László Györfi

László Györfi, Tamás Linder, Harro Walk

We study the excess minimum risk in statistical inference, defined as the difference between the minimum expected loss in estimating a random variable from an observed feature vector and the minimum expected loss in estimating the same random variable from a transformation (statistic) of the feature vector. After characterizing lossless transformations, i.e., transformations for which the excess risk is zero for all loss functions, we construct a partitioning test statistic for the hypothesis that a given transformation is lossless and show that for i.i.d. data the test is strongly consistent. More generally, we develop information-theoretic upper bounds on the excess risk that uniformly hold over fairly general classes of loss functions. Based on these bounds, we introduce the notion of a delta-lossless transformation and give sufficient conditions for a given transformation to be universally delta-lossless. Applications to classification, nonparametric regression, portfolio strategies, information bottleneck, and deep learning, are also surveyed.

Hüseyin Afşer, László Györfi, Harro Walk

We study the problem nonparametric classification with repeated observations. Let $\bX$ be the $d$ dimensional feature vector and let $Y$ denote the label taking values in $\{1,\dots ,M\}$. In contrast to usual setup with large sample size $n$ and relatively low dimension $d$, this paper deals with the situation, when instead of observing a single feature vector $\bX$ we are given $t$ repeated feature vectors $\bV_1,\dots ,\bV_t $. Some simple classification rules are presented such that the conditional error probabilities have exponential convergence rate of convergence as $t\to\infty$. In the analysis, we investigate particular models like robust detection by nominal densities, prototype classification, linear transformation, linear classification, scaling.

Balázs Csanád Csáji, László Györfi, Ambrus Tamás et al.

In this paper we revisit the classical method of partitioning classification and study its convergence rate under relaxed conditions, both for observable (non-privatised) and for privatised data. We consider the problem of classification in a $d$ dimensional Euclidean space. Previous results on the partitioning classifier worked with the strong density assumption, which is restrictive, as we demonstrate through simple examples. Here, we study the problem under much milder assumptions. We presuppose that the distribution of the inputs is a mixture of an absolutely continuous and a discrete distribution, such that the absolutely continuous component is concentrated to a $d_a$ dimensional subspace. In addition to the standard Lipschitz and margin conditions, a novel characteristic of the absolutely continuous component is introduced, by which the exact convergence rate of the classification error probability is computed, both for the binary and for the multi-label cases. Interestingly, this rate of convergence depends only on the intrinsic dimension of the inputs, $d_a$. The privacy constraints mean that the independent identically distributed data cannot be directly observed, and the classifiers are functions of the randomised outcome of a suitable local differential privacy mechanism. In this paper we add Laplace distributed noises to the discontinuations of all possible locations of the feature vector and to its label. Again, tight upper bounds on the rate of convergence of the classification error probability are derived, without the strong density assumption, such that this rate depends on $2d_a$.

László Györfi, Aryeh Kontorovich, Roi Weiss

We study the problem of estimating the density $f(\boldsymbol x)$ of a random vector ${\boldsymbol X}$ in $\mathbb R^d$. For a spanning tree $T$ defined on the vertex set $\{1,\dots ,d\}$, the tree density $f_{T}$ is a product of bivariate conditional densities. An optimal spanning tree minimizes the Kullback-Leibler divergence between $f$ and $f_{T}$. From i.i.d. data we identify an optimal tree $T^*$ and efficiently construct a tree density estimate $f_n$ such that, without any regularity conditions on the density $f$, one has $\lim_{n\to \infty} \int |f_n(\boldsymbol x)-f_{T^*}(\boldsymbol x)|d\boldsymbol x=0$ a.s. For Lipschitz $f$ with bounded support, $\mathbb E \left\{ \int |f_n(\boldsymbol x)-f_{T^*}(\boldsymbol x)|d\boldsymbol x\right\}=O\big(n^{-1/4}\big)$, a dimension-free rate.

Thomas Berrett, László Györfi, Harro Walk

In this paper we revisit the classical problem of nonparametric regression, but impose local differential privacy constraints. Under such constraints, the raw data $(X_1,Y_1),\ldots,(X_n,Y_n)$, taking values in $\mathbb{R}^d \times \mathbb{R}$, cannot be directly observed, and all estimators are functions of the randomised output from a suitable privacy mechanism. The statistician is free to choose the form of the privacy mechanism, and here we add Laplace distributed noise to a discretisation of the location of a feature vector $X_i$ and to the value of its response variable $Y_i$. Based on this randomised data, we design a novel estimator of the regression function, which can be viewed as a privatised version of the well-studied partitioning regression estimator. The main result is that the estimator is strongly universally consistent. Our methods and analysis also give rise to a strongly universally consistent binary classification rule for locally differentially private data.

László Györfi, Roi Weiss

We study universal consistency and convergence rates of simple nearest-neighbor prototype rules for the problem of multiclass classification in metric paces. We first show that a novel data-dependent partitioning rule, named Proto-NN, is universally consistent in any metric space that admits a universally consistent rule. Proto-NN is a significant simplification of OptiNet, a recently proposed compression-based algorithm that, to date, was the only algorithm known to be universally consistent in such a general setting. Practically, Proto-NN is simpler to implement and enjoys reduced computational complexity. We then proceed to study convergence rates of the excess error probability. We first obtain rates for the standard $k$-NN rule under a margin condition and a new generalized-Lipschitz condition. The latter is an extension of a recently proposed modified-Lipschitz condition from $\mathbb R^d$ to metric spaces. Similarly to the modified-Lipschitz condition, the new condition avoids any boundness assumptions on the data distribution. While obtaining rates for Proto-NN is left open, we show that a second prototype rule that hybridizes between $k$-NN and Proto-NN achieves the same rates as $k$-NN while enjoying similar computational advantages as Proto-NN. However, as $k$-NN, this hybrid rule is not consistent in general.