We show that, for finite-sum minimization problems, incorporating partial second-order information of the objective function can dramatically improve the robustness to mini-batch size of variance-reduced stochastic gradient methods, making them more scalable while retaining their benefits over traditional Newton-type approaches. We demonstrate this phenomenon on a prototypical stochastic second-order algorithm, called Mini-Batch Stochastic Variance-Reduced Newton ($\texttt{Mb-SVRN}$), which combines variance-reduced gradient estimates with access to an approximate Hessian oracle. In particular, we show that when the data size $n$ is sufficiently large, i.e., $n\gg \alpha^2\kappa$, where $\kappa$ is the condition number and $\alpha$ is the Hessian approximation factor, then $\texttt{Mb-SVRN}$ achieves a fast linear convergence rate that is independent of the gradient mini-batch size $b$, as long $b$ is in the range between $1$ and $b_{\max}=O(n/(\alpha \log n))$. Only after increasing the mini-batch size past this critical point $b_{\max}$, the method begins to transition into a standard Newton-type algorithm which is much more sensitive to the Hessian approximation quality. We verify this empirically on benchmark tasks showing that, after tuning the step size, the convergence rate of $\texttt{Mb-SVRN}$ remains fast for a wide range of mini-batch sizes, and the dependence of the phase transition point $b_{\max}$ on the Hessian approximation factor $\alpha$ agrees with our theory.

Optimal design is a critical yet challenging task within many applications. This challenge arises from the need for extensive trial and error, often done through simulations or running field experiments. Fortunately, sequential optimal design, also referred to as Bayesian optimization when using surrogates with a Bayesian flavor, has played a key role in accelerating the design process through efficient sequential sampling strategies. However, a key opportunity exists nowadays. The increased connectivity of edge devices sets forth a new collaborative paradigm for Bayesian optimization. A paradigm whereby different clients collaboratively borrow strength from each other by effectively distributing their experimentation efforts to improve and fast-track their optimal design process. To this end, we bring the notion of consensus to Bayesian optimization, where clients agree (i.e., reach a consensus) on their next-to-sample designs. Our approach provides a generic and flexible framework that can incorporate different collaboration mechanisms. In lieu of this, we propose transitional collaborative mechanisms where clients initially rely more on each other to maneuver through the early stages with scant data, then, at the late stages, focus on their own objectives to get client-specific solutions. Theoretically, we show the sub-linear growth in regret for our proposed framework. Empirically, through simulated datasets and a real-world collaborative sensor design experiment, we show that our framework can effectively accelerate and improve the optimal design process and benefit all participants. Note to Practitioners—The proposed algorithm allows multiple clients to collaboratively distribute their trial-and-error efforts to fast-track and improve the optimal design process. In the algorithm, each client performs a test locally and then shares the results with an orchestrator. Using the information from all clients, the orchestrator then finds the best new experiment that each client should undertake and sends those back for the next round of experiments. Through this process, all clients can leverage each other’s strengths and optimize their designs with far fewer experiments than each client operating in isolation. This is confirmed through many simulation examples, along with a real-life sensor design experiment where multiple collaborating agents seqeuntially coordinate their experimentation efforts. The goal is to rapidly discover the biosensor design and measurement format parameters that find the maximum amount of captured target analyte.

This work presents a new optimization algorithm for empirical risk minimization. The algorithm bridges the gap between first- and second-order methods by computing a search direction that uses a second-order-type update in one subspace, coupled with a scaled steepest descent step in the orthogonal complement. To this end, partial curvature information is incorporated to help with ill-conditioning, while simultaneously allowing the algorithm to scale to the large problem dimensions often encountered in machine learning applications. Theoretical results are presented to confirm that the algorithm converges to a stationary point in both the strongly convex and nonconvex cases. A stochastic variant of the algorithm is also presented, along with corresponding theoretical guarantees. Numerical results confirm the strengths of the new approach on standard machine learning problems.

Discovering the underlying behavior of complex systems is an important topic in many science and engineering disciplines. In this paper, we propose a novel neural network framework, finite difference neural networks (FD-Net), to learn partial differential equations from data. Specifically, our proposed finite difference inspired network is designed to learn the underlying governing partial differential equations from trajectory data, and to iteratively estimate the future dynamical behavior using only a few trainable parameters. We illustrate the performance (predictive power) of our framework on the heat equation, with and without noise and/or forcing, and compare our results to the Forward Euler method. Moreover, we show the advantages of using a Hessian-Free Trust Region method to train the network.

In this work we present a general framework for robust error estimation in face recognition. The proposed formulation allows the simultaneous use of various loss functions for modeling the residual in face images, which usually follows non-standard distributions, depending on the image capturing conditions. Our method extends the current vast literature offering flexibility in the selection of the residual modeling characteristics but, at the same time, considering many existing algorithms as special cases. As such, it proves robust for a range of error inducing factors, such as, varying illumination, occlusion, pixel corruption, disguise or their combinations. Extensive simulations document the superiority of selecting multiple models for representing the noise term in face recognition problems, allowing the algorithm to achieve near-optimal performance in most of the tested face databases. Finally, the multi-model residual representation offers useful insights into understanding how different noise types affect face recognition rates.

The question of how to parallelize the stochastic gradient descent (SGD) method has received much attention in the literature. In this paper, we focus instead on batch methods that use a sizeable fraction of the training set at each iteration to facilitate parallelism, and that employ second-order information. In order to improve the learning process, we follow a multi-batch approach in which the batch changes at each iteration. This can cause difficulties because L-BFGS employs gradient differences to update the Hessian approximations, and when these gradients are computed using different data points the process can be unstable. This paper shows how to perform stable quasi-Newton updating in the multi-batch setting, illustrates the behavior of the algorithm in a distributed computing platform, and studies its convergence properties for both the convex and nonconvex cases.