Here, the functionf() is applied toAx in a component-wise manner. The above model arises in many applications of signal processing [13, 10, 41], communications [56, 9, 25], and machine learning [48, 40].

We examine the relationship between the mutual information between the output model and the empirical sample and the algorithm's generalization in the context of stochastic convex optimization. Despite increasing interest in information-theoretic generalization bounds, it is uncertain if these bounds can provide insight into the exceptional performance of various learning algorithms. Our study of stochastic convex optimization reveals that, for true risk minimization, dimension-dependent mutual information is necessary. This indicates that existing information-theoretic generalization bounds fall short in capturing the generalization capabilities of algorithms like SGD and regularized ERM, which have dimension-independent sample complexity.

Formally, we will consider the following scenario.

We consider the problem of exact recovery of a $k$-sparse binary vector from generalized linear measurements (such as logistic regression). We analyze the linear estimation algorithm (Plan, Vershynin, Yudovina, 2017), and also show information theoretic lower bounds on the number of required measurements. As a consequence of our results, for noisy one bit quantized linear measurements ($\mathsf{1bCSbinary}$), we obtain a sample complexity of $O((k+\sigma^2)\log{n})$, where $\sigma^2$ is the noise variance. This is shown to be optimal due to the information theoretic lower bound. We also obtain tight sample complexity characterization for logistic regression. Since $\mathsf{1bCSbinary}$ is a strictly harder problem than noisy linear measurements ($\mathsf{SparseLinearReg}$) because of added quantization, the same sample complexity is achievable for $\mathsf{SparseLinearReg}$. While this sample complexity can be obtained via the popular lasso algorithm, linear estimation is computationally more efficient. Our lower bound holds for any set of measurements for $\mathsf{SparseLinearReg}$, (similar bound was known for Gaussian measurement matrices) and is closely matched by the maximum-likelihood upper bound. For $\mathsf{SparseLinearReg}$, it was conjectured in Gamarnik and Zadik, 2017 that there is a statistical-computational gap and the number of measurements should be at least $(2k+\sigma^2)\log{n}$ for efficient algorithms to exist. It is worth noting that our results imply that there is no such statistical-computational gap for $\mathsf{1bCSbinary}$ and logistic regression.

We examine the relationship between the mutual information between the output model and the empirical sample and the algorithm's generalization in the context of stochastic convex optimization. Despite increasing interest in information-theoretic generalization bounds, it is uncertain if these bounds can provide insight into the exceptional performance of various learning algorithms. Our study of stochastic convex optimization reveals that, for true risk minimization, dimension-dependent mutual information is necessary. This indicates that existing information-theoretic generalization bounds fall short in capturing the generalization capabilities of algorithms like SGD and regularized ERM, which have dimension-independent sample complexity.

This article proves information theoretic lower bounds for the community detection problem in a range of network models. For the SBM, this has been previously achieved in works of Mossel-Neeman-Sly, Coja-Oghlan, Abbe-Sandon etc. The authors of the current paper plant community structure (two randomly assigned communities) in various other models such as latent space models, preferential attachment models, small-world models etc. and prove similar information theoretic lower bounds. The proofs employ (not surprisingly) Fano's inequality in carefully restricted submodels. However, the authors only prove lower bounds, it is not clear if these are tight (it is for the SBM), and if so what (preferably polynomial time) algorithms can achieve them. There are a host of other interesting questions to ask.

In recent years, a rapidly increasing number of applications in practice requires optimizing non-convex objectives, like training neural networks, learning graphical models, maximum likelihood estimation. Though simple heuristics such as gradient descent with very few modifications tend to work well, theoretical understanding is very weak. We consider possibly the most natural class of non-convex functions where one could hope to obtain provable guarantees: functions that are "approximately convex", i.e. functions f: R

We consider a population, partitioned into a set of communities, and study the problem of identifying the largest community within the population via sequential, random sampling of individuals. There are multiple sampling domains, referred to as \emph{boxes}, which also partition the population. Each box may consist of individuals of different communities, and each community may in turn be spread across multiple boxes. The learning agent can, at any time, sample (with replacement) a random individual from any chosen box; when this is done, the agent learns the community the sampled individual belongs to, and also whether or not this individual has been sampled before. The goal of the agent is to minimize the probability of mis-identifying the largest community in a \emph{fixed budget} setting, by optimizing both the sampling strategy as well as the decision rule. We propose and analyse novel algorithms for this problem, and also establish information theoretic lower bounds on the probability of error under any algorithm. In several cases of interest, the exponential decay rates of the probability of error under our algorithms are shown to be optimal up to constant factors. The proposed algorithms are further validated via simulations on real-world datasets.

Paired estimation of change in parameters of interest over a population plays a central role in several application domains including those in the social sciences, epidemiology, medicine and biology. In these domains, the size of the population under study is often very large, however, the number of observations available per individual in the population is very small (\emph{sparse observations}) which makes the problem challenging. Consider the setting with $N$ independent individuals, each with unknown parameters $(p_i, q_i)$ drawn from some unknown distribution on $[0, 1]^2$. We observe $X_i \sim \text{Bin}(t, p_i)$ before an event and $Y_i \sim \text{Bin}(t, q_i)$ after the event. Provided these paired observations, $\{(X_i, Y_i) \}_{i=1}^N$, our goal is to accurately estimate the \emph{distribution of the change in parameters}, $\delta_i := q_i - p_i$, over the population and properties of interest like the \emph{$\ell_1$-magnitude of the change} with sparse observations ($t\ll N$). We provide \emph{information theoretic lower bounds} on the error in estimating the distribution of change and the $\ell_1$-magnitude of change. Furthermore, we show that the following two step procedure achieves the optimal error bounds: first, estimate the full joint distribution of the paired parameters using the maximum likelihood estimator (MLE) and then estimate the distribution of change and the $\ell_1$-magnitude of change using the joint MLE. Notably, and perhaps surprisingly, these error bounds are of the same order as the minimax optimal error bounds for learning the \emph{full} joint distribution itself (in Wasserstein-1 distance); in other words, estimating the magnitude of the change of parameters over the population is, in a minimax sense, as difficult as estimating the full joint distribution itself.

We analyze the necessary number of samples for sparse vector recovery in a noisy linear prediction setup. This model includes problems such as linear regression and classification. We focus on structured graph models. In particular, we prove that sufficient number of samples for the weighted graph model proposed by Hegde and others is also necessary. We use the Fano's inequality on well constructed ensembles as our main tool in establishing information theoretic lower bounds.