We present a continuous-time probabilistic approach for estimating the chirp signal and its instantaneous frequency function when the true forms of these functions are not accessible. Our model represents these functions by non-linearly cascaded Gaussian processes represented as non-linear stochastic differential equations. The posterior distribution of the functions is then estimated with stochastic filters and smoothers. We compute a (posterior) Cram\'er--Rao lower bound for the Gaussian process model, and derive a theoretical upper bound for the estimation error in the mean squared sense. The experiments show that the proposed method outperforms a number of state-of-the-art methods on a synthetic data. We also show that the method works out-of-the-box for two real-world datasets.

Approximating Stochastic Gradient Descent (SGD) as a Stochastic Differential Equation (SDE) has allowed researchers to enjoy the benefits of studying a continuous optimization trajectory while carefully preserving the stochasticity of SGD. Analogous study of adaptive gradient methods, such as RMSprop and Adam, has been challenging because there were no rigorously proven SDE approximations for these methods. This paper derives the SDE approximations for RMSprop and Adam, giving theoretical guarantees of their correctness as well as experimental validation of their applicability to common large-scaling vision and language settings. A key practical result is the derivation of a $\textit{square root scaling rule}$ to adjust the optimization hyperparameters of RMSprop and Adam when changing batch size, and its empirical validation in deep learning settings.

The study of stability and sensitivity of statistical methods or algorithms with respect to their data is an important problem in machine learning and statistics. The performance of the algorithm under resampling of the data is a fundamental way to measure its stability and is closely related to generalization or privacy of the algorithm. In this paper, we study the resampling sensitivity for the principal component analysis (PCA). Given an $ n \times p $ random matrix $ \mathbf{X} $, let $ \mathbf{X}^{[k]} $ be the matrix obtained from $ \mathbf{X} $ by resampling $ k $ randomly chosen entries of $ \mathbf{X} $. Let $ \mathbf{v} $ and $ \mathbf{v}^{[k]} $ denote the principal components of $ \mathbf{X} $ and $ \mathbf{X}^{[k]} $. In the proportional growth regime $ p/n \to \xi \in (0,1] $, we establish the sharp threshold for the sensitivity/stability transition of PCA. When $ k \gg n^{5/3} $, the principal components $ \mathbf{v} $ and $ \mathbf{v}^{[k]} $ are asymptotically orthogonal. On the other hand, when $ k \ll n^{5/3} $, the principal components $ \mathbf{v} $ and $ \mathbf{v}^{[k]} $ are asymptotically colinear. In words, we show that PCA is sensitive to the input data in the sense that resampling even a negligible portion of the input may completely change the output.

With an increased focus on incorporating fairness in machine learning models, it becomes imperative not only to assess and mitigate bias at each stage of the machine learning pipeline but also to understand the downstream impacts of bias across stages. Here we consider a general, but realistic, scenario in which a predictive model is learned from (potentially biased) training data, and model predictions are assessed post-hoc for fairness by some auditing method. We provide a theoretical analysis of how a specific form of data bias, differential sampling bias, propagates from the data stage to the prediction stage. Unlike prior work, we evaluate the downstream impacts of data biases quantitatively rather than qualitatively and prove theoretical guarantees for detection. Under reasonable assumptions, we quantify how the amount of bias in the model predictions varies as a function of the amount of differential sampling bias in the data, and at what point this bias becomes provably detectable by the auditor. Through experiments on two criminal justice datasets -- the well-known COMPAS dataset and historical data from NYPD's stop and frisk policy -- we demonstrate that the theoretical results hold in practice even when our assumptions are relaxed.

Predicting the demand for electricity with uncertainty helps in planning and operation of the grid to provide reliable supply of power to the consumers. Machine learning (ML)-based demand forecasting approaches can be categorized into (1) sample-based approaches, where each forecast is made independently, and (2) time series regression approaches, where some historical load and other feature information is used. When making a short-to-mid-term electricity demand forecast, some future information is available, such as the weather forecast and calendar variables. However, in existing forecasting models this future information is not fully incorporated. To overcome this limitation of existing approaches, we propose Masked Multi-Step Multivariate Probabilistic Forecasting (MMMPF), a novel and general framework to train any neural network model capable of generating a sequence of outputs, that combines both the temporal information from the past and the known information about the future to make probabilistic predictions. Experiments are performed on a real-world dataset for short-to-mid-term electricity demand forecasting for multiple regions and compared with various ML methods. They show that the proposed MMMPF framework outperforms not only sample-based methods but also existing time-series forecasting models with the exact same base models. Models trainded with MMMPF can also generate desired quantiles to capture uncertainty and enable probabilistic planning for grid of the future.

We consider the problem of Learning from Label Proportions (LLP), a weakly supervised classification setup where instances are grouped into "bags", and only the frequency of class labels at each bag is available. Albeit, the objective of the learner is to achieve low task loss at an individual instance level. Here we propose Easyllp: a flexible and simple-to-implement debiasing approach based on aggregate labels, which operates on arbitrary loss functions. Our technique allows us to accurately estimate the expected loss of an arbitrary model at an individual level. We showcase the flexibility of our approach by applying it to popular learning frameworks, like Empirical Risk Minimization (ERM) and Stochastic Gradient Descent (SGD) with provable guarantees on instance level performance. More concretely, we exhibit a variance reduction technique that makes the quality of LLP learning deteriorate only by a factor of k (k being bag size) in both ERM and SGD setups, as compared to full supervision. Finally, we validate our theoretical results on multiple datasets demonstrating our algorithm performs as well or better than previous LLP approaches in spite of its simplicity.

This paper presents a new multi-carrier non-orthogonal multiple-access (MC-NOMA)-empowered WFL system under an adaptive learning setting of Flexible Aggregation. Since a WFL round accommodates both local model training and uploading for each user, the use of Flexible Aggregation allows the users to train different numbers of iterations per round, adapting to their channel conditions and computing resources. The key idea is to use MC-NOMA to concurrently upload the local models of the users, thereby extending the local model training times of the users and increasing participating users. A new metric, namely, Weighted Global Proportion of Trained Mini-batches (WGPTM), is analytically established to measure the convergence of the new system. Another important aspect is that we maximize the WGPTM to harness the convergence of the new system by jointly optimizing the transmit powers and subchannel bandwidths. This nonconvex problem is converted equivalently to a tractable convex problem and solved efficiently using variable substitution and Cauchy's inequality. As corroborated experimentally using a convolutional neural network and an 18-layer residential network, the proposed MC-NOMA WFL can efficiently reduce communication delay, increase local model training times, and accelerate the convergence by over 40%, compared to its existing alternative.

A new random forest based model for solving the Multiple Instance Learning (MIL) problem under small tabular data, called Soft Tree Ensemble MIL (STE-MIL), is proposed. A new type of soft decision trees is considered, which is similar to the well-known soft oblique trees, but with a smaller number of trainable parameters. In order to train the trees, it is proposed to convert them into neural networks of a specific form, which approximate the tree functions. It is also proposed to aggregate the instance and bag embeddings (output vectors) by using the attention mechanism. The whole STE-MIL model, including soft decision trees, neural networks, the attention mechanism and a classifier, is trained in an end-to-end manner. Numerical experiments with tabular datasets illustrate STE-MIL. The corresponding code implementing the model is publicly available.

The sparse Johnson-Lindenstrauss transform is one of the central techniques in dimensionality reduction. It supports embedding a set of $n$ points in $\mathbb{R}^d$ into $m=O(\varepsilon^{-2} \lg n)$ dimensions while preserving all pairwise distances to within $1 \pm \varepsilon$. Each input point $x$ is embedded to $Ax$, where $A$ is an $m \times d$ matrix having $s$ non-zeros per column, allowing for an embedding time of $O(s \|x\|_0)$. Since the sparsity of $A$ governs the embedding time, much work has gone into improving the sparsity $s$. The current state-of-the-art by Kane and Nelson (JACM'14) shows that $s = O(\varepsilon ^{-1} \lg n)$ suffices. This is almost matched by a lower bound of $s = \Omega(\varepsilon ^{-1} \lg n/\lg(1/\varepsilon))$ by Nelson and Nguyen (STOC'13). Previous work thus suggests that we have near-optimal embeddings. In this work, we revisit sparse embeddings and identify a loophole in the lower bound. Concretely, it requires $d \geq n$, which in many applications is unrealistic. We exploit this loophole to give a sparser embedding when $d = o(n)$, achieving $s = O(\varepsilon^{-1}(\lg n/\lg(1/\varepsilon)+\lg^{2/3}n \lg^{1/3} d))$. We also complement our analysis by strengthening the lower bound of Nelson and Nguyen to hold also when $d \ll n$, thereby matching the first term in our new sparsity upper bound. Finally, we also improve the sparsity of the best oblivious subspace embeddings for optimal embedding dimensionality.

Clustering in high dimension spaces is a difficult task; the usual distance metrics may no longer be appropriate under the curse of dimensionality. Indeed, the choice of the metric is crucial, and it is highly dependent on the dataset characteristics. However a single metric could be used to correctly perform clustering on multiple datasets of different domains. We propose to do so, providing a framework for learning a transferable metric. We show that we can learn a metric on a labelled dataset, then apply it to cluster a different dataset, using an embedding space that characterises a desired clustering in the generic sense. We learn and test such metrics on several datasets of variable complexity (synthetic, MNIST, SVHN, omniglot) and achieve results competitive with the state-of-the-art while using only a small number of labelled training datasets and shallow networks.