A machine learning model may have several features, but some feature might have a higher impact on the output than others. For example, if a model is predicting apartment prices, the locality of the apartment might have a higher impact on the output than the number of floors the apartment building has. Hence, we come up with the concept of weights. Each feature is associated with a weight (a number) i.e. the higher the feature has an impact on the output, the larger the weight associated with it. But how do you decide what weight should be assigned to each feature?

Non-convex optimization problems are ubiquitous in machine learning, especially in Deep Learning. While such complex problems can often be successfully optimized in practice by using stochastic gradient descent (SGD), theoretical analysis cannot adequately explain this success. In particular, the standard analyses do not show global convergence of SGD on non-convex functions, and instead show convergence to stationary points (which can also be local minima or saddle points). We identify a broad class of nonconvex functions for which we can show that perturbed SGD (gradient descent perturbed by stochastic noise -- covering SGD as a special case) converges to a global minimum (or a neighborhood thereof), in contrast to gradient descent without noise that can get stuck in local minima far from a global solution. For example, on non-convex functions that are relatively close to a convex-like (strongly convex or PL) function we show that SGD can converge linearly to a global optimum.

We consider Ising models on the hypercube with a general interaction matrix $J$, and give a polynomial time sampling algorithm when all but $O(1)$ eigenvalues of $J$ lie in an interval of length one, a situation which occurs in many models of interest. This was previously known for the Glauber dynamics when *all* eigenvalues fit in an interval of length one; however, a single outlier can force the Glauber dynamics to mix torpidly. Our general result implies the first polynomial time sampling algorithms for low-rank Ising models such as Hopfield networks with a fixed number of patterns and Bayesian clustering models with low-dimensional contexts, and greatly improves the polynomial time sampling regime for the antiferromagnetic/ferromagnetic Ising model with inconsistent field on expander graphs. It also improves on previous approximation algorithm results based on the naive mean-field approximation in variational methods and statistical physics. Our approach is based on a new fusion of ideas from the MCMC and variational inference worlds. As part of our algorithm, we define a new nonconvex variational problem which allows us to sample from an exponential reweighting of a distribution by a negative definite quadratic form, and show how to make this procedure provably efficient using stochastic gradient descent. On top of this, we construct a new simulated tempering chain (on an extended state space arising from the Hubbard-Stratonovich transform) which overcomes the obstacle posed by large positive eigenvalues, and combine it with the SGD-based sampler to solve the full problem.

We consider the distributed stochastic gradient descent problem, where a main node distributes gradient calculations among $n$ workers from which at most $b \leq n$ can be utilized in parallel. By assigning tasks to all the workers and waiting only for the $k$ fastest ones, the main node can trade-off the error of the algorithm with its runtime by gradually increasing $k$ as the algorithm evolves. However, this strategy, referred to as adaptive k sync, can incur additional costs since it ignores the computational efforts of slow workers. We propose a cost-efficient scheme that assigns tasks only to $k$ workers and gradually increases $k$. As the response times of the available workers are unknown to the main node a priori, we utilize a combinatorial multi-armed bandit model to learn which workers are the fastest while assigning gradient calculations, and to minimize the effect of slow workers. Assuming that the mean response times of the workers are independent and exponentially distributed with different means, we give empirical and theoretical guarantees on the regret of our strategy, i.e., the extra time spent to learn the mean response times of the workers. Compared to adaptive k sync, our scheme achieves significantly lower errors with the same computational efforts while being inferior in terms of speed.

This work shows that a diverse collection of linear optimization methods, when run on general data, fail to overfit, despite lacking any explicit constraints or regularization: with high probability, their trajectories stay near the curve of optimal constrained solutions over the population distribution. This analysis is powered by an elementary but flexible proof scheme which can handle many settings, summarized as follows. Firstly, the data can be general: unlike other implicit bias works, it need not satisfy large margin or other structural conditions, and moreover can arrive sequentially IID, sequentially following a Markov chain, as a batch, and lastly it can have heavy tails. Secondly, while the main analysis is for mirror descent, rates are also provided for the Temporal-Difference fixed-point method from reinforcement learning; all prior high probability analyses in these settings required bounded iterates, bounded updates, bounded noise, or some equivalent. Thirdly, the losses are general, and for instance the logistic and squared losses can be handled simultaneously, unlike other implicit bias works. In all of these settings, not only is low population error guaranteed with high probability, but moreover low sample complexity is guaranteed so long as there exists any low-complexity near-optimal solution, even if the global problem structure and in particular global optima have high complexity.

We develop a new continuous-time stochastic gradient descent method for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm continuously updates the SDE model's parameters using an estimate for the gradient of the stationary distribution. The gradient estimate is simultaneously updated, asymptotically converging to the direction of steepest descent. We rigorously prove convergence of our online algorithm for linear SDE models and present numerical results for nonlinear examples. The proof requires analysis of the fluctuations of the parameter evolution around the direction of steepest descent. Bounds on the fluctuations are challenging to obtain due to the online nature of the algorithm (e.g., the stationary distribution will continuously change as the parameters change). We prove bounds for the solutions of a new class of Poisson partial differential equations, which are then used to analyze the parameter fluctuations in the algorithm.

We study convergence rates of AdaGrad-Norm as an exemplar of adaptive stochastic gradient methods (SGD), where the step sizes change based on observed stochastic gradients, for minimizing non-convex, smooth objectives. Despite their popularity, the analysis of adaptive SGD lags behind that of non adaptive methods in this setting. Specifically, all prior works rely on some subset of the following assumptions: (i) uniformly-bounded gradient norms, (ii) uniformly-bounded stochastic gradient variance (or even noise support), (iii) conditional independence between the step size and stochastic gradient. In this work, we show that AdaGrad-Norm exhibits an order optimal convergence rate of $\mathcal{O}\left(\frac{\mathrm{poly}\log(T)}{\sqrt{T}}\right)$ after $T$ iterations under the same assumptions as optimally-tuned non adaptive SGD (unbounded gradient norms and affine noise variance scaling), and crucially, without needing any tuning parameters. We thus establish that adaptive gradient methods exhibit order-optimal convergence in much broader regimes than previously understood.

Federated learning (FedL) has emerged as a popular technique for distributing model training over a set of wireless devices, via iterative local updates (at devices) and global aggregations (at the server). In this paper, we develop parallel successive learning (PSL), which expands the FedL architecture along three dimensions: (i) Network, allowing decentralized cooperation among the devices via device-to-device (D2D) communications. (ii) Heterogeneity, interpreted at three levels: (ii-a) Learning: PSL considers heterogeneous number of stochastic gradient descent iterations with different mini-batch sizes at the devices; (ii-b) Data: PSL presumes a dynamic environment with data arrival and departure, where the distributions of local datasets evolve over time, captured via a new metric for model/concept drift. (ii-c) Device: PSL considers devices with different computation and communication capabilities. (iii) Proximity, where devices have different distances to each other and the access point. PSL considers the realistic scenario where global aggregations are conducted with idle times in-between them for resource efficiency improvements, and incorporates data dispersion and model dispersion with local model condensation into FedL. Our analysis sheds light on the notion of cold vs. warmed up models, and model inertia in distributed machine learning. We then propose network-aware dynamic model tracking to optimize the model learning vs. resource efficiency tradeoff, which we show is an NP-hard signomial programming problem. We finally solve this problem through proposing a general optimization solver. Our numerical results reveal new findings on the interdependencies between the idle times in-between the global aggregations, model/concept drift, and D2D cooperation configuration.

Benign overfitting, the phenomenon where interpolating models generalize well in the presence of noisy data, was first observed in neural network models trained with gradient descent. To better understand this empirical observation, we consider the generalization error of two-layer neural networks trained to interpolation by gradient descent on the logistic loss following random initialization. We assume the data comes from well-separated class-conditional log-concave distributions and allow for a constant fraction of the training labels to be corrupted by an adversary. We show that in this setting, neural networks exhibit benign overfitting: they can be driven to zero training error, perfectly fitting any noisy training labels, and simultaneously achieve test error close to the Bayes-optimal error. In contrast to previous work on benign overfitting that require linear or kernel-based predictors, our analysis holds in a setting where both the model and learning dynamics are fundamentally nonlinear.

It is known that when the statistical models are singular, i.e., the Fisher information matrix at the true parameter is degenerate, the fixed step-size gradient descent algorithm takes polynomial number of steps in terms of the sample size $n$ to converge to a final statistical radius around the true parameter, which can be unsatisfactory for the application. To further improve that computational complexity, we consider the utilization of the second-order information in the design of optimization algorithms. Specifically, we study the normalized gradient descent (NormGD) algorithm for solving parameter estimation in parametric statistical models, which is a variant of gradient descent algorithm whose step size is scaled by the maximum eigenvalue of the Hessian matrix of the empirical loss function of statistical models. When the population loss function, i.e., the limit of the empirical loss function when $n$ goes to infinity, is homogeneous in all directions, we demonstrate that the NormGD iterates reach a final statistical radius around the true parameter after a logarithmic number of iterations in terms of $n$. Therefore, for fixed dimension $d$, the NormGD algorithm achieves the optimal overall computational complexity $\mathcal{O}(n)$ to reach the final statistical radius. This computational complexity is cheaper than that of the fixed step-size gradient descent algorithm, which is of the order $\mathcal{O}(n^{\tau})$ for some $\tau > 1$, to reach the same statistical radius. We illustrate our general theory under two statistical models: generalized linear models and mixture models, and experimental results support our prediction with general theory.