The weights w11, w12,... w23 and biases b1 & b2 can also be represented as matrices, initialized as random values. The first row of w and the first element of b are used to predict the first target variable, i.e., the yield of apples, and similarly, the second for the yield of oranges. Our model is simply a function that performs a matrix multiplication of the inputs and the weights w (transposed) and adds the bias b for each observation. Let's create a function called model that will predict output by calculating the above operation when the inputs is passed as a parameter: Now, let's compare the predictions with the actual targets: As you can see, the difference between our model's predictions and the actual target values is vast. So, we need to improve our model to reduce the difference.

This paper proposes a confidence interval construction for heterogeneous treatment effects in the context of multi-stage experiments with $N$ samples and high-dimensional, $d$, confounders. Our focus is on the case of $d\gg N$, but the results obtained also apply to low-dimensional cases. We showcase that the bias of regularized estimation, unavoidable in high-dimensional covariate spaces, is mitigated with a simple double-robust score. In this way, no additional bias removal is necessary, and we obtain root-$N$ inference results while allowing multi-stage interdependency of the treatments and covariates. Memoryless property is also not assumed; treatment can possibly depend on all previous treatment assignments and all previous multi-stage confounders. Our results rely on certain sparsity assumptions of the underlying dependencies. We discover new product rate conditions necessary for robust inference with dynamic treatments.

Greedy algorithms have long been a workhorse for learning graphical models, and more broadly for learning statistical models with sparse structure. In the context of learning directed acyclic graphs, greedy algorithms are popular despite their worst-case exponential runtime. In practice, however, they are very efficient. We provide new insight into this phenomenon by studying a general greedy score-based algorithm for learning DAGs. Unlike edge-greedy algorithms such as the popular GES and hill-climbing algorithms, our approach is vertex-greedy and requires at most a polynomial number of score evaluations. We then show how recent polynomial-time algorithms for learning DAG models are a special case of this algorithm, thereby illustrating how these order-based algorithms can be rigourously interpreted as score-based algorithms. This observation suggests new score functions and optimality conditions based on the duality between Bregman divergences and exponential families, which we explore in detail. Explicit sample and computational complexity bounds are derived. Finally, we provide extensive experiments suggesting that this algorithm indeed optimizes the score in a variety of settings.

False data injection attacks (FDIAs) represent a major class of attacks that aim to break the integrity of measurements by injecting false data into the smart metering devices in power grids. To the best of authors' knowledge, no study has attempted to design a detector that automatically models the underlying graph topology and spatially correlated measurement data of the smart grids to better detect cyber attacks. The contributions of this paper to detect and mitigate FDIAs are twofold. First, we present a generic, localized, and stealth (unobservable) attack generation methodology and publicly accessible datasets for researchers to develop and test their algorithms. Second, we propose a Graph Neural Network (GNN) based, scalable and real-time detector of FDIAs that efficiently combines model-driven and data-driven approaches by incorporating the inherent physical connections of modern AC power grids and exploiting the spatial correlations of the measurement. It is experimentally verified by comparing the proposed GNN based detector with the currently available FDIA detectors in the literature that our algorithm outperforms the best available solutions by 3.14%, 4.25%, and 4.41% in F1 score for standard IEEE testbeds with 14, 118, and 300 buses, respectively.

The use of contrastive loss for representation learning has become prominent in computer vision, and it is now getting attention in Natural Language Processing (NLP). Here, we explore the idea of using a batch-softmax contrastive loss when fine-tuning large-scale pre-trained transformer models to learn better task-specific sentence embeddings for pairwise sentence scoring tasks. We introduce and study a number of variations in the calculation of the loss as well as in the overall training procedure; in particular, we find that data shuffling can be quite important. Our experimental results show sizable improvements on a number of datasets and pairwise sentence scoring tasks including classification, ranking, and regression. Finally, we offer detailed analysis and discussion, which should be useful for researchers aiming to explore the utility of contrastive loss in NLP. Recent years have seen a revolution in Natural Language Processing (NLP) thanks to the advances in machine learning. While a lot of attention has been paid to the architectures, especially for deep learning, there has been less focus on studying loss functions. At the same time, loss functions based on similar or on the same ideas were reinvented multiple times under different names. This can cause difficulties when solving new problems or when designing new experiments based on previous results. To a greater extent, this applies to "universal" loss functions, which can be applied in different machine learning areas and tasks such as Computer Vision (CV), Recommendation Systems, and NLP.

In contrast to single-objective optimization (SOO), multi-objective optimization (MOO) requires an optimizer to find the Pareto frontier, a subset of feasible solutions that are not dominated by other feasible solutions. In this paper, we propose LaMOO, a novel multi-objective optimizer that learns a model from observed samples to partition the search space and then focus on promising regions that are likely to contain a subset of the Pareto frontier. The partitioning is based on the dominance number, which measures "how close" a data point is to the Pareto frontier among existing samples. To account for possible partition errors due to limited samples and model mismatch, we leverage Monte Carlo Tree Search (MCTS) to exploit promising regions while exploring suboptimal regions that may turn out to contain good solutions later. Theoretically, we prove the efficacy of learning space partitioning via LaMOO under certain assumptions. Empirically, on the HyperVolume (HV) benchmark, a popular MOO metric, LaMOO substantially outperforms strong baselines on multiple real-world MOO tasks, by up to 225% in sample efficiency for neural architecture search on Nasbench201, and up to 10% for molecular design. Multi-objective optimization (MOO) has been extensively used in many practical scenarios involving trade-offs between multiple objectives. For example, in automobile design (Chang, 2015), we must maximize the performance of the engine while simultaneously minimizing emissions and fuel consumption.

Self-supervised learning recently emerges as a promising direction to learn representations without manual labels. While contrastive learning (Oord et al., 2018; Tian et al., 2019; Bachman et al., 2019; He et al., 2020; Chen et al., 2020a) minimizes the distance of representation between positive pairs, and maximizes such distances between negative pairs, recently, non-contrastive self-supervised learning (abbreviated as nc-SSL) is able to learn nontrivial representation with only positive pairs, using an extra predictor and a stop-gradient operation. Furthermore, the learned representation shows comparable (or even better) performance for downstream tasks (e.g., image classification) (Grill et al., 2020; Chen & He, 2020). This brings about two fundamental questions: (1) why the learned representation does not collapse to trivial (i.e., constant) solutions, and (2) without negative pairs, what representation nc-SSL learns from the training and how the learned representation reduces the sample complexity in downstream tasks. While many theoretical results on contrastive SSL (Arora et al., 2019; Lee et al., 2020; Tosh et al., 2020; Wen & Li, 2021) do exist, similar study on nc-SSL has been very rare. As one of the first work towards this direction, Tian et al. (2021) show that while the global optimum of the non-contrastive loss is indeed a trivial one, following gradient direction in nc-SSL, one can find a local optimum that admits a nontrivial representation. Based on their theoretical findings on gradient-based methods, they proposed a new approach, DirectPred, that directly sets the predictor using the eigen-decomposition of the correlation matrix of input before the predictor, rather than updating it with gradient methods. As a method for nc-SSL, DirectPred shows comparable or better performance in multiple datasets, including CIFAR-10 (Krizhevsky et al., 2009), STL-10 (Coates et al., 2011) and ImageNet (Deng et al., 2009), compared to BYOL (Grill et al., 2020) and SimSiam (Chen & He, 2020) that optimize the predictor using gradient descent.

In this paper we present a new classification model in machine learning. Our result is threefold: 1) The model produces comparable predictive accuracy to that of most common classification models. 2) It runs significantly faster than most common classification models. 3) It has the ability to identify a portion of unseen samples for which class labels can be found with much higher predictive accuracy. Currently there are several patents pending on the proposed model.

Traditional methods for unsupervised learning of finite mixture models require to evaluate the likelihood of all components of the mixture. This becomes computationally prohibitive when the number of components is large, as it is, for example, in the sum-product (transform) networks. Therefore, we propose to apply a combination of the expectation maximization and the Metropolis-Hastings algorithm to evaluate only a small number of, stochastically sampled, components, thus substantially reducing the computational cost. The Markov chain of component assignments is sequentially generated across the algorithm's iterations, having a non-stationary target distribution whose parameters vary via a gradient-descent scheme. We put emphasis on generality of our method, equipping it with the ability to train both shallow and deep mixture models which involve complex, and possibly nonlinear, transformations. The performance of our method is illustrated in a variety of synthetic and real-data contexts, considering deep models, such as mixtures of normalizing flows and sum-product (transform) networks.

Multiclass logistic regression is a fundamental task in machine learning with applications in classification and boosting. Previous work (Foster et al., 2018) has highlighted the importance of improper predictors for achieving "fast rates" in the online multiclass logistic regression problem without suffering exponentially from secondary problem parameters, such as the norm of the predictors in the comparison class. While Foster et al. (2018) introduced a statistically optimal algorithm, it is in practice computationally intractable due to its run-time complexity being a large polynomial in the time horizon and dimension of input feature vectors. In this paper, we develop a new algorithm, FOLKLORE, for the problem which runs significantly faster than the algorithm of Foster et al. (2018) - the running time per iteration scales quadratically in the dimension - at the cost of a linear dependence on the norm of the predictors in the regret bound. This yields the first practical algorithm for online multiclass logistic regression, resolving an open problem of Foster et al. (2018). Furthermore, we show that our algorithm can be applied to online bandit multiclass prediction and online multiclass boosting, yielding more practical algorithms for both problems compared to the ones in (Foster et al., 2018) with similar performance guarantees. Finally, we also provide an online-to-batch conversion result for our algorithm.