Bilevel optimization has recently regained interest owing to its applications in emerging machine learning fields such as hyperparameter optimization, meta-learning, and reinforcement learning. Recent results have shown that simple alternating (implicit) gradient-based algorithms can achieve the same convergence rate of single-level gradient descent (GD) for bilevel problems with a strongly convex lower-level objective. However, it remains unclear whether this result can be generalized to bilevel problems beyond this basic setting. In this paper, we propose a Generalized ALternating mEthod for bilevel opTimization (GALET) with a nonconvex lower-level objective that satisfies the Polyak-{\L}ojasiewicz (PL) condition. We first introduce a stationary metric for the considered bilevel problems, which generalizes the existing metric. We then establish that GALET achieves an $\epsilon$-stationary metric for the considered problem within $\tilde{\cal O}(\epsilon^{-1})$ iterations, which matches the iteration complexity of GD for smooth nonconvex problems.
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In this paper, we show that the constant-dimensional Weisfeiler-Leman algorithm for groups (Brachter & Schweitzer, LICS 2020) can be fruitfully used to improve parallel complexity upper bounds on isomorphism testing for several families of groups. In particular, we show: - Groups with an Abelian normal Hall subgroup whose complement is $O(1)$-generated are identified by constant-dimensional Weisfeiler-Leman using only a constant number of rounds. This places isomorphism testing for this family of groups into $\textsf{L}$; the previous upper bound for isomorphism testing was $\textsf{P}$ (Qiao, Sarma, & Tang, STACS 2011). - We use the individualize-and-refine paradigm to obtain a $\textsf{quasiSAC}^{1}$ isomorphism test for groups without Abelian normal subgroups, previously only known to be in $\textsf{P}$ (Babai, Codenotti, & Qiao, ICALP 2012). - We extend a result of Brachter & Schweitzer (arXiv, 2021) on direct products of groups to the parallel setting. Namely, we also show that Weisfeiler-Leman can identify direct products in parallel, provided it can identify each of the indecomposable direct factors in parallel. They previously showed the analogous result for $\textsf{P}$. We finally consider the count-free Weisfeiler-Leman algorithm, where we show that count-free WL is unable to even distinguish Abelian groups in polynomial-time. Nonetheless, we use count-free WL in tandem with bounded non-determinism and limited counting to obtain a new upper bound of $\beta_{1}\textsf{MAC}^{0}(\textsf{FOLL})$ for isomorphism testing of Abelian groups. This improves upon the previous $\textsf{TC}^{0}(\textsf{FOLL})$ upper bound due to Chattopadhyay, Tor\'an, & Wagner (ACM Trans. Comput. Theory, 2013).
In this paper we give the first efficient algorithms for the $k$-center problem on dynamic graphs undergoing edge updates. In this problem, the goal is to partition the input into $k$ sets by choosing $k$ centers such that the maximum distance from any data point to the closest center is minimized. It is known that it is NP-hard to get a better than $2$ approximation for this problem. While in many applications the input may naturally be modeled as a graph, all prior works on $k$-center problem in dynamic settings are on metrics. In this paper, we give a deterministic decremental $(2+\epsilon)$-approximation algorithm and a randomized incremental $(4+\epsilon)$-approximation algorithm, both with amortized update time $kn^{o(1)}$ for weighted graphs. Moreover, we show a reduction that leads to a fully dynamic $(2+\epsilon)$-approximation algorithm for the $k$-center problem, with worst-case update time that is within a factor $k$ of the state-of-the-art upper bound for maintaining $(1+\epsilon)$-approximate single-source distances in graphs. Matching this bound is a natural goalpost because the approximate distances of each vertex to its center can be used to maintain a $(2+\epsilon)$-approximation of the graph diameter and the fastest known algorithms for such a diameter approximation also rely on maintaining approximate single-source distances.
Motivated by the wide range of modern applications of the Erlang-B blocking model beyond communication networks and call centers to sizing and pricing in design production systems, messaging systems, and app-based parking systems, we study admission control for such a system but with unknown arrival and service rates. In our model, at every job arrival, a dispatcher decides to assign the job to an available server or block it. Every served job yields a fixed reward for the dispatcher, but it also results in a cost per unit time of service. Our goal is to design a dispatching policy that maximizes the long-term average reward for the dispatcher based on observing only the arrival times and the state of the system at each arrival that reflects a realistic sampling of such systems. Critically, the dispatcher observes neither the service times nor departure times so that standard reinforcement learning-based approaches that use reward signals do not apply. Hence, we develop our learning-based dispatch scheme as a parametric learning problem a'la self-tuning adaptive control. In our problem, certainty equivalent control switches between an always admit if room policy (explore infinitely often) and a never admit policy (immediately terminate learning), which is distinct from the adaptive control literature. Hence, our learning scheme judiciously uses the always admit if room policy so that learning doesn't stall. We prove that for all service rates, the proposed policy asymptotically learns to take the optimal action and present finite-time regret guarantees. The extreme contrast in the certainty equivalent optimal control policies leads to difficulties in learning that show up in our regret bounds for different parameter regimes: constant regret in one regime versus regret growing logarithmically in the other.
Weighted low rank approximation is a fundamental problem in numerical linear algebra, and it has many applications in machine learning. Given a matrix $M \in \mathbb{R}^{n \times n}$, a weight matrix $W \in \mathbb{R}_{\geq 0}^{n \times n}$, a parameter $k$, the goal is to output two matrices $U, V \in \mathbb{R}^{n \times k}$ such that $\| W \circ (M - U V^\top) \|_F$ is minimized, where $\circ$ denotes the Hadamard product. Such a problem is known to be NP-hard and even hard to approximate assuming Exponential Time Hypothesis [GG11, RSW16]. Meanwhile, alternating minimization is a good heuristic solution for approximating weighted low rank approximation. The work [LLR16] shows that, under mild assumptions, alternating minimization does provide provable guarantees. In this work, we develop an efficient and robust framework for alternating minimization. For weighted low rank approximation, this improves the runtime of [LLR16] from $n^2 k^2$ to $n^2k$. At the heart of our work framework is a high-accuracy multiple response regression solver together with a robust analysis of alternating minimization.
Smart contracts are small but highly error-prone programs that implement agreements between multiple parties. We present a reactive synthesis approach for the automatic construction of smart contract state machines. Towards this end, we extend temporal stream logic (TSL) with universally quantified parameters over infinite domains. Parameterized TSL is a convenient logic to specify the temporal control flow, i.e., the correct order of transactions, as well as the data flow of the contract's fields. We develop a two-step approach that 1) synthesizes a finite representation of the - in general - infinite-state system and 2) splits the system into a compact hierarchical architecture that enables the implementation of the state machine in Solidity. We implement the approach in our prototype tool SCSynt, which - within seconds - automatically constructs Solidity code that realizes the specified control flow.
The utilization of finite field multipliers is pervasive in contemporary digital systems, with hardware implementation for bit parallel operation often necessitating millions of logic gates. However, various digital design issues, whether inherent or stemming from soft errors, can result in gate malfunction, ultimately can cause gates to malfunction, which in turn results in incorrect multiplier outputs. Thus, to prevent susceptibility to error, it is imperative to employ a reliable finite field multiplier implementation that boasts a robust fault detection capability. In order to achieve the best fault detection performance for finite field detection performance for finite field multipliers while maintaining a low-complexity implementation, this study proposes a novel fault detection scheme for a recent bit-parallel polynomial basis over GF(2m). The primary concept behind the proposed approach is centered on the implementation of an efficient BCH decoder that utilize Berlekamp-Rumsey-Solomon (BRS) algorithm and Chien-search method to effectively locate errors with minimal delay. The results of our synthesis indicate that our proposed error detection and correction architecture for a 45-bit multiplier with 5-bit errors achieves a 37% and 49% reduction in critical path delay compared to existing designs. Furthermore, a 45-bit multiplicand with five errors has hardware complexity that is only 80%, which is significantly less complex than the most advanced BCH-based fault recognition techniques, such as TMR, Hamming's single error correction, and LDPC-based methods for finite field multiplication which is desirable in constrained applications, such as smart cards, IoT devices, and implantable medical devices.
Federated bilevel optimization (FBO) has shown great potential recently in machine learning and edge computing due to the emerging nested optimization structure in meta-learning, fine-tuning, hyperparameter tuning, etc. However, existing FBO algorithms often involve complicated computations and require multiple sub-loops per iteration, each of which contains a number of communication rounds. In this paper, we propose a simple and flexible FBO framework named SimFBO, which is easy to implement without sub-loops, and includes a generalized server-side aggregation and update for improving communication efficiency. We further propose System-level heterogeneity robust FBO (ShroFBO) as a variant of SimFBO with stronger resilience to heterogeneous local computation. We show that SimFBO and ShroFBO provably achieve a linear convergence speedup with partial client participation and client sampling without replacement, as well as improved sample and communication complexities. Experiments demonstrate the effectiveness of the proposed methods over existing FBO algorithms.
We consider estimation of parameters defined as linear functionals of solutions to linear inverse problems. Any such parameter admits a doubly robust representation that depends on the solution to a dual linear inverse problem, where the dual solution can be thought as a generalization of the inverse propensity function. We provide the first source condition double robust inference method that ensures asymptotic normality around the parameter of interest as long as either the primal or the dual inverse problem is sufficiently well-posed, without knowledge of which inverse problem is the more well-posed one. Our result is enabled by novel guarantees for iterated Tikhonov regularized adversarial estimators for linear inverse problems, over general hypothesis spaces, which are developments of independent interest.
Deep reinforcement learning algorithms can perform poorly in real-world tasks due to the discrepancy between source and target environments. This discrepancy is commonly viewed as the disturbance in transition dynamics. Many existing algorithms learn robust policies by modeling the disturbance and applying it to source environments during training, which usually requires prior knowledge about the disturbance and control of simulators. However, these algorithms can fail in scenarios where the disturbance from target environments is unknown or is intractable to model in simulators. To tackle this problem, we propose a novel model-free actor-critic algorithm -- namely, state-conservative policy optimization (SCPO) -- to learn robust policies without modeling the disturbance in advance. Specifically, SCPO reduces the disturbance in transition dynamics to that in state space and then approximates it by a simple gradient-based regularizer. The appealing features of SCPO include that it is simple to implement and does not require additional knowledge about the disturbance or specially designed simulators. Experiments in several robot control tasks demonstrate that SCPO learns robust policies against the disturbance in transition dynamics.
Knowledge graph completion aims to predict missing relations between entities in a knowledge graph. While many different methods have been proposed, there is a lack of a unifying framework that would lead to state-of-the-art results. Here we develop PathCon, a knowledge graph completion method that harnesses four novel insights to outperform existing methods. PathCon predicts relations between a pair of entities by: (1) Considering the Relational Context of each entity by capturing the relation types adjacent to the entity and modeled through a novel edge-based message passing scheme; (2) Considering the Relational Paths capturing all paths between the two entities; And, (3) adaptively integrating the Relational Context and Relational Path through a learnable attention mechanism. Importantly, (4) in contrast to conventional node-based representations, PathCon represents context and path only using the relation types, which makes it applicable in an inductive setting. Experimental results on knowledge graph benchmarks as well as our newly proposed dataset show that PathCon outperforms state-of-the-art knowledge graph completion methods by a large margin. Finally, PathCon is able to provide interpretable explanations by identifying relations that provide the context and paths that are important for a given predicted relation.
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