Gating is a key feature in modern neural networks including LSTMs, GRUs and sparsely-gated deep neural networks. The backbone of such gated networks is a mixture-of-experts layer, where several experts make regression decisions and gating controls how to weigh the decisions in an input-dependent manner. Despite having such a prominent role in both modern and classical machine learning, very little is understood about parameter recovery of mixture-of-experts since gradient descent and EM algorithms are known to be stuck in local optima in such models. In this paper, we perform a careful analysis of the optimization landscape and show that with appropriately designed loss functions, gradient descent can indeed learn the parameters accurately. A key idea underpinning our results is the design of two {\em distinct} loss functions, one for recovering the expert parameters and another for recovering the gating parameters. We demonstrate the first sample complexity results for parameter recovery in this model for any algorithm and demonstrate significant performance gains over standard loss functions in numerical experiments.

Despite remarkable empirical success, the training dynamics of generative adversarial networks (GAN), which involves solving a minimax game using stochastic gradients, is still poorly understood. In this work, we analyze last-iterate convergence of simultaneous gradient descent (simGD) and its variants under the assumption of convex-concavity, guided by a continuous-time analysis with differential equations. First, we show that simGD, as is, converges with stochastic sub-gradients under strict convexity in the primal variable. Second, we generalize optimistic simGD to accommodate an optimism rate separate from the learning rate and show its convergence with full gradients. Finally, we present anchored simGD, a new method, and show convergence with stochastic subgradients.

We show that under mild conditions, Estimation of Distribution Algorithms (EDAs) can be written as variational Expectation-Maximization (EM) that uses a mixture of weighted particles as the approximate posterior. In the infinite particle limit, EDAs can be viewed as exact EM. Because EM sits on a rigorous statistical foundation and has been thoroughly analyzed, this connection provides a coherent framework with which to reason about EDAs. Importantly, the connection also suggests avenues for possible improvements to EDAs owing to our ability to leverage general statistical tools and generalizations of EM. For example, we make use of results about known EM convergence properties to propose an adaptive, hybrid EDA-gradient descent algorithm; this hybrid demonstrates better performance than either component of the hybrid on several canonical, non-convex test functions. We also demonstrate empirically that although one might hypothesize that reducing the variational gap could prove useful, it actually degrades performance of EDAs. Finally, we show that the connection between EM and EDAs provides us with a new perspective on why EDAs are performing approximate natural gradient descent.

Determining the optimal location of control cabinet components requires the exploration of a large configuration space. For real-world control cabinets it is impractical to evaluate all possible cabinet configurations. Therefore, we need to apply methods for intelligent exploration of cabinet configuration space that enable to find a near-optimal configuration without evaluation of all possible configurations. In this paper, we describe an approach for multi-objective optimization of control cabinet layout that is based on Pareto Simulated Annealing. Optimization aims at minimizing the total wire length used for interconnection of components and the heat convection within the cabinet. We simulate heat convection to study the warm air flow within the control cabinet and determine the optimal position of components that generate heat during the operation. We evaluate and demonstrate the effectiveness of our approach empirically for various control cabinet sizes and usage scenarios.

Data-driven model training is increasingly relying on finding Nash equilibria with provable techniques, e.g., for GANs and multi-agent RL. In this paper, we analyse a new extra-gradient method, that performs gradient extrapolations and updates on a random subset of players at each iteration. This approach provably exhibits the same rate of convergence as full extra-gradient in non-smooth convex games. We propose an additional variance reduction mechanism for this to hold for smooth convex games. Our approach makes extrapolation amenable to massive multiplayer settings, and brings empirical speed-ups, in particular when using cyclic sampling schemes. We demonstrate the efficiency of player sampling on large-scale non-smooth and non-strictly convex games. We show that the joint use of extrapolation and player sampling allows to train better GANs on CIFAR10.

Tensor decomposition is a well-known tool for multiway data analysis. This work proposes using stochastic gradients for efficient generalized canonical polyadic (GCP) tensor decomposition of large-scale tensors. GCP tensor decomposition is a recently proposed version of tensor decomposition that allows for a variety of loss functions such as logistic loss for binary data or Huber loss for robust estimation. The stochastic gradient is formed from randomly sampled elements of the tensor. For dense tensors, we simply use uniform sampling. For sparse tensors, we propose two types of stratified sampling that give precedence to sampling nonzeros. Numerical results demonstrate the advantages of the proposed approach and its scalability to large-scale problems.

Policy gradients methods are perhaps the most widely used class of reinforcement learning algorithms. These methods apply to complex, poorly understood, control problems by performing stochastic gradient descent over a parameterized class of polices. Unfortunately, even for simple control problems solvable by classical techniques, policy gradient algorithms face non-convex optimization problems and are widely understood to converge only to local minima. This work identifies structural properties -- shared by finite MDPs and several classic control problems -- which guarantee that policy gradient objective function has no suboptimal local minima despite being non-convex. When these assumptions are relaxed, our work gives conditions under which any local minimum is near-optimal, where the error bound depends on a notion of the expressive capacity of the policy class.

We propose a new metaheuristic training scheme that combines Stochastic Gradient Descent (SGD) and Discrete Optimization in an unconventional way. Our idea is to define a discrete neighborhood of the current SGD point containing a number of "potentially good moves" that exploit gradient information, and to search this neighborhood by using a classical metaheuristic scheme borrowed from Discrete Optimization. In the present paper we investigate the use of a simple Simulated Annealing (SA) metaheuristic that accepts/rejects a candidate new solution in the neighborhood with a probability that depends both on the new solution quality and on a parameter (the temperature) which is modified over time to lower the probability of accepting worsening moves. We use this scheme as an automatic way to perform hyper-parameter tuning, hence the title of the paper. A distinctive feature of our scheme is that hyper-parameters are modified within a single SGD execution (and not in an external loop, as customary) and evaluated on the fly on the current minibatch, i.e., their tuning is fully embedded within the SGD algorithm. The use of SA for training is not new, but previous proposals were mainly intended for non-differentiable objective functions for which SGD is not applied due to the lack of gradients. On the contrary, our SA method requires differentiability of (a proxy of) the loss function, and leverages on the availability of a gradient direction to define local moves that have a large probability to improve the current solution. Computational results on image classification (CIFAR-10) are reported, showing that the proposed approach leads to an improvement of the final validation accuracy for modern Deep Neural Networks such as ResNet34 and VGG16.

We introduce and analyze a parallel sequential Monte Carlo methodology for the numerical solution of optimization problems that involve the minimization of a cost function that consists of the sum of many individual components. The proposed scheme is a stochastic zeroth order optimization algorithm which demands only the capability to evaluate small subsets of components of the cost function. It can be depicted as a bank of samplers that generate particle approximations of several sequences of probability measures. These measures are constructed in such a way that they have associated probability density functions whose global maxima coincide with the global minima of the original cost function. The algorithm selects the best performing sampler and uses it to approximate a global minimum of the cost function. We prove analytically that the resulting estimator converges to a global minimum of the cost function almost surely and provide explicit convergence rates in terms of the number of generated Monte Carlo samples. We show, by way of numerical examples, that the algorithm can tackle cost functions with multiple minima or with broad "flat" regions which are hard to minimize using gradient-based techniques.

We study the relationship between the speed at which a neural network learns a function and the frequency of the function. We build on recent results that show that the dynamics of overparameterized neural networks trained with gradient descent can be well approximated by a linear system. When normalized training data is uniformly distributed on a hypersphere, the eigenfunctions of this linear system are spherical harmonic functions. We derive the corresponding eigenvalues for each frequency after introducing a bias term in the model. This bias term had been omitted from the linear network model without significantly affecting previous theoretical results. However, we show theoretically and experimentally that a shallow neural network without bias cannot learn simple, low frequency functions with odd frequencies, in the limit of large amounts of data. Our results enable us to make specific predictions of the time it will take a network with bias to learn functions of varying frequency. These predictions match the behavior of real shallow and deep networks.