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Julien Chhor

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Machine Learning math.ST Statistics Theory Information Theory

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preprint2026arXiv

Minimax optimal submatrix detection: Sharp non-asymptotic rates

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.

preprint2022arXiv

Benign overfitting and adaptive nonparametric regression

In the nonparametric regression setting, we construct an estimator which is a continuous function interpolating the data points with high probability, while attaining minimax optimal rates under mean squared risk on the scale of Hölder classes adaptively to the unknown smoothness.

preprint2022arXiv

Robust Estimation of Discrete Distributions under Local Differential Privacy

Although robust learning and local differential privacy are both widely studied fields of research, combining the two settings is just starting to be explored. We consider the problem of estimating a discrete distribution in total variation from $n$ contaminated data batches under a local differential privacy constraint. A fraction $1-ε$ of the batches contain $k$ i.i.d. samples drawn from a discrete distribution $p$ over $d$ elements. To protect the users' privacy, each of the samples is privatized using an $α$ -locally differentially private mechanism. The remaining $εn$ batches are an adversarial contamination. The minimax rate of estimation under contamination alone, with no privacy, is known to be $ε/\sqrt{k}+\sqrt{d/kn}$ , up to a $\sqrt{\log(1/ε)}$ factor. Under the privacy constraint alone, the minimax rate of estimation is $\sqrt{d^2/α^2 kn}$ . We show that combining the two constraints leads to a minimax estimation rate of $ε\sqrt{d/α^2 k}+\sqrt{d^2/α^2 kn}$ up to a $\sqrt{\log(1/ε)}$ factor, larger than the sum of the two separate rates. We provide a polynomial-time algorithm achieving this bound, as well as a matching information theoretic lower bound.

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