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Hierarchical SGD (H-SGD) has emerged as a new distributed SGD algorithm for multi-level communication networks. In H-SGD, before each global aggregation, workers send their updated local models to local servers for aggregations. Despite recent research efforts, the effect of local aggregation on global convergence still lacks theoretical understanding. In this work, we first introduce a new notion of &quot;upward&quot; and &quot;downward&quot; divergences. We then use it to conduct a novel analysis to obtain a worst-case convergence upper bound for two-level H-SGD with non-IID data, non-convex objective function, and stochastic gradient. By extending this result to the case with random grouping, we observe that this convergence upper bound of H-SGD is between the upper bounds of two single-level local SGD settings, with the number of local iterations equal to the local and global update periods in H-SGD, respectively. We refer to this as the &quot;sandwich behavior&quot;. Furthermore, we extend our analytical approach based on &quot;upward&quot; and &quot;downward&quot; divergences to study the convergence for the general case of H-SGD with more than two levels, where the &quot;sandwich behavior&quot; still holds. Our theoretical results provide key insights of why local aggregation can be beneficial in improving the convergence of H-SGD.

AAAI 2022

Demystifying Why Local Aggregation Helps: Convergence Analysis of Hierarchical SGD

The purpose of the AAAI conference is to promote research in artificial intelligence (AI) and scientific exchange among AI researchers, practitioners, scientists, and engineers in affiliated disciplines. 

AAAI 2021

Hierarchical SGD (H-SGD) has emerged as a new distributed SGD algorithm for multi-level communication networks. In H-SGD, before each global aggregation, workers send their updated local models to local servers for aggregations. Despite recent research efforts, the effect of local aggregation on global convergence still lacks theoretical understanding. In this work, we first introduce a new notion of "upward" and "downward" divergences. We then use it to conduct a novel analysis to obtain a worst-case convergence upper bound for two-level H-SGD with non-IID data, non-convex objective function, and stochastic gradient. By extending this result to the case with random grouping, we observe that this convergence upper bound of H-SGD is between the upper bounds of two single-level local SGD settings, with the number of local iterations equal to the local and global update periods in H-SGD, respectively. We refer to this as the "sandwich behavior". Furthermore, we extend our analytical approach based on "upward" and "downward" divergences to study the convergence for the general case of H-SGD with more than two levels, where the "sandwich behavior" still holds. Our theoretical results provide key insights of why local aggregation can be beneficial in improving the convergence of H-SGD.

technical paper

Policy gradient methods have been frequently applied to problems in control and reinforcement learning with great success, yet existing convergence analysis still relies on non-intuitive, impractical and often opaque conditions. In particular, existing rates are achieved in limited settings, under strict regularity conditions. In this work, we establish explicit convergence rates of policy gradient methods, extending the convergence regime to weakly smooth policy classes with L2 integrable gradient. We provide intuitive examples to illustrate the insight behind these new conditions. Notably, our analysis also shows that convergence rates are achievable for both the standard policy gradient and the natural policy gradient algorithms under these assumptions. Lastly we provide performance guarantees for the converged policies.

Convergence and Optimality of Policy Gradient Methods in Weakly Smooth Settings

We propose a new approach, the calibrated nonparametric scan statistic (CNSS), for more accurate detection of anomalous patterns in large-scale, real-world graphs. Scan statistics identify connected subgraphs that are interesting or unexpected through maximization of a likelihood ratio statistic; in particular, nonparametric scan statistics (NPSSs) identify subgraphs with a higher than expected proportion of individually significant nodes. However, we show that recently proposed NPSS methods are miscalibrated, failing to account for the maximization of the statistic over the multiplicity of subgraphs. This results in both reduced detection power for subtle signals, and low precision of the detected subgraph even for stronger signals. Thus we develop a new statistical approach to recalibrate NPSSs, correctly adjusting for multiple hypothesis testing and taking the underlying graph structure into account. While the recalibration, based on randomization testing, is computationally expensive, we propose both an efficient (approximate) algorithm and new, closed-form lower bounds (on the expected maximum proportion of significant nodes for subgraphs of a given size, under the null hypothesis of no anomalous patterns). These advances, along with the integration of recent core-tree decomposition methods, enable CNSS to scale to large real-world graphs, with substantial improvement in the accuracy of detected subgraphs. Extensive experiments on both semi-synthetic and real-world datasets are demonstrated to validate the effectiveness of our proposed methods, in comparison with state-of-the-art counterparts.

Calibrated Nonparametric Scan Statistics for Anomalous Pattern Detection in Graphs

In black-box optimization problems, we aim to maximize an unknown objective function, where the function is only accessible through feedbacks of an evaluation or simulation oracle. In real-life, the feedbacks of such oracles are often noisy and available after some unknown delay that may depend on the computation time of the oracle. Additionally, if the exact evaluations are expensive but coarse approximations are available at a lower cost, the feedbacks can have multi-fidelity. In order to address this problem, we propose a generic extension of hierarchical optimistic tree search (HOO), called ProCrastinated Tree Search (PCTS), that flexibly accommodates a delay and noise-tolerant bandit algorithm. We provide a generic proof technique to quantify regret of PCTS under delayed, noisy, and multi-fidelity feedbacks. Specifically, we derive regret bounds of PCTS enabled with delayed-UCB1 (DUCB1) and delayed-UCB-V (DUCBV) algorithms. Given a horizon T, PCTS retains the regret bound of non-delayed HOO for expected delay of O(log T), and worsens by T^((1-Î±)/(d+2)) for expected delays of O(T^(1-Î±)) for Î± âˆˆ (0,1]. We experimentally validate on multiple synthetic functions and hyperparameter tuning problems that PCTS outperforms the state-of-the-art black-box optimization methods for feedbacks with different noise levels, delays, and fidelity.

Procrastinated Tree Search: Black-Box Optimization with Delayed, Noisy, and Multi-Fidelity Feedback

Despite their popularity for graph structured data, existing Graph Neural Networks (GNNs) have inherent limitations for fundamental graph problems such as shortest paths, k-connectivity, minimum spanning tree and minimum cuts. In these instances, it is known that one needs GNNs of high depth, scaling at a polynomial rate with the number of nodes n, to provably encode the solution space, in turn affecting their statistical efficiency.
 
 In this work we propose a new hybrid architecture to overcome this limitation. Our proposed architecture that we call as GNNplus networks involve a combination of multiple parallel low depth GNNs along with simple pooling layers involving low depth fully connected networks. We provably demonstrate that for many graph problems, the solution space can be encoded by GNNplus networks using depth that scales only poly-logarithmically in the number of nodes. This also has statistical advantages that we demonstrate via generalization bounds for GNNplus networks. We empirically show the effectiveness of our proposed architecture for a variety of graph problems and real world classification problems.

Beyond GNNs: An Efficient Architecture for Graph Problems

Modeling the preferences of agents over a set of alternatives is a principal concern in many areas. The dominant approach has been to find a single reward/utility function with the property that alternatives yielding higher rewards are preferred over alternatives yielding lower rewards. However, in many settings, preferences are based on multipleâ€”often competingâ€”objectives; a single reward function is not adequate to represent such preferences. This paper proposes a method for inferring multi-objective reward-based representations of an agent's observed preferences. We model the agent's priorities over different objectives as entering lexicographically, so that objectives with lower priorities matter only when the agent is indifferent with respect to objectives with higher priorities. We offer two example applications in healthcareâ€”one inspired by cancer treatment, the other inspired by organ transplantationâ€”to illustrate how the lexicographically-ordered rewards we learn can provide a better understanding of a decision-maker's preferences and help improve policies when used in reinforcement learning.

Inferring Lexicographically-Ordered Rewards from Preferences

Mixed-integer programming (MIP) technology offers a generic way of formulating and solving combinatorial optimization problems. While generally reliable, state-of-the-art MIP solvers base many crucial decisions on hand-crafted heuristics, largely ignoring common patterns within a given instance distribution of the problem of interest. Here, we propose MIP-GNN, a general framework for enhancing such solvers with data-driven insights. By encoding the variable-constraint interactions of a given mixed-integer linear program (MILP) as a bipartite graph, we leverage state-of-the-art graph neural network architectures to predict variable biases, i.e., component-wise averages of (near) optimal solutions, indicating how likely a variable will be set to 0 or 1 in (near) optimal solutions of binary MILPs. In turn, the predicted biases stemming from a single, once-trained model are used to guide the solver, replacing heuristic components. We integrate MIP-GNN into a state-of-the-art MIP solver, applying it to tasks such as node selection and warm-starting, showing significant improvements compared to the default setting of the solver on two classes of challenging binary MILPs. Our code and appendix are publicly available at https://github.com/lyeskhalil/mipGNN.

MIP-GNN: A Data-Driven Framework for Guiding Combinatorial Solvers

Optimization of real-world black-box functions defined over purely categorical variables is an active area of research. In particular, optimization and design of biological sequences with specific functional or structural properties have a profound impact in medicine, materials science, and biotechnology. Standalone search algorithms, such as simulated annealing (SA) and Monte Carlo tree search (MCTS), are typically used for such optimization problems. In order to improve the performance and sample efficiency of such algorithms, we propose to use existing methods in conjunction with a surrogate model for the black-box evaluations over purely categorical variables. To this end, we present two different representations, a group-theoretic Fourier expansion and an abridged one-hot encoded Boolean Fourier expansion. To learn such representations, we consider two different settings to update our surrogate model. First, we utilize an adversarial online regression setting where Fourier characters of each representation are considered as experts and their respective coefficients are updated via an exponential weight update rule each time the black box is evaluated. Second, we consider a Bayesian setting where queries are selected via Thompson sampling and the posterior is updated via a sparse Bayesian regression model (over our proposed representation) with a regularized horseshoe prior. Numerical experiments over synthetic benchmarks as well as real-world RNA sequence optimization and design problems demonstrate the representational power of the proposed methods, which achieve competitive or superior performance compared to state-of-the-art counterparts, while improving the computation cost and/or sample efficiency, substantially.

Fourier Representations for Black-Box Optimization over Categorical Variables

Machine learning methods such as deep neural networks (DNNs), despite their success across different domains, are known to often generate incorrect predictions with high confidence on inputs outside their training distribution. The deployment of DNNs in safety-critical domains requires detection of out-of-distribution (OOD) data so that DNNs can abstain from making predictions on those. A number of methods have been recently developed for OOD detection, but there is still room for improvement. We propose the new method iDECODe, leveraging in-distribution equivariance for conformal OOD detection. It relies on a novel base non-conformity measure and a new aggregation method, used in the inductive conformal anomaly detection framework, thereby guaranteeing a bounded false detection rate. We demonstrate the efficacy of iDECODe by experiments on image and audio datasets, obtaining state-of-the-art results. We also show that iDECODe can detect adversarial examples. Code, pre-trained models, and data are available at  https://github.com/ramneetk/iDECODe.

iDECODe: In-Distribution Equivariance for Conformal Out-of-Distribution Detection

Formulating real-world optimization problems often begins with making predictions from historical data (e.g., an optimizer that aims to recommend fast routes relies upon travel-time predictions). Typically, learning the prediction model used to generate the optimization problem and solving that problem are performed in two separate stages. Recent work has showed how such prediction models can be learned end-to-end by differentiating through the optimization task. Such methods often yield empirical improvements, which are typically attributed to end-to-end making better error tradeoffs than the standard loss function used in a two-stage solution. We refine this explanation and more precisely characterize when end-to-end can improve performance. When prediction targets are stochastic, a two-stage solution must make an a priori choice about which statistics of the target distribution to model---we consider expectations over prediction targets---while an end-to-end solution can make this choice adaptively. We show that the performance gap between a two-stage and end-to-end approach is closely related to the \emph{price of correlation} concept in stochastic optimization and show the implications of some existing POC results for the predict-then-optimize problem. We then consider a novel and particularly practical setting, where multiple prediction targets are combined to obtain each of the objective function's coefficients. We give explicit constructions where (1) two-stage performs unboundedly worse than end-to-end; and (2) two-stage is optimal. 
 We use simulations to experimentally quantify performance gaps and identify a wide range of real-world applications from the literature whose objective functions rely on multiple prediction targets, suggesting that end-to-end learning could yield significant improvements.

The Perils of Learning Before Optimizing

This paper studies bandit algorithms under data poisoning attacks in a bounded reward setting. We consider a strong attacker model in which the attacker can observe both the selected actions and their corresponding rewards, and can contaminate the rewards with additive noise. We show that any bandit algorithm with regret O(log T) can be forced to suffer a regret O(T) with an expected amount of contamination O(log T). This amount of contamination is also necessary, as we prove that there exists an O(log T) regret bandit algorithm, specifically the classical UCB, that requires Omega(log T) amount of contamination to suffer regret Omega(T). To combat such poisoning attacks, our second main contribution is to propose verification based mechanisms, which use limited verification to access a limited number of uncontaminated rewards. In particular, for the case of unlimited verifications, we show that with O(log T) expected number of verifications, a simple modified version of the Explore-then-Commit type bandit algorithm can restore the order optimal O(log T) regret irrespective of the amount of contamination used by the attacker. 
 
We also provide a UCB-like verification scheme, called Secure-UCB, that also enjoys full recovery from any attacks, also with O(log T) expected number of verifications. To derive a matching lower bound on the number of verifications, we also prove that for any order-optimal bandit algorithm, this number of verifications O(log T) is necessary to recover the order-optimal regret. On the other hand, when the number of verifications is bounded above by a budget B, we propose a novel algorithm, Secure-BARBAR, which provably achieves O(min(C,T/sqrt(B))) regret with high probability against weak attackers (i.e., attackers who have to place the contamination before seeing the actual pulls of the bandit algorithm), where C is the total amount of contamination by the attacker, which breaks the known Omega(C) lower bound of the non-verified setting if C is large.

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