arXiv subjects

Shahar Mendelson

Publications and source records attributed to Shahar Mendelson.

At least 55 records · Page 3Linked to original sources

arXiv2017

Regularization, sparse recovery, and median-of-means tournaments

A regularized risk minimization procedure for regression function estimation is introduced that achieves near optimal accuracy and confidence under general conditions, including heavy-tailed predictor and response variables. The procedure is based on median-of-means tournaments, introduced by the authors in [8]. It is shown that the new procedure outperforms standard regularized empirical risk minimization procedures such as lasso or slope in heavy-tailed problems.

math.ST↗

arXiv2017

Column normalization of a random measurement matrix

In this note we answer a question of G. Lecué, by showing that column normalization of a random matrix with iid entries need not lead to good sparse recovery properties, even if the generating random variable has a reasonable moment growth. Specifically, for every $2 \leq p \leq c_1\log d$ we construct a random vector $X \in R^d$ with iid, mean-zero, variance $1$ coordinates, that satisfies $\sup_{t \in S^{d-1}} \| \|_{L_q} \leq c_2\sqrt{q}$ for every $2\leq q \leq p$. We show that if $m \leq c_3\sqrt{p}d^{1/p}$ and $\tildeΓ:R^d \to R^m$ is the column-normalized matrix generated by $m$ independent copies of $X$, then with probability at least $1-2\exp(-c_4m)$, $\tildeΓ$ does not satisfy the exact reconstruction property of order $2$.

stat.ML↗

arXiv2017

Sub-Gaussian estimators of the mean of a random vector

We study the problem of estimating the mean of a random vector $X$ given a sample of $N$ independent, identically distributed points. We introduce a new estimator that achieves a purely sub-Gaussian performance under the only condition that the second moment of $X$ exists. The estimator is based on a novel concept of a multivariate median.

math.ST↗

arXiv2017

Regularization and the small-ball method I: sparse recovery

We obtain bounds on estimation error rates for regularization procedures of the form \begin{equation*} \hat f \in {\rm argmin}_{f\in F}\left(\frac{1}{N}\sum_{i=1}^N\left(Y_i-f(X_i)\right)^2+λΨ(f)\right) \end{equation*} when $Ψ$ is a norm and $F$ is convex. Our approach gives a common framework that may be used in the analysis of learning problems and regularization problems alike. In particular, it sheds some light on the role various notions of sparsity have in regularization and on their connection with the size of subdifferentials of $Ψ$ in a neighbourhood of the true minimizer. As `proof of concept' we extend the known estimates for the LASSO, SLOPE and trace norm regularization.

math.ST↗

arXiv2016

Learning subgaussian classes : Upper and minimax bounds

We obtain sharp oracle inequalities for the empirical risk minimization procedure in the regression model under the assumption that the target Y and the model F are subgaussian. The bound we obtain is sharp in the minimax sense if F is convex. Moreover, under mild assumptions on F, the error rate of ERM remains optimal even if the procedure is allowed to perform with constant probability. A part of our analysis is a new proof of minimax results for the gaussian regression model.

math.ST↗

arXiv2016

Regularization and the small-ball method II: complexity dependent error rates

For a convex class of functions $F$, a regularization functions $Ψ(\cdot)$ and given the random data $(X_i, Y_i)_{i=1}^N$, we study estimation properties of regularization procedures of the form \begin{equation*} \hat f \in {\rm argmin}_{f\in F}\Big(\frac{1}{N}\sum_{i=1}^N\big(Y_i-f(X_i)\big)^2+λΨ(f)\Big) \end{equation*} for some well chosen regularization parameter $λ$. We obtain bounds on the $L_2$ estimation error rate that depend on the complexity of the "true model" $F^*:=\{f\in F: Ψ(f)\leqΨ(f^*)\}$, where $f^*\in {\rm argmin}_{f\in F}\mathbb{E}(Y-f(X))^2$ and the $(X_i,Y_i)$'s are independent and distributed as $(X,Y)$. Our estimate holds under weak stochastic assumptions -- one of which being a small-ball condition satisfied by $F$ -- and for rather flexible choices of regularization functions $Ψ(\cdot)$. Moreover, the result holds in the learning theory framework: we do not assume any a-priori connection between the output $Y$ and the input $X$. As a proof of concept, we apply our general estimation bound to various choices of $Ψ$, for example, the $\ell_p$ and $S_p$-norms (for $p\geq1$), weak-$\ell_p$, atomic norms, max-norm and SLOPE. In many cases, the estimation rate almost coincides with the minimax rate in the class $F^*$.

math.ST↗

arXiv2016

Risk minimization by median-of-means tournaments

We consider the classical statistical learning/regression problem, when the value of a real random variable Y is to be predicted based on the observation of another random variable X. Given a class of functions F and a sample of independent copies of (X, Y ), one needs to choose a function f from F such that f(X) approximates Y as well as possible, in the mean-squared sense. We introduce a new procedure, the so-called median-of-means tournament, that achieves the optimal tradeoff between accuracy and confidence under minimal assumptions, and in particular outperforms classical methods based on empirical risk minimization.

math.ST↗

arXiv2016

Performance of empirical risk minimization in linear aggregation

We study conditions under which, given a dictionary $F=\{f_1,\ldots ,f_M\}$ and an i.i.d. sample $(X_i,Y_i)_{i=1}^N$, the empirical minimizer in $\operatorname {span}(F)$ relative to the squared loss, satisfies that with high probability \[R\bigl(\tilde{f}^{\mathrm{ERM}}\bigr)\leq\inf_{f\in\operatorname {span}(F)}R(f)+r_N(M),\] where $R(\cdot)$ is the squared risk and $r_N(M)$ is of the order of $M/N$. Among other results, we prove that a uniform small-ball estimate for functions in $\operatorname {span}(F)$ is enough to achieve that goal when the noise is independent of the design.

math.ST↗

arXiv2016

On multiplier processes under weak moment assumptions

We show that if $V \subset \R^n$ satisfies a certain symmetry condition (closely related to unconditionaity) and if $X$ is an isotropic random vector for which $\|\inr{X,t}\|_{L_p} \leq L \sqrt{p}$ for every $t \in S^{n-1}$ and $p \lesssim \log n$, then the corresponding empirical and multiplier processes indexed by $V$ behave as if $X$ were $L$-subgaussian.

math.ST↗

arXiv2015

Upper bounds on product and multiplier empirical processes

We study two empirical process of special structure: firstly, the centred multiplier process indexed by a class $F$, $f \to \left|\sum_{i=1}^N (ξ_i f(X_i) - \E ξf)\right|$, where the i.i.d. multipliers $(ξ_i)_{i=1}^N$ need not be independent of $(X_i)_{i=1}^N$, and secondly, $(f,h) \to \left|\sum_{i=1}^N (f(X_i)h(X_i)-\E f h) \right|$, the centred product process indexed by the classes $F$ and $H$. We use chaining methods to obtain high probability upper bounds on the suprema of the two processes using a natural variation of Talagrand's $γ$-functionals.

math.PR↗

arXiv2015

`local' vs. `global' parameters -- breaking the gaussian complexity barrier

We show that if $F$ is a convex class of functions that is $L$-subgaussian, the error rate of learning problems generated by independent noise is equivalent to a fixed point determined by `local' covering estimates of the class, rather than by the gaussian averages. To that end, we establish new sharp upper and lower estimates on the error rate for such problems.

stat.ML↗

arXiv2015

Sparse recovery under weak moment assumptions

We prove that iid random vectors that satisfy a rather weak moment assumption can be used as measurement vectors in Compressed Sensing, and the number of measurements required for exact reconstruction is the same as the best possible estimate -- exhibited by a random gaussian matrix. We also prove that this moment condition is necessary, up to a $\log \log $ factor. Applications to the Compatibility Condition and the Restricted Eigenvalue Condition in the noisy setup and to properties of neighbourly random polytopes are also discussed.

math.ST↗

arXiv2015

On aggregation for heavy-tailed classes

We introduce an alternative to the notion of `fast rate' in Learning Theory, which coincides with the optimal error rate when the given class happens to be convex and regular in some sense. While it is well known that such a rate cannot always be attained by a learning procedure (i.e., a procedure that selects a function in the given class), we introduce an aggregation procedure that attains that rate under rather minimal assumptions -- for example, that the $L_q$ and $L_2$ norms are equivalent on the linear span of the class for some $q>2$, and the target random variable is square-integrable.

math.ST↗

arXiv2014

Dvoretzky type theorems for subgaussian coordinate projections

Given a class of functions $F$ on a probability space $(Ω,μ)$, we study the structure of a typical coordinate projection of the class, defined by $\{(f(X_i))_{i=1}^N : f \in F\}$, where $X_1,...,X_N$ are independent, selected according to $μ$. This notion of projection generalizes the standard linear random projection used in Asymptotic Geometric Analysis. We show that when $F$ is a subgaussian class of functions, a typical coordinate projection satisfies a Dvoretzky type theorem.

math.FA↗

arXiv2014

Learning without Concentration

We obtain sharp bounds on the performance of Empirical Risk Minimization performed in a convex class and with respect to the squared loss, without assuming that class members and the target are bounded functions or have rapidly decaying tails. Rather than resorting to a concentration-based argument, the method used here relies on a `small-ball' assumption and thus holds for classes consisting of heavy-tailed functions and for heavy-tailed targets. The resulting estimates scale correctly with the `noise level' of the problem, and when applied to the classical, bounded scenario, always improve the known bounds.

cs.LG↗

arXiv2014

Learning without Concentration for General Loss Functions

We study prediction and estimation problems using empirical risk minimization, relative to a general convex loss function. We obtain sharp error rates even when concentration is false or is very restricted, for example, in heavy-tailed scenarios. Our results show that the error rate depends on two parameters: one captures the intrinsic complexity of the class, and essentially leads to the error rate in a noise-free (or realizable) problem; the other measures interactions between class members the target and the loss, and is dominant when the problem is far from realizable. We also explain how one may deal with outliers by choosing the loss in a way that is calibrated to the intrinsic complexity of the class and to the noise-level of the problem (the latter is measured by the distance between the target and the class).

stat.ML↗

arXiv2014

Necessary moment conditions for exact reconstruction via basis pursuit

Let $X=(x_1,...,x_n)$ be a random vector that satisfies a weak small ball property and whose coordinates $x_i$ satisfy that $\|x_i\|_{L_p} \lesssim \sqrt{p} \|x_i\|_{L_2}$ for $p \sim \log n$. In \cite{LM_compressed}, it was shown that $N$ independent copies of $X$ can be used as measurement vectors in Compressed Sensing (using the basis pursuit algorithm) to reconstruct any $d$-sparse vector with the optimal number of measurements $N\gtrsim d \log\big(e n/d\big)$. In this note we show that the result is almost optimal. We construct a random vector $X$ with iid, mean-zero, variance one coordinates that satisfies the same weak small ball property and whose coordinates satisfy that $\|x_i\|_{L_p} \lesssim \sqrt{p} \|x_i\|_{L_2}$ for $p \sim (\log n)/(\log N)$, but the basis pursuit algorithm fails to recover even $1$-sparse vectors. The construction shows that `spiky' measurement vectors may lead to a poor performance by the basis pursuit algorithm, but on the other hand may still perform in an optimal way if one chooses a different reconstruction algorithm (like $\ell_0$-minimization). This exhibits the fact that the convex relaxation of $\ell_0$-minimization comes at a significant cost when using `spiky' measurement vectors.

math.ST↗

arXiv2013

Bounding the smallest singular value of a random matrix without concentration

Given $X$ a random vector in ${\mathbb{R}}^n$, set $X_1,...,X_N$ to be independent copies of $X$ and let $Γ=\frac{1}{\sqrt{N}}\sum_{i=1}^N e_i$ be the matrix whose rows are $\frac{X_1}{\sqrt{N}},\dots, \frac{X_N}{\sqrt{N}}$. We obtain sharp probabilistic lower bounds on the smallest singular value $λ_{\min}(Γ)$ in a rather general situation, and in particular, under the assumption that $X$ is an isotropic random vector for which $\sup_{t\in S^{n-1}}{\mathbb{E}}| |^{2+η} \leq L$ for some $L,η>0$. Our results imply that a Bai-Yin type lower bound holds for $η>2$, and, up to a log-factor, for $η=2$ as well. The bounds hold without any additional assumptions on the Euclidean norm $\|X\|_{\ell_2^n}$. Moreover, we establish a nontrivial lower bound even without any higher moment assumptions (corresponding to the case $η=0$), if the linear forms satisfy a weak `small ball' property.

math.PR↗