are estimated through a variety of descriptive analytics. We test this machine learning approach on fulfillment data collected from Walmart U.S. eCommerce, and our approach is shown to be capable of improving the box utilization rate by more than 10%. Keywords: Shipping box design, k-medoids clustering, eCommerce, packaging science, operations research 1 Introduction The assortment of shipping boxes utilized by the fulfillment warehouse to pack and ship customer's online orders is a critical component of nowadays eCommerce business, as it will directly affect not only profit margins but also customer's experience.

A key strategy for speeding up computation is to run in parallel on multiple cores. However, on hard combinatorial problems, exploiting parallelism has been surprisingly challenging. It appears that traditional divide-and-conquer strategies do not work well, due to the intricate non-local nature of the interactions between the problem variables. In this paper, we introduce a novel way in which parallelism can be used to exploit hidden structure of hard combinatorial problems. We demonstrate the success of this approach on minimal set basis problem, which has a wide range of applications in machine learning and system security, etc. We also show the effectiveness on a related application problem from materials discovery. In our approach, a large number of smaller sub-problems are identified and solved concurrently. We then aggregate the information from those solutions, and use this to initialize the search of a global, complete solver. We show that this strategy leads to a significant speed-up over a sequential approach. The strategy also greatly outperforms state-of-the-art incomplete solvers in terms of solution quality. Our work opens up a novel angle for using parallelism to solve hard combinatorial problems.

In this paper there is proposed a generalized version of the SVM for binary classification problems in the case of using an arbitrary transformation x -> y. An approach similar to the classic SVM method is used. The problem is widely explained. Various formulations of primal and dual problems are proposed. For one of the most important cases the formulae are derived in detail. A simple computational example is demonstrated. The algorithm and its implementation is presented in Octave language.

Of the common commutative binary logical connectives, only and and or may be used as operators that take arbitrary numbers of arguments with order and multiplicity being irrelevant, that is, as connectives that take sets of arguments. This is especially evident in the Common Logic Interchange Format, in which it is easy for operators to be given arbitrary numbers of arguments. The reason is that and and or are associative and idempotent, as well as commutative. We extend the ability of taking sets of arguments to the other common commutative connectives by defining generalized versions of nand , nor , xor ,and iff , as well as the additional, parameterized connectives andor and thresh . We prove that andor is expressively complete — all the other connectives may be considered abbreviations of it.