Compressed Stochastic Gradient Descent (SGD) algorithms have been recently proposed to address the communication bottleneck in distributed and decentralized optimization problems, such as those that arise in federated machine learning. Existing compressed SGD algorithms assume the use of non-adaptive step-sizes(constant or diminishing) to provide theoretical convergence guarantees. Typically, the step-sizes are fine-tuned in practice to the dataset and the learning algorithm to provide good empirical performance. Such fine-tuning might be impractical in many learning scenarios, and it is therefore of interest to study compressed SGD using adaptive step-sizes. Motivated by prior work on adaptive step-size methods for SGD to train neural networks efficiently in the uncompressed setting, we develop an adaptive step-size method for compressed SGD. In particular, we introduce a scaling technique for the descent step in compressed SGD, which we use to establish order-optimal convergence rates for convex-smooth and strong convex-smooth objectives under an interpolation condition and for non-convex objectives under a strong growth condition. We also show through simulation examples that without this scaling, the algorithm can fail to converge. We present experimental results on deep neural networks for real-world datasets, and compare the performance of our proposed algorithm with previously proposed compressed SGD methods in literature, and demonstrate improved performance on ResNet-18, ResNet-34 and DenseNet architectures for CIFAR-100 and CIFAR-10 datasets at various levels of compression.

We present a new scheme to compensate for the small-scales approximations resulting from Particle-Mesh (PM) schemes for cosmological N-body simulations. This kind of simulations are fast and low computational cost realizations of the large scale structures, but lack resolution on small scales. To improve their accuracy, we introduce an additional effective force within the differential equations of the simulation, parameterized by a Fourier-space Neural Network acting on the PM-estimated gravitational potential. We compare the results for the matter power spectrum obtained to the ones obtained by the PGD scheme (Potential gradient descent scheme). We notice a similar improvement in term of power spectrum, but we find that our approach outperforms PGD for the cross-correlation coefficients, and is more robust to changes in simulation settings (different resolutions, different cosmologies).

This paper introduces a novel motion planner, incrementally stochastic and accelerated gradient information mixed optimization (iSAGO), for robotic manipulators in a narrow workspace. Primarily, we propose the overall scheme of iSAGO informed by the mixed momenta for an efficient constrained optimization based on the penalty method. In the stochastic part, we generate the adaptive stochastic momenta via the random selection of sub-functionals based on the adaptive momentum (Adam) method to solve the body-obstacle stuck case. Due to the slow convergence of the stochastic part, we integrate the accelerated gradient descent (AGD) to improve the planning efficiency. Moreover, we adopt the Bayesian tree inference (BTI) to transform the whole trajectory optimization (SAGO) into an incremental sub-trajectory optimization (iSAGO), which improves the computation efficiency and success rate further. Finally, we tune the key parameters and benchmark iSAGO against the other 5 planners on LBR-iiwa in a bookshelf and AUBO-i5 on a storage shelf. The result shows the highest success rate and moderate solving efficiency of iSAGO.

This work considers optimization of composition of functions in a nested form over Riemannian manifolds where each function contains an expectation. This type of problems is gaining popularity in applications such as policy evaluation in reinforcement learning or model customization in meta-learning. The standard Riemannian stochastic gradient methods for non-compositional optimization cannot be directly applied as stochastic approximation of inner functions create bias in the gradients of the outer functions. For two-level composition optimization, we present a Riemannian Stochastic Composition Gradient Descent (R-SCGD) method that finds an approximate stationary point, with expected squared Riemannian gradient smaller than $\epsilon$, in $O(\epsilon^{-2})$ calls to the stochastic gradient oracle of the outer function and stochastic function and gradient oracles of the inner function. Furthermore, we generalize the R-SCGD algorithms for problems with multi-level nested compositional structures, with the same complexity of $O(\epsilon^{-2})$ for the first-order stochastic oracle. Finally, the performance of the R-SCGD method is numerically evaluated over a policy evaluation problem in reinforcement learning.

Macro placement is the problem of placing memory blocks on a chip canvas. It can be formulated as a combinatorial optimization problem over sequence pairs, a representation which describes the relative positions of macros. Solving this problem is particularly challenging since the objective function is expensive to evaluate. In this paper, we develop a novel approach to macro placement using Bayesian optimization (BO) over sequence pairs. BO is a machine learning technique that uses a probabilistic surrogate model and an acquisition function that balances exploration and exploitation to efficiently optimize a black-box objective function. BO is more sample-efficient than reinforcement learning and therefore can be used with more realistic objectives. Additionally, the ability to learn from data and adapt the algorithm to the objective function makes BO an appealing alternative to other black-box optimization methods such as simulated annealing, which relies on problem-dependent heuristics and parameter-tuning. We benchmark our algorithm on the fixed-outline macro placement problem with the half-perimeter wire length objective and demonstrate competitive performance.

We introduce SPIRAL, a SuPerlinearly convergent Incremental pRoximal ALgorithm, for solving nonconvex regularized finite sum problems under a relative smoothness assumption. In the spirit of SVRG and SARAH, each iteration of SPIRAL consists of an inner and an outer loop. It combines incremental and full (proximal) gradient updates with a linesearch. It is shown that when using quasi-Newton directions, superlinear convergence is attained under mild assumptions at the limit points. More importantly, thanks to said linesearch, global convergence is ensured while it is shown that unit stepsize will be eventually always accepted. Simulation results on different convex, nonconvex, and non-Lipschitz differentiable problems show that our algorithm as well as its adaptive variant are competitive to the state of the art.

The manifold assumption for high-dimensional data assumes that the data is generated by varying a set of parameters obtained from a low-dimensional latent space. Deep generative models (DGMs) are widely used to learn data representations in an unsupervised way. DGMs parameterize the underlying low-dimensional manifold in the data space using bottleneck architectures such as variational autoencoders (VAEs). The bottleneck dimension for VAEs is treated as a hyperparameter that depends on the dataset and is fixed at design time after extensive tuning. As the intrinsic dimensionality of most real-world datasets is unknown, often, there is a mismatch between the intrinsic dimensionality and the latent dimensionality chosen as a hyperparameter. This mismatch can negatively contribute to the model performance for representation learning and sample generation tasks. This paper proposes relevance encoding networks (RENs): a novel probabilistic VAE-based framework that uses the automatic relevance determination (ARD) prior in the latent space to learn the data-specific bottleneck dimensionality. The relevance of each latent dimension is directly learned from the data along with the other model parameters using stochastic gradient descent and a reparameterization trick adapted to non-Gaussian priors. We leverage the concept of DeepSets to capture permutation invariant statistical properties in both data and latent spaces for relevance determination. The proposed framework is general and flexible and can be used for the state-of-the-art VAE models that leverage regularizers to impose specific characteristics in the latent space (e.g., disentanglement). With extensive experimentation on synthetic and public image datasets, we show that the proposed model learns the relevant latent bottleneck dimensionality without compromising the representation and generation quality of the samples.

In this paper, we tackle a significant challenge in PCA: heterogeneity. When data are collected from different sources with heterogeneous trends while still sharing some congruency, it is critical to extract shared knowledge while retaining unique features of each source. To this end, we propose personalized PCA (PerPCA), which uses mutually orthogonal global and local principal components to encode both unique and shared features. We show that, under mild conditions, both unique and shared features can be identified and recovered by a constrained optimization problem, even if the covariance matrices are immensely different. Also, we design a fully federated algorithm inspired by distributed Stiefel gradient descent to solve the problem. The algorithm introduces a new group of operations called generalized retractions to handle orthogonality constraints, and only requires global PCs to be shared across sources. We prove the linear convergence of the algorithm under suitable assumptions. Comprehensive numerical experiments highlight PerPCA's superior performance in feature extraction and prediction from heterogeneous datasets. As a systematic approach to decouple shared and unique features from heterogeneous datasets, PerPCA finds applications in several tasks including video segmentation, topic extraction, and distributed clustering.

We consider stochastic convex optimization for heavy-tailed data with the guarantee of being differentially private (DP). Prior work on this problem is restricted to the gradient descent (GD) method, which is inefficient for large-scale problems. In this paper, we resolve this issue and derive the first high-probability bounds for the private stochastic method with clipping. For general convex problems, we derive excess population risks $\Tilde{O}\left(\frac{d^{1/7}\sqrt{\ln\frac{(n \epsilon)^2}{\beta d}}}{(n\epsilon)^{2/7}}\right)$ and $\Tilde{O}\left(\frac{d^{1/7}\ln\frac{(n\epsilon)^2}{\beta d}}{(n\epsilon)^{2/7}}\right)$ under bounded or unbounded domain assumption, respectively (here $n$ is the sample size, $d$ is the dimension of the data, $\beta$ is the confidence level and $\epsilon$ is the private level). Then, we extend our analysis to the strongly convex case and non-smooth case (which works for generalized smooth objectives with H$\ddot{\text{o}}$lder-continuous gradients). We establish new excess risk bounds without bounded domain assumption. The results above achieve lower excess risks and gradient complexities than existing methods in their corresponding cases. Numerical experiments are conducted to justify the theoretical improvement.

We introduce data structures for solving robust regression through stochastic gradient descent (SGD) by sampling gradients with probability proportional to their norm, i.e., importance sampling. Although SGD is widely used for large scale machine learning, it is well-known for possibly experiencing slow convergence rates due to the high variance from uniform sampling. On the other hand, importance sampling can significantly decrease the variance but is usually difficult to implement because computing the sampling probabilities requires additional passes over the data, in which case standard gradient descent (GD) could be used instead. In this paper, we introduce an algorithm that approximately samples $T$ gradients of dimension $d$ from nearly the optimal importance sampling distribution for a robust regression problem over $n$ rows. Thus our algorithm effectively runs $T$ steps of SGD with importance sampling while using sublinear space and just making a single pass over the data. Our techniques also extend to performing importance sampling for second-order optimization.