Run Procrustes, Run! On the convergence of accelerated Procrustes Flow

In this work, we present theoretical results on the convergence of non-convex accelerated gradient descent in matrix factorization models. The technique is applied to matrix sensing problems with squared loss, for the estimation of a rank $r$ optimal solution $X^\star \in \mathbb{R}^{n \times n}$. We show that the acceleration leads to linear convergence rate, even under non-convex settings where the variable $X$ is represented as $U U^\top$ for $U \in \mathbb{R}^{n \times r}$. Our result has the same dependence on the condition number of the objective --and the optimal solution-- as that of the recent results on non-accelerated algorithms. However, acceleration is observed in practice, both in synthetic examples and in two real applications: neuronal multi-unit activities recovery from single electrode recordings, and quantum state tomography on quantum computing simulators.

Implicit regularization and solution uniqueness in over-parameterized matrix sensing

We consider whether algorithmic choices in over-parameterized linear matrix factorization introduce implicit regularization. We focus on noiseless matrix sensing over rank-$r$ positive semi-definite (PSD) matrices in $\mathbb{R}^{n \times n}$, with a sensing mechanism that satisfies the restricted isometry property (RIP). The algorithm we study is that of \emph{factored gradient descent}, where we model the low-rankness and PSD constraints with the factorization $UU^\top$, where $U \in \mathbb{R}^{n \times r}$. Surprisingly, recent work argues that the choice of $r \leq n$ is not pivotal: even setting $U \in \mathbb{R}^{n \times n}$ is sufficient for factored gradient descent to find the rank-$r$ solution, which suggests that operating over the factors leads to an implicit regularization. In this note, we provide a different perspective. We show that, in the noiseless case, under certain conditions, the PSD constraint by itself is sufficient to lead to a unique rank-$r$ matrix recovery, without implicit or explicit low-rank regularization. \emph{I.e.}, under assumptions, the set of PSD matrices, that are consistent with the observed data, is a singleton, irrespective of the algorithm used.

Simple and practical algorithms for $\ell_p$-norm low-rank approximation

We propose practical algorithms for entrywise $\ell_p$-norm low-rank approximation, for $p = 1$ or $p = \infty$. The proposed framework, which is non-convex and gradient-based, is easy to implement and typically attains better approximations, faster, than state of the art. From a theoretical standpoint, we show that the proposed scheme can attain $(1 + \varepsilon)$-OPT approximations. Our algorithms are not hyperparameter-free: they achieve the desiderata only assuming algorithm's hyperparameters are known a priori---or are at least approximable. I.e., our theory indicates what problem quantities need to be known, in order to get a good solution within polynomial time, and does not contradict to recent inapproximabilty results, as in [46].

Approximate Newton-based statistical inference using only stochastic gradients

We present a novel inference framework for convex empirical risk minimization, using approximate stochastic Newton steps. The proposed algorithm is based on the notion of finite differences and allows the approximation of a Hessian-vector product from first-order information. In theory, our method efficiently computes the statistical error covariance in $M$-estimation, both for unregularized convex learning problems and high-dimensional LASSO regression, without using exact second order information, or resampling the entire data set. In practice, we demonstrate the effectiveness of our framework on large-scale machine learning problems, that go even beyond convexity: as a highlight, our work can be used to detect certain adversarial attacks on neural networks.

IHT dies hard: Provable accelerated Iterative Hard Thresholding

We study --both in theory and practice-- the use of momentum motions in classic iterative hard thresholding (IHT) methods. By simply modifying plain IHT, we investigate its convergence behavior on convex optimization criteria with non-convex constraints, under standard assumptions. In diverse scenaria, we observe that acceleration in IHT leads to significant improvements, compared to state of the art projected gradient descent and Frank-Wolfe variants. As a byproduct of our inspection, we study the impact of selecting the momentum parameter: similar to convex settings, two modes of behavior are observed --"rippling" and linear-- depending on the level of momentum.

Statistical inference using SGD

We present a novel method for frequentist statistical inference in $M$-estimation problems, based on stochastic gradient descent (SGD) with a fixed step size: we demonstrate that the average of such SGD sequences can be used for statistical inference, after proper scaling. An intuitive analysis using the Ornstein-Uhlenbeck process suggests that such averages are asymptotically normal. From a practical perspective, our SGD-based inference procedure is a first order method, and is well-suited for large scale problems. To show its merits, we apply it to both synthetic and real datasets, and demonstrate that its accuracy is comparable to classical statistical methods, while requiring potentially far less computation.

Algorithms for Learning Sparse Additive Models with Interactions in High Dimensions

A function $f: \mathbb{R}^d \rightarrow \mathbb{R}$ is a Sparse Additive Model (SPAM), if it is of the form $f(\mathbf{x}) = \sum_{l \in \mathcal{S}}\phi_{l}(x_l)$ where $\mathcal{S} \subset [d]$, $|\mathcal{S}| \ll d$. Assuming $\phi$'s, $\mathcal{S}$ to be unknown, there exists extensive work for estimating $f$ from its samples. In this work, we consider a generalized version of SPAMs, that also allows for the presence of a sparse number of second order interaction terms. For some $\mathcal{S}_1 \subset [d], \mathcal{S}_2 \subset {[d] \choose 2}$, with $|\mathcal{S}_1| \ll d, |\mathcal{S}_2| \ll d^2$, the function $f$ is now assumed to be of the form: $\sum_{p \in \mathcal{S}_1}\phi_{p} (x_p) + \sum_{(l,l^{\prime}) \in \mathcal{S}_2}\phi_{(l,l^{\prime})} (x_l,x_{l^{\prime}})$. Assuming we have the freedom to query $f$ anywhere in its domain, we derive efficient algorithms that provably recover $\mathcal{S}_1,\mathcal{S}_2$ with finite sample bounds. Our analysis covers the noiseless setting where exact samples of $f$ are obtained, and also extends to the noisy setting where the queries are corrupted with noise. For the noisy setting in particular, we consider two noise models namely: i.i.d Gaussian noise and arbitrary but bounded noise. Our main methods for identification of $\mathcal{S}_2$ essentially rely on estimation of sparse Hessian matrices, for which we provide two novel compressed sensing based schemes. Once $\mathcal{S}_1, \mathcal{S}_2$ are known, we show how the individual components $\phi_p$, $\phi_{(l,l^{\prime})}$ can be estimated via additional queries of $f$, with uniform error bounds. Lastly, we provide simulation results on synthetic data that validate our theoretical findings.

A single-phase, proximal path-following framework

We propose a new proximal, path-following framework for a class of constrained convex problems. We consider settings where the nonlinear---and possibly non-smooth---objective part is endowed with a proximity operator, and the constraint set is equipped with a self-concordant barrier. Our approach relies on the following two main ideas. First, we re-parameterize the optimality condition as an auxiliary problem, such that a good initial point is available; by doing so, a family of alternative paths towards the optimum is generated. Second, we combine the proximal operator with path-following ideas to design a single-phase, proximal, path-following algorithm. Our method has several advantages. First, it allows handling non-smooth objectives via proximal operators; this avoids lifting the problem dimension in order to accommodate non-smooth components in optimization. Second, it consists of only a \emph{single phase}: While the overall convergence rate of classical path-following schemes for self-concordant objectives does not suffer from the initialization phase, proximal path-following schemes undergo slow convergence, in order to obtain a good starting point \cite{TranDinh2013e}. In this work, we show how to overcome this limitation in the proximal setting and prove that our scheme has the same $\mathcal{O}(\sqrt{\nu}\log(1/\varepsilon))$ worst-case iteration-complexity with standard approaches \cite{Nesterov2004,Nesterov1994} without requiring an initial phase, where $\nu$ is the barrier parameter and $\varepsilon$ is a desired accuracy. Finally, our framework allows errors in the calculation of proximal-Newton directions, without sacrificing the worst-case iteration complexity. We demonstrate the merits of our algorithm via three numerical examples, where proximal operators play a key role.

Finding Low-Rank Solutions via Non-Convex Matrix Factorization, Efficiently and Provably

A rank-$r$ matrix $X \in \mathbb{R}^{m \times n}$ can be written as a product $U V^\top$, where $U \in \mathbb{R}^{m \times r}$ and $V \in \mathbb{R}^{n \times r}$. One could exploit this observation in optimization: e.g., consider the minimization of a convex function $f(X)$ over rank-$r$ matrices, where the set of rank-$r$ matrices is modeled via the factorization $UV^\top$. Though such parameterization reduces the number of variables, and is more computationally efficient (of particular interest is the case $r \ll \min\{m, n\}$), it comes at a cost: $f(UV^\top)$ becomes a non-convex function w.r.t. $U$ and $V$. We study such parameterization for optimization of generic convex objectives $f$, and focus on first-order, gradient descent algorithmic solutions. We propose the Bi-Factored Gradient Descent (BFGD) algorithm, an efficient first-order method that operates on the $U, V$ factors. We show that when $f$ is (restricted) smooth, BFGD has local sublinear convergence, and linear convergence when $f$ is both (restricted) smooth and (restricted) strongly convex. For several key applications, we provide simple and efficient initialization schemes that provide approximate solutions good enough for the above convergence results to hold.

Provable Burer-Monteiro factorization for a class of norm-constrained matrix problems

We study the projected gradient descent method on low-rank matrix problems with a strongly convex objective. We use the Burer-Monteiro factorization approach to implicitly enforce low-rankness; such factorization introduces non-convexity in the objective. We focus on constraint sets that include both positive semi-definite (PSD) constraints and specific matrix norm-constraints. Such criteria appear in quantum state tomography and phase retrieval applications. We show that non-convex projected gradient descent favors local linear convergence in the factored space. We build our theory on a novel descent lemma, that non-trivially extends recent results on the unconstrained problem. The resulting algorithm is Projected Factored Gradient Descent, abbreviated as ProjFGD, and shows superior performance compared to state of the art on quantum state tomography and sparse phase retrieval applications.