There is a stark disparity between the learning rate schedules used in the practice of large scale machine learning and what are considered admissible learning rate schedules prescribed in the theory of stochastic approximation. Recent results, such as in the 'super-convergence' methods which use oscillating learning rates, serve to emphasize this point even more.
One plausible explanation is that non-convex neural network training procedures are better suited to the use of fundamentally different learning rate  schedules, such as the ``cut the learning rate every constant number of epochs'' method (which more closely resembles an exponentially decaying learning rate schedule); note that this widely used schedule is in stark contrast to the polynomial decay schemes prescribed in the stochastic approximation literature, which are indeed shown to be (worst case) optimal for classes of convex optimization problems.

The main contribution of this work shows that the picture is far more nuanced, where we do not even need to move to non-convex optimization to show other learning rate schemes can be far more effective. In fact, even for the simple case of stochastic linear regression with a fixed time horizon, the rate achieved by any polynomial decay scheme is sub-optimal compared to the statistical minimax rate (by a factor of condition number); in contrast the ```''cut the learning rate every constant number of epochs'' provides an exponential improvement (depending only logarithmically on the condition number) compared to any polynomial decay scheme.  Finally, it is important to ask if our theoretical insights are somehow fundamentally tied to quadratic loss minimization (where we have circumvented minimax lower bounds for more general convex optimization problems)? Here, we conjecture that recent results which make the gradient norm small at a near optimal rate, for both convex and non-convex optimization, may also provide more insights into learning rate schedules used in practice.

Population risk---the expectation of the loss over the sampling mechanism---is always of primary interest in machine learning. However, learning algorithms only have access to empirical risk, which is the average loss over training examples. Although the two risks are typically guaranteed to be pointwise close, for applications with nonconvex nonsmooth losses (such as modern deep networks), the effects of sampling can transform a well-behaved population risk into an empirical risk with a landscape that is problematic for optimization. The empirical risk can be nonsmooth, and it may have many additional local minima. This paper considers a general optimization framework which aims to find approximate local minima of a smooth nonconvex function F (population risk) given only access to the function value of another function f (empirical risk), which is pointwise close to F. We propose a simple algorithm based on stochastic gradient descent (SGD) on a smoothed version of f which is guaranteed to find a second-order stationary point if f and F are close enough, thus escaping all saddle points of F and all the additional local minima introduced by f. We also provide an almost matching lower bound showing that our SGD-based approach achieves the optimal trade-off between the relevant parameters, as well as the optimal dependence on problem dimension d, among all algorithms making a polynomial number of queries. As a concrete example, we show that our results can be directly used to give sample complexities for learning a ReLU unit, whose empirical risk is nonsmooth.

Matrix completion is a well-studied problem with many machine learning applications. In practice, the problem is often solved by non-convex optimization algorithms. However, the current theoretical analysis for non-convex algorithms relies heavily on the assumption that every entry is observed with exactly the same probability p, which is not realistic in practice.
In this paper, we investigate a more realistic semi-random model, where the probability of observing each entry is at least p. Even with this mild semi-random perturbation, we can construct counter-examples where existing non-convex algorithms get stuck in bad local optima.
In light of the negative results, we propose a pre-processing step that tries to re-weight the semi-random input, so that it becomes "similar" to a random input. We give a nearly-linear time algorithm for this problem, and show that after our pre-processing, all the local minima of the non-convex objective can be used to approximately recover the underlying ground-truth matrix.

A key task in Bayesian statistics is sampling from distributions that are only specified up to a partition function (i.e., constant of proportionality). However, without any assumptions, sampling (even approximately) can be #P-hard, and few works have provided "beyond worst-case" guarantees for such settings. For log-concave distributions, classical results going back to Bakry and \'Emery (1985) show that natural continuous-time Markov chains called Langevin diffusions mix in polynomial time. The most salient feature of log-concavity violated in practice is uni-modality: commonly, the distributions we wish to sample from are multi-modal. In the presence of multiple deep and well-separated modes, Langevin diffusion suffers from torpid mixing. We address this problem by combining Langevin diffusion with simulated tempering. The result is a Markov chain that mixes more rapidly by transitioning between different temperatures of the distribution. We analyze this Markov chain for the canonical multi-modal distribution: a mixture of gaussians (of equal variance). The algorithm based on our Markov chain provably samples from distributions that are close to mixtures of gaussians, given access to the gradient of the log-pdf. For the analysis, we use a spectral decomposition theorem for graphs (Gharan and Trevisan, 2014) and a Markov chain decomposition technique (Madras and Randall, 2002).

Direct policy gradient methods for reinforcement learning and continuous control problems are a popular approach for a variety of reasons: 1) they are easy to implement without explicit knowledge of the underlying model 2) they are an "end-to-end" approach, directly optimizing the performance metric of interest 3) they inherently allow for richly parameterized policies. A notable drawback is that even in the most basic continuous control problem (that of linear quadratic regulators), these methods must solve a non-convex optimization problem, where little is understood about their efficiency from both computational and statistical perspectives. In contrast, system identification and model based planning in optimal control theory have a much more solid theoretical footing, where much is known with regards to their computational and statistical properties. This work bridges this gap showing that (model free) policy gradient methods globally converge to the optimal solution and are efficient (polynomially so in relevant problem dependent quantities) with regards to their sample and computational complexities.

We consider the problem of learning a one-hidden-layer neural network: we assume the input x is from Gaussian distribution and the label y = a*\sigma(Bx), where a is a nonnegative vector in m dimensions, B is a full-rank weight matrix. We first give an analytic formula for the population risk of the standard squared loss and demonstrate that it implicitly attempts to decompose a sequence of low-rank tensors simultaneously. Inspired by the formula, we design a non-convex objective function G whose landscape is guaranteed to have the following properties:
1. All local minima of G are also global minima.
2. All global minima of G correspond to the ground truth parameters.
3. The value and gradient of G can be estimated using samples.
With these properties, stochastic gradient descent on G provably converges to the global minimum and learn the ground-truth parameters. We also prove finite sample complexity result and validate the results by simulations.

In this paper we develop a new framework that captures the common landscape underlying the common non-convex low-rank matrix problems including matrix sensing, matrix completion and robust PCA. In particular, we show for all above problems (including asymmetric cases): 1) all local minima are also globally optimal; 2) no high-order saddle points exists. These results explain why simple algorithms such as stochastic gradient descent have global converge, and efficiently optimize these non-convex objective functions in practice. Our framework connects and simplifies the existing analyses on optimization landscapes for matrix sensing and symmetric matrix completion. The framework naturally leads to new results for asymmetric matrix completion and robust PCA.

This paper shows that a perturbed form of gradient descent converges to a second-order stationary point in a number iterations which depends only poly-logarithmically on dimension (i.e., it is almost "dimension-free"). The convergence rate of this procedure matches the well-known convergence rate of gradient descent to first-order stationary points, up to log factors. When all saddle points are non-degenerate, all second-order stationary points are local minima, and our result thus shows that perturbed gradient descent can escape saddle points almost for free. Our results can be directly applied to many machine learning applications, including deep learning. As a particular concrete example of such an application, we show that our results can be used directly to establish sharp global convergence rates for matrix factorization. Our results rely on a novel characterization of the geometry around saddle points, which may be of independent interest to the non-convex optimization community.

Non-convex optimization with local search heuristics has been widely used in machine learning, achieving many state-of-art results. It becomes increasingly important to understand why they can work for these NP-hard problems on typical data. The landscape of many objective functions in learning have been conjectured to have the geometric property that “all local optima are (approximately) global optima”, and thus they can be solved efficiently by local search algorithms. However, establishing such property can be very difficult.
In this paper we analyze the optimization landscape of the random over-complete tensor decomposition problem, which has many applications in unsupervised leaning, especially in learning latent variable models. In practice, it can be efficiently solved by gradient ascent on a non-convex objective. We show that for any small constant c > 0, among the set of points with function values (1 + c)-factor larger than the expectation of the function, all the local maxima are approximate global maxima. Previously, the best known result only characterizes the geometry in small neighborhoods around the true components. Our result implies that even with an initialization that is barely better than the random guess, the gradient ascent algorithm is guaranteed to solve this problem

Developing efficient and guaranteed nonconvex algorithms has been an important challenge in modern machine learning. Algorithms with good empirical performance such as stochastic gradient descent often lack theoretical guarantees. In this paper, we analyze the class of homotopy or continuation methods for global optimization of nonconvex functions. These methods start from an objective function that is efficient to optimize (e.g. convex), and progressively modify it to obtain the required objective, and the solutions are passed along the homotopy path. For the challenging problem of tensor PCA, we prove global convergence of the homotopy method in the "high noise" regime. The signal-to-noise requirement for our algorithm is tight in the sense that it matches the recovery guarantee for the best degree-4 sum-of-squares algorithm. In addition, we prove a phase transition along the homotopy path for tensor PCA. This allows to simplify the homotopy method to a local search algorithm, viz., tensor power iterations, with a specific initialization and a noise injection procedure, while retaining the theoretical guarantees.

Matrix completion is a basic machine learning problem that has wide applications, especially in collaborative filtering and recommender systems. Simple non-convex optimization algorithms are popular and effective in practice. Despite recent progress in proving various non-convex algorithms converge from a good initial point, it remains unclear why random or arbitrary initialization suffices in practice. We prove that the commonly used non-convex objective function for matrix completion has no spurious local minima -- all local minima must also be global. Therefore, many popular optimization algorithms such as (stochastic) gradient descent can provably solve matrix completion with arbitrary initialization in polynomial time.

Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have degenerate saddle points such that the first and second order derivatives cannot distinguish them with local optima. In this paper we use higher order derivatives to escape these saddle points: we design the first efficient algorithm guaranteed to converge to a third order local optimum (while existing techniques are at most second order). We also show that it is NP-hard to extend this further to finding fourth order local optima.

This paper considers the problem of canonical-correlation analysis (CCA) (Hotelling, 1936) and, more broadly, the generalized eigenvector problem for a pair of symmetric matrices. These are two fundamental problems in data analysis and scientific computing with numerous applications in machine learning and statistics (Shi and Malik, 2000; Hardoon et al., 2004; Witten et al., 2009).

We provide simple iterative algorithms, with improved runtimes, for solving these problems that are globally linearly convergent with moderate dependencies on the condition numbers and eigenvalue gaps of the matrices involved.

We obtain our results by reducing CCA to the top-k generalized eigenvector problem. We solve this problem through a general framework that simply requires black box access to an approximate linear system solver. Instantiating this framework with accelerated gradient descent we obtain a near linear running time.

Our algorithm is linear in the input size and the number of components k up to a log(k) factor. This is essential for handling large-scale matrices that appear in practice. To the best of our knowledge this is the first such algorithm with global linear convergence. We hope that our results prompt further research and ultimately improve the practical running time for performing these important data analysis procedures on large data sets.

We develop a family of accelerated stochastic algorithms that minimize sums of convex functions. Our algorithms improve upon the fastest running time for empirical risk minimization (ERM), and in particular linear least-squares regression, across a wide range of problem settings. To achieve this, we establish a framework based on the classical proximal point algorithm. Namely, we provide several algorithms that reduce the minimization of a strongly convex function to approximate minimizations of regularizations of the function. Using these results, we accelerate recent fast stochastic algorithms in a black-box fashion. Empirically, we demonstrate that the resulting algorithms exhibit notions of stability that are advantageous in practice. Both in theory and in practice, the provided algorithms reap the computational benefits of adding a large strongly convex regularization term, without incurring a corresponding bias to the original problem.

We analyze stochastic gradient descent for optimizing non-convex functions. In many cases for non-convex functions the goal is to find a reasonable local minimum, and the main concern is that gradient updates are trapped in saddle points. In this paper we identify strict saddle property for non-convex problem that allows for efficient optimization. Using this property we show that stochastic gradient descent converges to a local minimum in a polynomial number of iterations. To the best of our knowledge this is the first work that gives global convergence guarantees for stochastic gradient descent on non-convex functions with exponentially many local minima and saddle points.

Our analysis can be applied to orthogonal tensor decomposition, which is widely used in learning a rich class of latent variable models. We propose a new optimization formulation for the tensor decomposition problem that has strict saddle property. As a result we get the first online algorithm for orthogonal tensor decomposition with global convergence guarantee.

Many optimization problems that arise in science and engineering are those in which we only have a stochastic approximation to the underlying objective (e.g. estimation problems such as linear regression). That is, given some distribution D over functions f(x), we wish to minimize P(x)=E[f(x)], using as few samples from D as possible. In the absence of computational constraints, the empirical risk minimizer (ERM) -- the minimizer on a sample average of observed data -- is widely regarded as the estimation strategy of choice due to its desirable statistical convergence properties. Our goal is to do as well as the empirical risk minimizer on every problem while minimizing the use of computational resources such as running time and space usage.

This work provides a simple streaming algorithm for solving this problem, with performance guarantees competitive with that of the ERM. In particular, under standard regularity assumptions on D, our algorithm enjoys the following properties:

• The algorithm can be implemented in linear time with a single pass of the observed data, using space linear in the size of a single sample.
• The algorithm achieves the same statistical rate of convergence as the empirical risk minimizer on every problem D (even considering constant factors).
• The algorithm's performance depends on the initial error at a rate that decreases super-polynomially.

Moreover, we quantify the rate of convergence of both our algorithm and that of the ERM, showing that, after a number of samples that depend only on the regularity parameters, the streaming algorithm rapidly approaches the leading order rate of ERM in expectation.

We give a new algorithm for learning a two-layer neural network under a general class of input distributions. Assuming there is a ground-truth two-layer network y=Aσ(Wx)+ξ, where A,W are weight matrices, ξ represents noise, and the number of neurons in the hidden layer is no larger than the input or output, our algorithm is guaranteed to recover the parameters A,W of the ground-truth network. The only requirement on the input x is that it is symmetric, which still allows highly complicated and structured input. Our algorithm is based on the method-of-moments framework and extends several results in tensor decompositions. We use spectral algorithms to avoid the complicated non-convex optimization in learning neural networks. Experiments show that our algorithm can robustly learn the ground-truth neural network with a small number of samples for many symmetric input distributions.

Word embedding is a powerful tool in natural language processing. In this paper we consider the problem of word embedding composition --- given vector representations of two words, compute a vector for the entire phrase. We give a generative model that can capture specific syntactic relations between words. Under our model, we prove that the correlations between three words (measured by their PMI) form a tensor that has an approximate low rank Tucker decomposition. The result of the Tucker decomposition gives the word embeddings as well as a core tensor, which can be used to produce better compositions of the word embeddings. We also complement our theoretical results with experiments that verify our assumptions, and demonstrate the effectiveness of the new composition method.

Non-convex optimization with local search heuristics has been widely used in machine learning, achieving many state-of-art results. It becomes increasingly important to understand why they can work for these NP-hard problems on typical data. The landscape of many objective functions in learning have been conjectured to have the geometric property that “all local optima are (approximately) global optima”, and thus they can be solved efficiently by local search algorithms. However, establishing such property can be very difficult.
In this paper we analyze the optimization landscape of the random over-complete tensor decomposition problem, which has many applications in unsupervised leaning, especially in learning latent variable models. In practice, it can be efficiently solved by gradient ascent on a non-convex objective. We show that for any small constant c > 0, among the set of points with function values (1 + c)-factor larger than the expectation of the function, all the local maxima are approximate global maxima. Previously, the best known result only characterizes the geometry in small neighborhoods around the true components. Our result implies that even with an initialization that is barely better than the random guess, the gradient ascent algorithm is guaranteed to solve this problem

Many machine learning applications use latent variable models to explain structure in data, whereby visible variables (= coordinates of the given datapoint) are explained as a probabilistic function of some hidden variables. Finding parameters with the maximum likelihood is NP-hard even in very simple settings. In recent years, provably efficient algorithms were nevertheless developed for models with linear structures: topic models, mixture models, hidden markov models, etc. These algorithms use matrix or tensor decomposition, and make some reasonable assumptions about the parameters of the underlying model.
But matrix or tensor decomposition seems of little use when the latent variable model has nonlinearities. The current paper shows how to make progress: tensor decomposition is applied for learning the single-layer {\em noisy or} network, which is a textbook example of a Bayes net, and used for example in the classic QMR-DT software for diagnosing which disease(s) a patient may have by observing the symptoms he/she exhibits.
The technical novelty here, which should be useful in other settings in future, is analysis of tensor decomposition in presence of systematic error (i.e., where the noise/error is correlated with the signal, and doesn't decrease as number of samples goes to infinity). This requires rethinking all steps of tensor decomposition methods from the ground up.
For simplicity our analysis is stated assuming that the network parameters were chosen from a probability distribution but the method seems more generally applicable.

Developing efficient and guaranteed nonconvex algorithms has been an important challenge in modern machine learning. Algorithms with good empirical performance such as stochastic gradient descent often lack theoretical guarantees. In this paper, we analyze the class of homotopy or continuation methods for global optimization of nonconvex functions. These methods start from an objective function that is efficient to optimize (e.g. convex), and progressively modify it to obtain the required objective, and the solutions are passed along the homotopy path. For the challenging problem of tensor PCA, we prove global convergence of the homotopy method in the "high noise" regime. The signal-to-noise requirement for our algorithm is tight in the sense that it matches the recovery guarantee for the best degree-4 sum-of-squares algorithm. In addition, we prove a phase transition along the homotopy path for tensor PCA. This allows to simplify the homotopy method to a local search algorithm, viz., tensor power iterations, with a specific initialization and a noise injection procedure, while retaining the theoretical guarantees.

In many settings, we have multiple data sets (also called views) that capture different and overlapping aspects of the same phenomenon. We are often interested in finding patterns that are unique to one or to a subset of the views. For example, we might have one set of molecular observations and one set of physiological observations on the same group of individuals, and we want to quantify molecular patterns that are uncorrelated with physiology. Despite being a common problem, this is highly challenging when the correlations come from complex distributions. In this paper, we develop the general framework of Rich Component Analysis (RCA) to model settings where the observations from different views are driven by different sets of latent components, and each component can be a complex, high-dimensional distribution. We introduce algorithms based on cumulant extraction that provably learn each of the components without having to model the other components. We show how to integrate RCA with stochastic gradient descent into a meta-algorithm for learning general models, and demonstrate substantial improvement in accuracy on several synthetic and real datasets in both supervised and unsupervised tasks. Our method makes it possible to learn latent variable models when we don't have samples from the true model but only samples after complex perturbations.

Tensor rank and low-rank tensor decompositions have many applications in learning and complexity theory. Most known algorithms use unfoldings of tensors and can only handle rank up to n^{p/2} for a p-th order tensor. Previously no efficient algorithm can decompose 3rd order tensors when the rank is super-linear in the dimension. Using ideas from sum-of-squares hierarchy, we give the first quasi-polynomial time algorithm that can decompose a random 3rd order tensor decomposition when the rank is as large as n^{3/2}/polylog n.

We also give a polynomial time algorithm for certifying the injective norm of random low rank tensors. Our tensor decomposition algorithm exploits the relationship between injective norm and the tensor components. The proof relies on interesting tools for decoupling random variables to prove better matrix concentration bounds, which can be useful in other settings.

We analyze stochastic gradient descent for optimizing non-convex functions. In many cases for non-convex functions the goal is to find a reasonable local minimum, and the main concern is that gradient updates are trapped in saddle points. In this paper we identify strict saddle property for non-convex problem that allows for efficient optimization. Using this property we show that stochastic gradient descent converges to a local minimum in a polynomial number of iterations. To the best of our knowledge this is the first work that gives global convergence guarantees for stochastic gradient descent on non-convex functions with exponentially many local minima and saddle points.

Our analysis can be applied to orthogonal tensor decomposition, which is widely used in learning a rich class of latent variable models. We propose a new optimization formulation for the tensor decomposition problem that has strict saddle property. As a result we get the first online algorithm for orthogonal tensor decomposition with global convergence guarantee.

Efficiently learning mixture of Gaussians is a fundamental problem in statistics and learning theory. Given samples coming from a random one out of k Gaussian distributions in R^n, the learning problem asks to estimate the means and the covariance matrices of these Gaussians. This learning problem arises in many areas ranging from the natural sciences to the social sciences, and has also found many machine learning applications.

Unfortunately, learning mixture of Gaussians is an information theoretically hard problem: in order to learn the parameters up to a reasonable accuracy, the number of samples required is exponential in the number of Gaussian components in the worst case. In this work, we show that provided we are in high enough dimensions, the class of Gaussian mixtures is learnable in its most general form under a smoothed analysis framework, where the parameters are randomly perturbed from an adversarial starting point. In particular, given samples from a mixture of Gaussians with randomly perturbed parameters, when n > {\Omega}(k^2), we give an algorithm that learns the parameters with polynomial running time and using polynomial number of samples.

The central algorithmic ideas consist of new ways to decompose the moment tensor of the Gaussian mixture by exploiting its structural properties. The symmetries of this tensor are derived from the combinatorial structure of higher order moments of Gaussian distributions (sometimes referred to as Isserlis' theorem or Wick's theorem). We also develop new tools for bounding smallest singular values of structured random matrices, which could be useful in other smoothed analysis settings.

This is a series of ongoing works where we try to understand how to decompose overcomplete third-order tensors. In general, decomposing third order tensors can be useful in learning many latent variable models (multi-view model, mixture of Gaussians, etc.) Previous algorithms usually can either only handle the undercomplete regime where the number of components is smaller than the number of of dimensions, or require access of higher order tensors which often results in higher sample complexity and running time. Surprisingly, many heuristics (such as Tensor Power Method or Alternating Least Squares) work very well in practice. In these works we try to understand when is decomposing third-order tensors tractable. Several parts are available on arxiv.

The first part [Arxiv] tries to analyze Tensor Power Method. In this paper we show if there is a good initialization (that is constant-close to some component), Tensor Power Method will converge to vectors that are very close to the true component. We also give an algorithm that converges to the exact true component given the (already reasonably close) results of the tensor power method. Good initializations can be found if there are some supervised information, or by spectral method if ther number of component k is only a constant factor larger than the dimension d.

The second part [Arxiv] tries to understand how many samples do we need in order to estimate the moment tensor accurately enough for several models. We show surprisingly in many natural cases the sample complexity is nearly linear in the number of components. In general the bounds in this paper is tighter than many previous works.

The third part (more independent from the first two) [Arxiv] tries to understand how well does Tensor Power Method perform without really good initialization. We use techniques from Approximate Message Passing to analyze the dynamics of the tensor power method, and show that the initialization does not need to be very close to the true component for tensor power method to perform well. This partly explains why tensor power method still works in the mildly overcomplete settings.

See also Majid Janzamin's page about this project for more information.