We consider the problem of detecting a community of densely connected vertices in a high-dimensional bipartite graph of size n1 n2. Under the null hypothesis, the observed graph is drawn from a bipartite Erd os-Renyi distribution with connection probability p0. Under the alternative hypothesis, there exists an unknown bipartite subgraph of size k1 k2 in which edges appear with probability p1 = p0 +δfor some δ > 0, while all other edges outside the subgraph appear with probability p0. Specifically, we provide non-asymptotic upper and lower bounds on the smallest signal strength δ that is both necessary and sufficient to ensure the existence of a test with small enough Type I and Type II errors. We also derive novel minimax-optimal tests achieving these fundamental limits when the underlying graph is sufficiently dense. Our proposed tests involve a combination of hardthresholded nonlinear statistics of the adjacency matrix, the analysis of which may be of independent interest. In contrast with previous work, our non-asymptotic upper and lower bounds match for any configuration of n1,n2,k1,k2.

We study the problem of robust estimation under heterogeneous corruption rates, where each sample may be independently corrupted with a known but non-identical probability. This setting arises naturally in distributed and federated learning, crowdsourcing, and sensor networks, yet existing robust estimators typically assume uniform or worst-case corruption, ignoring structural heterogeneity. For mean estimation for multivariate bounded distributions and univariate gaussian distributions, we give tight minimax rates for all heterogeneous corruption patterns. For multivariate gaussian mean estimation and linear regression, we establish the minimax rate for squared error up to a factor of d, where d is the dimension. Roughly, our findings suggest that samples beyond a certain corruption threshold may be discarded by the optimal estimators - this threshold is determined by the empirical distribution of the corruption rates given.

We consider the problem of estimating the correlation of two random variables X and Y, where the pairs (X, Y) are not observed together, but are instead separated co-ordinate-wise at two servers: server 1 contains all the X observations, and server 2 contains the corresponding Y observations. In this vertically distributed setting, we assume that each server has its own privacy constraints, owing to which they can only share suitably privatized statistics of their own component observations. We consider differing privacy budgets (ε1, δ1) and (ε2, δ2) for the two servers and determine the minimax optimal rates for correlation estimation allowing for both noninteractive and interactive mechanisms. We also provide correlation estimators that achieve these rates and further develop inference procedures, namely, confidence intervals, for the estimated correlations. Our results are characterized by an interesting rate in terms of the sample size n, ε1, ε2, which is strictly slower than the usual central privacy estimation rates. More interestingly, we find that the interactive mechanism is always better than its non-interactive counterpart whenever the two privacy budgets are different. Results from extensive numerical experiments support our theoretical findings.

We study the problem of robust estimation under heterogeneous corruption rates, where each sample may be independently corrupted with a known but non-identical probability. This setting arises naturally in distributed and federated learning, crowdsourcing, and sensor networks, yet existing robust estimators typically assume uniform or worst-case corruption, ignoring structural heterogeneity. For mean estimation for multivariate bounded distributions and univariate gaussian distributions, we give tight minimax rates for all heterogeneous corruption patterns. For multivariate gaussian mean estimation and linear regression, we establish the minimax rate for squared error up to a factor of $\sqrt{d}$, where $d$ is the dimension. Roughly, our findings suggest that samples beyond a certain corruption threshold may be discarded by the optimal estimators -- this threshold is determined by the empirical distribution of the corruption rates given.

Given an observation $\mathbf Y \in \mathbb{R}^{d_1\times d_2}$ from the model $\mathbf Y = \mathbf X + \mathbf E$ where $\mathbf X$ is constant and $\mathbf E$ has i.i.d. $N(0,1)$ entries, we consider the problem of detecting a planted submatrix in the mean matrix $\mathbf X$. Specifically, we aim to distinguish the null hypothesis $\mathbf X = 0$ from the alternative hypothesis in which $\mathbf X$ is non-zero only on a submatrix of size $s_1 \times s_2$ with elevated entries bounded below by $μ>0$. We establish a minimax lower bound characterizing how large $μ$ must be to ensure that the two hypotheses are distinguishable with high probability. Furthermore, we derive novel minimax-optimal tests achieving the lower bound, and describe extensions of these tests that are adaptive to unknown sparsity levels $s_1$ and $s_2$. In contrast with previous work, which required restrictive assumptions on $s_1,s_2, d_1$ and $d_2$, our non-asymptotic upper and lower bounds match for any configuration of these parameters.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

We consider the problem of estimating a function defined over nlocations on a d-dimensional grid (having all side lengths equal to n1/d). When the function is constrained to have discrete total variation bounded by Cn, we derive the minimax optimal (squared) `2 estimation error rate, parametrized by n,Cn. Total variation denoising, also known as the fused lasso, is seen to be rate optimal. Several simpler estimators exist, such as Laplacian smoothing and Laplacian eigenmaps. A natural question is: can these simpler estimators perform just as well?

Indeed, the fact that this parameter is fixed actually hinders statistical consistency of the original procedure. Our modified Mondrian Forest algorithm grows trees with increasing lifetime parameters $\lambda_n$, and uses an alternative updating rule, allowing to work also in an online fashion. Second, we provide a theoretical analysis establishing simple conditions for consistency. Our theoretical analysis also exhibits a surprising fact: our algorithm achieves the minimax rate (optimal rate) for the estimation of a Lipschitz regression function, which is a strong extension of previous results~\cite{arlot2014purf_bias} to an \emph{arbitrary dimension}.

We consider the problem of global optimization of an unknown non-convex smooth function with noisy zeroth-order feedback. We propose a local minimax framework to study the fundamental difficulty of optimizing smooth functions with adaptive function evaluations. We show that for functions with fast growth around their global minima, carefully designed optimization algorithms can identify a near global minimizer with many fewer queries than worst-case global minimax theory predicts. For the special case of strongly convex and smooth functions, our implied convergence rates match the ones developed for zeroth-order convex optimization problems. On the other hand, we show that in the worst case no algorithm can converge faster than the minimax rate of estimating an unknown functions in linf-norm. Finally, we show that non-adaptive algorithms, although optimal in a global minimax sense, do not attain the optimal local minimax rate.