我们研究社会上公平$（\ ell_p，k）$的近似算法 - $ m $组的聚类问题，其特殊案例包括社会公平的$ k $ -Median（$ p = 1 $）和社会公平的$ k $ - 均值（$ p = 2 $）问题。我们提出（1）一个多项式时间$（5+2 \ sqrt {6}）^p $ - approximation，最多$ k+m $中心（2）a $（5+2 \ sqrt {6}+\ \ \ \ \ \ \ \ \ \ \ \ \ \\ epsilon）^p $ - approximation with $ k $中心$ n^{2^{o（p）} \ cdot m^2} $，和（3）a $（15+6 \ sqrt {6}） ^p $ k $中心的时间$ k^{m} \ cdot \ text {poly}（n）$。第一个结果是通过使用一系列线性程序的迭代圆形方法的细化来获得的。后两个结果是通过将最多$ K+M $中心的解决方案转换为使用（2）的稀疏方法的$ K $中心的解决方案，并通过详尽的搜索（3）。我们还将算法的性能与现有的双色算法以及基准数据集中的$ K $中心近似算法的恰好比较，并发现我们的算法在实践中也优于现有方法。

ML的梯度下降的成功尤其是学习神经网络是显着的和稳健的。在大脑如何学习的背景下，似乎在生物学上难以实现（如果不是难以判断）的梯度下降的一个方面是，其更新依赖于通过相同的连接到更早层的反馈。这种双向链路在脑网络中相对较少，即使存在互易连接时，它们也可能不等级。随机反馈对准（LillicRap等，2016），后向后重量是随机的和固定的，已经提出作为生物合理的替代品，并发现凭经验有效。我们调查如何以及当反馈对齐（FA）工作的方式，重点关注分层结构的最基本问题之一 - 低秩矩阵分解。在这个问题中，给定矩阵$ y_ {n \ times m} $，目标是找到低秩分解$ z_ {n \ times r} w_ {r \ times m} $，从而最小化错误$ \ | zw - 我\ | _f $。梯度血压最佳地解决了这个问题。我们显示FA收敛于当$ r \ ge \ mbox {rank}（y）$时收敛到最佳解决方案。我们还阐明了Fa工作的方式。经验上观察到前进权重矩阵和（随机）反馈矩阵在FA更新期间更接近。我们的分析严格地源地源于这种现象，并展示了如何促进FA的收敛。我们还表明，当$ r <\ mbox {rank}（y）$时，FA可能远非最佳。这是梯度下降和FA之间的第一个可提供的分离结果。此外，即使当它们的错误$ \ | zw-y \ | _f $大致相等时，梯度下降和fa发现的表示也可能是几乎正交的。

组件是大量的神经元，其同步射击被假设代表记忆，概念，单词和其他认知类别。据信，组件可以在高级认知现象和低级神经活动之间提供桥梁。最近，已经显示出一种称为“大会微积分（AC）”的计算系统，其曲目在集会上具有生物学上合理的操作，能够模拟由任意空间的计算模拟，但也可以模拟复杂的认知现象，例如语言，推理和规划和计划。但是，尚不清楚组装可以调解学习的机制。在这里，我们提出了这样一种机制，并严格证明，对于标记组件的分布定义的简单分类问题，可以可靠地形成代表每个类别的新组装，以响应类中的一些刺激。因此，该组件是对同一类的新刺激的响应可靠地召回的。此外，只要相应的类是相似组件的簇时，这些类组件就可以区分。为了证明这些结果，我们利用具有动态边缘权重的随机图理论来估计激活顶点的序列，在过去五年中对该领域的先前计算和定理产生了强烈的概括。这些定理通过实验证明了组件的成功形成，这些定理代表了从此类分布中绘制的合成数据以及MNIST上的概念类别的形成，并在MNIST上，这可以通过每个数字的一​​个组装来分类。该机制被视为一种学习算法，完全是在线上，从很少的样本中概括，只需要轻度的监督 - 在大脑模型中学习的所有关键属性。

Designing experiments often requires balancing between learning about the true treatment effects and earning from allocating more samples to the superior treatment. While optimal algorithms for the Multi-Armed Bandit Problem (MABP) provide allocation policies that optimally balance learning and earning, they tend to be computationally expensive. The Gittins Index (GI) is a solution to the MABP that can simultaneously attain optimality and computationally efficiency goals, and it has been recently used in experiments with Bernoulli and Gaussian rewards. For the first time, we present a modification of the GI rule that can be used in experiments with exponentially-distributed rewards. We report its performance in simulated 2- armed and 3-armed experiments. Compared to traditional non-adaptive designs, our novel GI modified design shows operating characteristics comparable in learning (e.g. statistical power) but substantially better in earning (e.g. direct benefits). This illustrates the potential that designs using a GI approach to allocate participants have to improve participant benefits, increase efficiencies, and reduce experimental costs in adaptive multi-armed experiments with exponential rewards.

Quadruped robots are currently used in industrial robotics as mechanical aid to automate several routine tasks. However, presently, the usage of such a robot in a domestic setting is still very much a part of the research. This paper discusses the understanding and virtual simulation of such a robot capable of detecting and understanding human emotions, generating its gait, and responding via sounds and expression on a screen. To this end, we use a combination of reinforcement learning and software engineering concepts to simulate a quadruped robot that can understand emotions, navigate through various terrains and detect sound sources, and respond to emotions using audio-visual feedback. This paper aims to establish the framework of simulating a quadruped robot that is emotionally intelligent and can primarily respond to audio-visual stimuli using motor or audio response. The emotion detection from the speech was not as performant as ERANNs or Zeta Policy learning, still managing an accuracy of 63.5%. The video emotion detection system produced results that are almost at par with the state of the art, with an accuracy of 99.66%. Due to its "on-policy" learning process, the PPO algorithm was extremely rapid to learn, allowing the simulated dog to demonstrate a remarkably seamless gait across the different cadences and variations. This enabled the quadruped robot to respond to generated stimuli, allowing us to conclude that it functions as predicted and satisfies the aim of this work.

Real-world robotic grasping can be done robustly if a complete 3D Point Cloud Data (PCD) of an object is available. However, in practice, PCDs are often incomplete when objects are viewed from few and sparse viewpoints before the grasping action, leading to the generation of wrong or inaccurate grasp poses. We propose a novel grasping strategy, named 3DSGrasp, that predicts the missing geometry from the partial PCD to produce reliable grasp poses. Our proposed PCD completion network is a Transformer-based encoder-decoder network with an Offset-Attention layer. Our network is inherently invariant to the object pose and point's permutation, which generates PCDs that are geometrically consistent and completed properly. Experiments on a wide range of partial PCD show that 3DSGrasp outperforms the best state-of-the-art method on PCD completion tasks and largely improves the grasping success rate in real-world scenarios. The code and dataset will be made available upon acceptance.

When robots learn reward functions using high capacity models that take raw state directly as input, they need to both learn a representation for what matters in the task -- the task ``features" -- as well as how to combine these features into a single objective. If they try to do both at once from input designed to teach the full reward function, it is easy to end up with a representation that contains spurious correlations in the data, which fails to generalize to new settings. Instead, our ultimate goal is to enable robots to identify and isolate the causal features that people actually care about and use when they represent states and behavior. Our idea is that we can tune into this representation by asking users what behaviors they consider similar: behaviors will be similar if the features that matter are similar, even if low-level behavior is different; conversely, behaviors will be different if even one of the features that matter differs. This, in turn, is what enables the robot to disambiguate between what needs to go into the representation versus what is spurious, as well as what aspects of behavior can be compressed together versus not. The notion of learning representations based on similarity has a nice parallel in contrastive learning, a self-supervised representation learning technique that maps visually similar data points to similar embeddings, where similarity is defined by a designer through data augmentation heuristics. By contrast, in order to learn the representations that people use, so we can learn their preferences and objectives, we use their definition of similarity. In simulation as well as in a user study, we show that learning through such similarity queries leads to representations that, while far from perfect, are indeed more generalizable than self-supervised and task-input alternatives.

and widely used information measurement metric, particularly popularized for SSVEP- based Brain-Computer (BCI) interfaces. By combining speed and accuracy into a single-valued parameter, this metric aids in the evaluation and comparison of various target identification algorithms across different BCI communities. To accurately depict performance and inspire an end-to-end design for futuristic BCI designs, a more thorough examination and definition of ITR is therefore required. We model the symbiotic communication medium, hosted by the retinogeniculate visual pathway, as a discrete memoryless channel and use the modified capacity expressions to redefine the ITR. We use graph theory to characterize the relationship between the asymmetry of the transition statistics and the ITR gain with the new definition, leading to potential bounds on data rate performance. On two well-known SSVEP datasets, we compared two cutting-edge target identification methods. Results indicate that the induced DM channel asymmetry has a greater impact on the actual perceived ITR than the change in input distribution. Moreover, it is demonstrated that the ITR gain under the new definition is inversely correlated with the asymmetry in the channel transition statistics. Individual input customizations are further shown to yield perceived ITR performance improvements. An algorithm is proposed to find the capacity of binary classification and further discussions are given to extend such results to ensemble techniques.We anticipate that the results of our study will contribute to the characterization of the highly dynamic BCI channel capacities, performance thresholds, and improved BCI stimulus designs for a tighter symbiosis between the human brain and computer systems while enhancing the efficiency of the underlying communication resources.

A step-search sequential quadratic programming method is proposed for solving nonlinear equality constrained stochastic optimization problems. It is assumed that constraint function values and derivatives are available, but only stochastic approximations of the objective function and its associated derivatives can be computed via inexact probabilistic zeroth- and first-order oracles. Under reasonable assumptions, a high-probability bound on the iteration complexity of the algorithm to approximate first-order stationarity is derived. Numerical results on standard nonlinear optimization test problems illustrate the advantages and limitations of our proposed method.

Differentiable Architecture Search (DARTS) has attracted considerable attention as a gradient-based Neural Architecture Search (NAS) method. Since the introduction of DARTS, there has been little work done on adapting the action space based on state-of-art architecture design principles for CNNs. In this work, we aim to address this gap by incrementally augmenting the DARTS search space with micro-design changes inspired by ConvNeXt and studying the trade-off between accuracy, evaluation layer count, and computational cost. To this end, we introduce the Pseudo-Inverted Bottleneck conv block intending to reduce the computational footprint of the inverted bottleneck block proposed in ConvNeXt. Our proposed architecture is much less sensitive to evaluation layer count and outperforms a DARTS network with similar size significantly, at layer counts as small as 2. Furthermore, with less layers, not only does it achieve higher accuracy with lower GMACs and parameter count, GradCAM comparisons show that our network is able to better detect distinctive features of target objects compared to DARTS.