Personalization of machine learning (ML) predictions for individual users/domains/enterprises is critical for practical recommendation systems. Standard personalization approaches involve learning a user/domain specific \emph{embedding} that is fed into a fixed global model which can be limiting. On the other hand, personalizing/fine-tuning model itself for each user/domain — a.k.a meta-learning — has high storage/infrastructure cost. Moreover, rigorous theoretical studies of scalable personalization approaches have been very limited. To address the above issues, we propose a novel meta-learning style approach that models network weights as a sum of low-rank and sparse components. This captures common information from multiple individuals/users together in the low-rank part while sparse part captures user-specific idiosyncrasies. We then study the framework in the linear setting, where the problem reduces to that of estimating the sum of a rank-$r$ and a $k$-column sparse matrix using a small number of linear measurements. We propose a computationally efficient alternating minimization method with iterative hard thresholding — AMHT-LRS — to learn the low-rank and sparse part. Theoretically, for the realizable Gaussian data setting, we show that AMHT-LRS solves the problem efficiently with nearly optimal sample complexity. Finally, a significant challenge in personalization is ensuring privacy of each user’s sensitive data. We alleviate this problem by proposing a differentially private variant of our method that also is equipped with strong generalization guarantees.

$\newcommand{\Tr}{\mathsf{Tr}}$ We consider the problem of high-dimensional heavy-tailed statistical estimation in the streaming setting, which is much harder than the traditional batch setting due to memory constraints. We cast this problem as stochastic convex optimization with heavy tailed stochastic gradients, and prove that the widely used Clipped-SGD algorithm attains near-optimal sub-Gaussian statistical rates whenever the second moment of the stochastic gradient noise is finite. More precisely, with $T$ samples, we show that Clipped-SGD, for smooth and strongly convex objectives, achieves an error of $\sqrt{\frac{\Tr(\Sigma)+\sqrt{\Tr(\Sigma)\\|\Sigma\\|_2}\ln(\tfrac{\ln(T)}{\delta})}{T}}$ with probability $1-\delta$, where $\Sigma$ is the covariance of the clipped gradient. Note that the fluctuations (depending on $\tfrac{1}{\delta}$) are of lower order than the term $\Tr(\Sigma)$. This improves upon the current best rate of $\sqrt{\frac{\Tr(\Sigma)\ln(\tfrac{1}{\delta})}{T}}$ for Clipped-SGD, known \emph{only} for smooth and strongly convex objectives. Our results also extend to smooth convex and lipschitz convex objectives. Key to our result is a novel iterative refinement strategy for martingale concentration, improving upon the PAC-Bayes approach of \citet{catoni2018dimension}.

We study the problem of differentially private linear regression where each of the data point is sampled from a fixed sub-Gaussian style distribution. We propose and analyze a one-pass mini-batch stochastic gradient descent method (DP-AMBSSGD) where points in each iteration are sampled without replacement. Noise is added for DP but the noise standard deviation is estimated online. Compared to existing $(\epsilon, \delta)$-DP techniques which have sub-optimal error bounds, DP-AMBSSGD is able to provide nearly optimal error bounds in terms of key parameters like dimensionality $d$, number of points $N$, and the standard deviation \sigma of the noise in observations. For example, when the $d$-dimensional covariates are sampled i.i.d. from the normal distribution, then the excess error of DP-AMBSSGD due to privacy is $\sigma^2 d/N(1+d/(\epsilon^2 N))$, i.e., the error is meaningful when number of samples N\geq d \log d which is the standard operative regime for linear regression. In contrast, error bounds for existing efficient methods in this setting are: $d^3/(\epsilon^2 N^2)$, even for $\sigma=0$. That is, for constant $\epsilon$, the existing techniques require $N=d^{1.5}$ to provide a non-trivial result.