Descent directions such as movement towards Descent directions, including movement towards Frank-Wolfe vertices, away-steps, in-face away-steps and pairwise directions, have been an important design consideration in conditional gradient descent (CGD) variants. In this work, we attempt to demystify the impact of the movement in these directions towards attaining constrained minimizers. The optimal local direction of descent is the directional derivative (i.e., shadow) of the projection of the negative gradient. We show that this direction is the best away-step possible, and the continuous-time dynamics of moving in the shadow is equivalent to the dynamics of projected gradient descent (PGD), although it's non-trivial to discretize. We also show that Frank-Wolfe (FW) vertices correspond to projecting onto the polytope using an "infinite" step in the direction of the negative gradient, thus providing a new perspective on these steps. We combine these insights into a novel Shadow-CG method that uses FW and shadow steps, while enjoying linear convergence, with a rate that depends on the number of breakpoints in its projection curve, rather than the pyramidal width. We provide a linear bound on the number of breakpoints for simple polytopes and present scaling-invariant upper bounds for general polytopes based on the number of facets. We exemplify the benefit of using Shadow-CG computationally for various applications, while raising an open question about tightening the bound on the number of breakpoints for general polytopes.

Minimizing error in prevalence estimates and diagnostic classifiers remains a challenging task in serology. In theory, these problems can be reduced to modeling class-conditional probability densities (PDFs) of measurement outcomes, which control all downstream analyses. However, this task quickly succumbs to the curse of dimensionality, even for assay outputs with only a few dimensions (e.g. target antigens). To address this problem, we propose a technique that uses empirical training data to classify samples and estimate prevalence in arbitrary dimension without direct access to the conditional PDFs. We motivate this method via a lemma that relates relative conditional probabilities to minimum-error classification boundaries. This leads us to formulate an optimization problem that: (i) embeds the data in a parameterized, curved space; (ii) classifies samples based on their position relative to a coordinate axis; and (iii) subsequently optimizes the space by minimizing the empirical classification error of pure training data, for which the classes are known. Interestingly, the solution to this problem requires use of a homotopy-type method to stabilize the optimization. We then extend the analysis to the case of impure training data, for which the classes are unknown. We find that two impure datasets suffice for both prevalence estimation and classification, provided they satisfy a linear independence property. Lastly, we discuss how our analysis unifies discriminative and generative learning techniques in a common framework based on ideas from set and measure theory. Throughout, we validate our methods in the context of synthetic data and a research-use SARS-CoV-2 enzyme-linked immunosorbent (ELISA) assay.

We consider trawl processes, which are stationary and infinitely divisible stochastic processes and can describe a wide range of statistical properties, such as heavy tails and long memory. In this paper, we develop the first likelihood-based methodology for the inference of real-valued trawl processes and introduce novel deterministic and probabilistic forecasting methods. Being non-Markovian, with a highly intractable likelihood function, trawl processes require the use of composite likelihood functions to parsimoniously capture their statistical properties. We formulate the composite likelihood estimation as a stochastic optimization problem for which it is feasible to implement iterative gradient descent methods. We derive novel gradient estimators with variances that are reduced by several orders of magnitude. We analyze both the theoretical properties and practical implementation details of these estimators and release a Python library which can be used to fit a large class of trawl processes. In a simulation study, we demonstrate that our estimators outperform the generalized method of moments estimators in terms of both parameter estimation error and out-of-sample forecasting error. Finally, we formalize a stochastic chain rule for our gradient estimators. We apply the new theory to trawl processes and provide a unified likelihood-based methodology for the inference of both real-valued and integer-valued trawl processes.

Given data from diverse sets of distinct distributions, domain generalization aims to learn models that generalize to unseen distributions. A common approach is designing a data-driven surrogate penalty to capture generalization and minimize the empirical risk jointly with the penalty. We argue that a significant failure mode of this recipe is an excess risk due to an erroneous penalty or hardness in joint optimization. We present an approach that eliminates this problem. Instead of jointly minimizing empirical risk with the penalty, we minimize the penalty under the constraint of optimality of the empirical risk. This change guarantees that the domain generalization penalty cannot impair optimization of the empirical risk, i.e., in-distribution performance. To solve the proposed optimization problem, we demonstrate an exciting connection to rate-distortion theory and utilize its tools to design an efficient method. Our approach can be applied to any penalty-based domain generalization method, and we demonstrate its effectiveness by applying it to three examplar methods from the literature, showing significant improvements.

Constrained Markov Decision Processes (CMDPs) are one of the common ways to model safe reinforcement learning problems, where constraint functions model the safety objectives. Lagrangian-based dual or primal-dual algorithms provide efficient methods for learning in CMDPs. For these algorithms, the currently known regret bounds in the finite-horizon setting allow for a "cancellation of errors"; one can compensate for a constraint violation in one episode with a strict constraint satisfaction in another. However, we do not consider such a behavior safe in practical applications. In this paper, we overcome this weakness by proposing a novel model-based dual algorithm OptAug-CMDP for tabular finite-horizon CMDPs. Our algorithm is motivated by the augmented Lagrangian method and can be performed efficiently. We show that during $K$ episodes of exploring the CMDP, our algorithm obtains a regret of $\tilde{O}(\sqrt{K})$ for both the objective and the constraint violation. Unlike existing Lagrangian approaches, our algorithm achieves this regret without the need for the cancellation of errors.

Seaweed biomass offers significant potential for climate mitigation, but large-scale, autonomous open-ocean farms are required to fully exploit it. Such farms typically have low propulsion and are heavily influenced by ocean currents. We want to design a controller that maximizes seaweed growth over months by taking advantage of the non-linear time-varying ocean currents for reaching high-growth regions. The complex dynamics and underactuation make this challenging even when the currents are known. This is even harder when only short-term imperfect forecasts with increasing uncertainty are available. We propose a dynamic programming-based method to efficiently solve for the optimal growth value function when true currents are known. We additionally present three extensions when as in reality only forecasts are known: (1) our methods resulting value function can be used as feedback policy to obtain the growth-optimal control for all states and times, allowing closed-loop control equivalent to re-planning at every time step hence mitigating forecast errors, (2) a feedback policy for long-term optimal growth beyond forecast horizons using seasonal average current data as terminal reward, and (3) a discounted finite-time Dynamic Programming (DP) formulation to account for increasing ocean current estimate uncertainty. We evaluate our approach through 30-day simulations of floating seaweed farms in realistic Pacific Ocean current scenarios. Our method demonstrates an achievement of 95.8% of the best possible growth using only 5-day forecasts. This confirms the feasibility of using low-power propulsion and optimal control for enhanced seaweed growth on floating farms under real-world conditions.

This paper considers a market for trading Internet of Things (IoT) data that is used to train machine learning models. The data, either raw or processed, is supplied to the market platform through a network and the price of such data is controlled based on the value it brings to the machine learning model. We explore the correlation property of data in a game-theoretical setting to eventually derive a simplified distributed solution for a data trading mechanism that emphasizes the mutual benefit of devices and the market. The key proposal is an efficient algorithm for markets that jointly addresses the challenges of availability and heterogeneity in participation, as well as the transfer of trust and the economic value of data exchange in IoT networks. The proposed approach establishes the data market by reinforcing collaboration opportunities between device with correlated data to avoid information leakage. Therein, we develop a network-wide optimization problem that maximizes the social value of coalition among the IoT devices of similar data types; at the same time, it minimizes the cost due to network externalities, i.e., the impact of information leakage due to data correlation, as well as the opportunity costs. Finally, we reveal the structure of the formulated problem as a distributed coalition game and solve it following the simplified split-and-merge algorithm. Simulation results show the efficacy of our proposed mechanism design toward a trusted IoT data market, with up to 32.72% gain in the average payoff for each seller.

Heavy Ball (HB) nowadays is one of the most popular momentum methods in non-convex optimization. It has been widely observed that incorporating the Heavy Ball dynamic in gradient-based methods accelerates the training process of modern machine learning models. However, the progress on establishing its theoretical foundation of acceleration is apparently far behind its empirical success. Existing provable acceleration results are of the quadratic or close-to-quadratic functions, as the current techniques of showing HB's acceleration are limited to the case when the Hessian is fixed. In this work, we develop some new techniques that help show acceleration beyond quadratics, which is achieved by analyzing how the change of the Hessian at two consecutive time points affects the convergence speed. Based on our technical results, a class of Polyak-\L{}ojasiewicz (PL) optimization problems for which provable acceleration can be achieved via HB is identified. Moreover, our analysis demonstrates a benefit of adaptively setting the momentum parameter. (Update: 08/29/2023) Erratum is added in Appendix J. This is an updated version that fixes an issue in the previous version. An additional condition needs to be satisfied for the acceleration result of HB beyond quadratics in this work, which naturally holds when the dimension is one or, more broadly, when the Hessian is diagonal. We elaborate on the issue in Appendix J.

A popular approach to model interactions is to represent them as a network with nodes being the agents and the interactions being the edges. Interactions are often timestamped, which leads to having timestamped edges. Many real-world temporal networks have a recurrent or possibly cyclic behaviour. For example, social network activity may be heightened during certain hours of day. In this paper, our main interest is to model recurrent activity in such temporal networks. As a starting point we use stochastic block model, a popular choice for modelling static networks, where nodes are split into $R$ groups. We extend this model to temporal networks by modelling the edges with a Poisson process. We make the parameters of the process dependent on time by segmenting the time line into $K$ segments. To enforce the recurring activity we require that only $H < K$ different set of parameters can be used, that is, several, not necessarily consecutive, segments must share their parameters. We prove that the searching for optimal blocks and segmentation is an NP-hard problem. Consequently, we split the problem into 3 subproblems where we optimize blocks, model parameters, and segmentation in turn while keeping the remaining structures fixed. We propose an iterative algorithm that requires $O(KHm + Rn + R^2H)$ time per iteration, where $n$ and $m$ are the number of nodes and edges in the network. We demonstrate experimentally that the number of required iterations is typically low, the algorithm is able to discover the ground truth from synthetic datasets, and show that certain real-world networks exhibit recurrent behaviour as the likelihood does not deteriorate when $H$ is lowered.

With the increasing practicality of deep learning applications, practitioners are inevitably faced with datasets corrupted by noise from various sources such as measurement errors, mislabeling, and estimated surrogate inputs/outputs that can adversely impact the optimization results. It is a common practice to improve the optimization algorithm's robustness to noise, since this algorithm is ultimately in charge of updating the network parameters. Previous studies revealed that the first-order moment used in Adam-like stochastic gradient descent optimizers can be modified based on the Student's t-distribution. While this modification led to noise-resistant updates, the other associated statistics remained unchanged, resulting in inconsistencies in the assumed models. In this paper, we propose AdaTerm, a novel approach that incorporates the Student's t-distribution to derive not only the first-order moment but also all the associated statistics. This provides a unified treatment of the optimization process, offering a comprehensive framework under the statistical model of the t-distribution for the first time. The proposed approach offers several advantages over previously proposed approaches, including reduced hyperparameters and improved robustness and adaptability. This noise-adaptive behavior contributes to AdaTerm's exceptional learning performance, as demonstrated through various optimization problems with different and/or unknown noise ratios. Furthermore, we introduce a new technique for deriving a theoretical regret bound without relying on AMSGrad, providing a valuable contribution to the field