EM for Mixture of Linear Regression with Clustered Data

Modern data-driven and distributed learning frameworks deal with diverse massive data generated by clients spread across heterogeneous environments. Indeed, data heterogeneity is a major bottleneck in scaling up many distributed learning paradigms. In many settings however, heterogeneous data may be generated in clusters with shared structures, as is the case in several applications such as federated learning where a common latent variable governs the distribution of all the samples generated by a client. It is therefore natural to ask how the underlying clustered structures in distributed data can be exploited to improve learning schemes. In this paper, we tackle this question in the special case of estimating $d$-dimensional parameters of a two-component mixture of linear regressions problem where each of $m$ nodes generates $n$ samples with a shared latent variable. We employ the well-known Expectation-Maximization (EM) method to estimate the maximum likelihood parameters from $m$ batches of dependent samples each containing $n$ measurements. Discarding the clustered structure in the mixture model, EM is known to require $O(\log(mn/d))$ iterations to reach the statistical accuracy of $O(\sqrt{d/(mn)})$. In contrast, we show that if initialized properly, EM on the structured data requires only $O(1)$ iterations to reach the same statistical accuracy, as long as $m$ grows up as $e^{o(n)}$. Our analysis establishes and combines novel asymptotic optimization and generalization guarantees for population and empirical EM with dependent samples, which may be of independent interest.

A Unified Approach to Controlling Implicit Regularization via Mirror Descent

Inspired by the remarkable success of deep neural networks, there has been significant interest in understanding the generalization performance of overparameterized models. Substantial efforts have been invested in characterizing how optimization algorithms impact generalization through their "preferred" solutions, a phenomenon commonly referred to as implicit regularization. In particular, it has been argued that gradient descent (GD) induces an implicit $\ell_2$-norm regularization in regression and classification problems. However, the implicit regularization of different algorithms are confined to either a specific geometry or a particular class of learning problems, indicating a gap in a general approach for controlling the implicit regularization. To address this, we present a unified approach using mirror descent (MD), a notable generalization of GD, to control implicit regularization in both regression and classification settings. More specifically, we show that MD with the general class of homogeneous potential functions converges in direction to a generalized maximum-margin solution for linear classification problems, thereby answering a long-standing question in the classification setting. Further, we show that MD can be implemented efficiently and under suitable conditions, enjoys fast convergence. Through comprehensive experiments, we demonstrate that MD is a versatile method to produce learned models with different regularizers, which in turn have different generalization performances.

The Inductive Bias of Flatness Regularization for Deep Matrix Factorization

Recent works on over-parameterized neural networks have shown that the stochasticity in optimizers has the implicit regularization effect of minimizing the sharpness of the loss function (in particular, the trace of its Hessian) over the family zero-loss solutions. More explicit forms of flatness regularization also empirically improve the generalization performance. However, it remains unclear why and when flatness regularization leads to better generalization. This work takes the first step toward understanding the inductive bias of the minimum trace of the Hessian solutions in an important setting: learning deep linear networks from linear measurements, also known as \emph{deep matrix factorization}. We show that for all depth greater than one, with the standard Restricted Isometry Property (RIP) on the measurements, minimizing the trace of Hessian is approximately equivalent to minimizing the Schatten 1-norm of the corresponding end-to-end matrix parameters (i.e., the product of all layer matrices), which in turn leads to better generalization. We empirically verify our theoretical findings on synthetic datasets.

Projection-Free Online Convex Optimization via Efficient Newton Iterations

This paper presents new projection-free algorithms for Online Convex Optimization (OCO) over a convex domain $\mathcal{K} \subset \mathbb{R}^d$. Classical OCO algorithms (such as Online Gradient Descent) typically need to perform Euclidean projections onto the convex set $\cK$ to ensure feasibility of their iterates. Alternative algorithms, such as those based on the Frank-Wolfe method, swap potentially-expensive Euclidean projections onto $\mathcal{K}$ for linear optimization over $\mathcal{K}$. However, such algorithms have a sub-optimal regret in OCO compared to projection-based algorithms. In this paper, we look at a third type of algorithms that output approximate Newton iterates using a self-concordant barrier for the set of interest. The use of a self-concordant barrier automatically ensures feasibility without the need for projections. However, the computation of the Newton iterates requires a matrix inverse, which can still be expensive. As our main contribution, we show how the stability of the Newton iterates can be leveraged to compute the inverse Hessian only a vanishing fraction of the rounds, leading to a new efficient projection-free OCO algorithm with a state-of-the-art regret bound.

When does Metropolized Hamiltonian Monte Carlo provably outperform Metropolis-adjusted Langevin algorithm?

We analyze the mixing time of Metropolized Hamiltonian Monte Carlo (HMC) with the leapfrog integrator to sample from a distribution on $\mathbb{R}^d$ whose log-density is smooth, has Lipschitz Hessian in Frobenius norm and satisfies isoperimetry. We bound the gradient complexity to reach $\epsilon$ error in total variation distance from a warm start by $\tilde O(d^{1/4}\text{polylog}(1/\epsilon))$ and demonstrate the benefit of choosing the number of leapfrog steps to be larger than 1. To surpass previous analysis on Metropolis-adjusted Langevin algorithm (MALA) that has $\tilde{O}(d^{1/2}\text{polylog}(1/\epsilon))$ dimension dependency in Wu et al. (2022), we reveal a key feature in our proof that the joint distribution of the location and velocity variables of the discretization of the continuous HMC dynamics stays approximately invariant. This key feature, when shown via induction over the number of leapfrog steps, enables us to obtain estimates on moments of various quantities that appear in the acceptance rate control of Metropolized HMC. Moreover, to deal with another bottleneck on the HMC proposal distribution overlap control in the literature, we provide a new approach to upper bound the Kullback-Leibler divergence between push-forwards of the Gaussian distribution through HMC dynamics initialized at two different points. Notably, our analysis does not require log-concavity or independence of the marginals, and only relies on an isoperimetric inequality. To illustrate the applicability of our result, several examples of natural functions that fall into our framework are discussed.

A Simple Proof of the Mixing of Metropolis-Adjusted Langevin Algorithm under Smoothness and Isoperimetry

We study the mixing time of Metropolis-Adjusted Langevin algorithm (MALA) for sampling a target density on $\mathbb{R}^d$. We assume that the target density satisfies $\psi_\mu$-isoperimetry and that the operator norm and trace of its Hessian are bounded by $L$ and $\Upsilon$ respectively. Our main result establishes that, from a warm start, to achieve $\epsilon$-total variation distance to the target density, MALA mixes in $O\left(\frac{(L\Upsilon)^{\frac12}}{\psi_\mu^2} \log\left(\frac{1}{\epsilon}\right)\right)$ iterations. Notably, this result holds beyond the log-concave sampling setting and the mixing time depends on only $\Upsilon$ rather than its upper bound $L d$. In the $m$-strongly logconcave and $L$-log-smooth sampling setting, our bound recovers the previous minimax mixing bound of MALA~\cite{wu2021minimax}.

Sampling with Barriers: Faster Mixing via Lewis Weights

We analyze Riemannian Hamiltonian Monte Carlo (RHMC) for sampling a polytope defined by $m$ inequalities in $\R^n$ endowed with the metric defined by the Hessian of a self-concordant convex barrier function. We use a hybrid of the $p$-Lewis weight barrier and the standard logarithmic barrier and prove that the mixing rate is bounded by $\tilde O(m^{1/3}n^{4/3})$, improving on the previous best bound of $\tilde O(mn^{2/3})$, based on the log barrier. Our analysis overcomes several technical challenges to establish this result, in the process deriving smoothness bounds on Hamiltonian curves and extending self-concordance notions to the infinity norm; both properties appear to be of independent interest.

Optimal algorithms for group distributionally robust optimization and beyond

Distributionally robust optimization (DRO) can improve the robustness and fairness of learning methods. In this paper, we devise stochastic algorithms for a class of DRO problems including group DRO, subpopulation fairness, and empirical conditional value at risk (CVaR) optimization. Our new algorithms achieve faster convergence rates than existing algorithms for multiple DRO settings. We also provide a new information-theoretic lower bound that implies our bounds are tight for group DRO. Empirically, too, our algorithms outperform known methods

Quasi-Newton Steps for Efficient Online Exp-Concave Optimization

The aim of this paper is to design computationally-efficient and optimal algorithms for the online and stochastic exp-concave optimization settings. Typical algorithms for these settings, such as the Online Newton Step (ONS), can guarantee a $O(d\ln T)$ bound on their regret after $T$ rounds, where $d$ is the dimension of the feasible set. However, such algorithms perform so-called generalized projections whenever their iterates step outside the feasible set. Such generalized projections require $\Omega(d^3)$ arithmetic operations even for simple sets such a Euclidean ball, making the total runtime of ONS of order $d^3 T$ after $T$ rounds, in the worst-case. In this paper, we side-step generalized projections by using a self-concordant barrier as a regularizer to compute the Newton steps. This ensures that the iterates are always within the feasible set without requiring projections. This approach still requires the computation of the inverse of the Hessian of the barrier at every step. However, using the stability properties of the Newton steps, we show that the inverse of the Hessians can be efficiently approximated via Taylor expansions for most rounds, resulting in a $O(d^2 T +d^\omega \sqrt{T})$ total computational complexity, where $\omega$ is the exponent of matrix multiplication. In the stochastic setting, we show that this translates into a $O(d^3/\epsilon)$ computational complexity for finding an $\epsilon$-suboptimal point, answering an open question by Koren 2013. We first show these new results for the simple case where the feasible set is a Euclidean ball. Then, to move to general convex set, we use a reduction to Online Convex Optimization over the Euclidean ball. Our final algorithm can be viewed as a more efficient version of ONS.

Bandit Algorithms for Prophet Inequality and Pandora's Box

The Prophet Inequality and Pandora's Box problems are fundamental stochastic problem with applications in Mechanism Design, Online Algorithms, Stochastic Optimization, Optimal Stopping, and Operations Research. A usual assumption in these works is that the probability distributions of the $n$ underlying random variables are given as input to the algorithm. Since in practice these distributions need to be learned, we initiate the study of such stochastic problems in the Multi-Armed Bandits model. In the Multi-Armed Bandits model we interact with $n$ unknown distributions over $T$ rounds: in round $t$ we play a policy $x^{(t)}$ and receive a partial (bandit) feedback on the performance of $x^{(t)}$. The goal is to minimize the regret, which is the difference over $T$ rounds in the total value of the optimal algorithm that knows the distributions vs. the total value of our algorithm that learns the distributions from the partial feedback. Our main results give near-optimal $\tilde{O}(\mathsf{poly}(n)\sqrt{T})$ total regret algorithms for both Prophet Inequality and Pandora's Box. Our proofs proceed by maintaining confidence intervals on the unknown indices of the optimal policy. The exploration-exploitation tradeoff prevents us from directly refining these confidence intervals, so the main technique is to design a regret upper bound that is learnable while playing low-regret Bandit policies.