Adversarial Combinatorial Bandits with Switching Costs

We study the problem of adversarial combinatorial bandit with a switching cost $\lambda$ for a switch of each selected arm in each round, considering both the bandit feedback and semi-bandit feedback settings. In the oblivious adversarial case with $K$ base arms and time horizon $T$, we derive lower bounds for the minimax regret and design algorithms to approach them. To prove these lower bounds, we design stochastic loss sequences for both feedback settings, building on an idea from previous work in Dekel et al. (2014). The lower bound for bandit feedback is $ \tilde{\Omega}\big( (\lambda K)^{\frac{1}{3}} (TI)^{\frac{2}{3}}\big)$ while that for semi-bandit feedback is $ \tilde{\Omega}\big( (\lambda K I)^{\frac{1}{3}} T^{\frac{2}{3}}\big)$ where $I$ is the number of base arms in the combinatorial arm played in each round. To approach these lower bounds, we design algorithms that operate in batches by dividing the time horizon into batches to restrict the number of switches between actions. For the bandit feedback setting, where only the total loss of the combinatorial arm is observed, we introduce the Batched-Exp2 algorithm which achieves a regret upper bound of $\tilde{O}\big((\lambda K)^{\frac{1}{3}}T^{\frac{2}{3}}I^{\frac{4}{3}}\big)$ as $T$ tends to infinity. In the semi-bandit feedback setting, where all losses for the combinatorial arm are observed, we propose the Batched-BROAD algorithm which achieves a regret upper bound of $\tilde{O}\big( (\lambda K)^{\frac{1}{3}} (TI)^{\frac{2}{3}}\big)$.

Multi-Armed Bandits with Abstention

We introduce a novel extension of the canonical multi-armed bandit problem that incorporates an additional strategic element: abstention. In this enhanced framework, the agent is not only tasked with selecting an arm at each time step, but also has the option to abstain from accepting the stochastic instantaneous reward before observing it. When opting for abstention, the agent either suffers a fixed regret or gains a guaranteed reward. Given this added layer of complexity, we ask whether we can develop efficient algorithms that are both asymptotically and minimax optimal. We answer this question affirmatively by designing and analyzing algorithms whose regrets meet their corresponding information-theoretic lower bounds. Our results offer valuable quantitative insights into the benefits of the abstention option, laying the groundwork for further exploration in other online decision-making problems with such an option. Numerical results further corroborate our theoretical findings.

Fixed-Budget Differentially Private Best Arm Identification

We study best arm identification (BAI) in linear bandits in the fixed-budget regime under differential privacy constraints, when the arm rewards are supported on the unit interval. Given a finite budget $T$ and a privacy parameter $\varepsilon>0$, the goal is to minimise the error probability in finding the arm with the largest mean after $T$ sampling rounds, subject to the constraint that the policy of the decision maker satisfies a certain {\em $\varepsilon$-differential privacy} ($\varepsilon$-DP) constraint. We construct a policy satisfying the $\varepsilon$-DP constraint (called {\sc DP-BAI}) by proposing the principle of {\em maximum absolute determinants}, and derive an upper bound on its error probability. Furthermore, we derive a minimax lower bound on the error probability, and demonstrate that the lower and the upper bounds decay exponentially in $T$, with exponents in the two bounds matching order-wise in (a) the sub-optimality gaps of the arms, (b) $\varepsilon$, and (c) the problem complexity that is expressible as the sum of two terms, one characterising the complexity of standard fixed-budget BAI (without privacy constraints), and the other accounting for the $\varepsilon$-DP constraint. Additionally, we present some auxiliary results that contribute to the derivation of the lower bound on the error probability. These results, we posit, may be of independent interest and could prove instrumental in proving lower bounds on error probabilities in several other bandit problems. Whereas prior works provide results for BAI in the fixed-budget regime without privacy constraints or in the fixed-confidence regime with privacy constraints, our work fills the gap in the literature by providing the results for BAI in the fixed-budget regime under the $\varepsilon$-DP constraint.

Towards Accurate Guided Diffusion Sampling through Symplectic Adjoint Method

Training-free guided sampling in diffusion models leverages off-the-shelf pre-trained networks, such as an aesthetic evaluation model, to guide the generation process. Current training-free guided sampling algorithms obtain the guidance energy function based on a one-step estimate of the clean image. However, since the off-the-shelf pre-trained networks are trained on clean images, the one-step estimation procedure of the clean image may be inaccurate, especially in the early stages of the generation process in diffusion models. This causes the guidance in the early time steps to be inaccurate. To overcome this problem, we propose Symplectic Adjoint Guidance (SAG), which calculates the gradient guidance in two inner stages. Firstly, SAG estimates the clean image via $n$ function calls, where $n$ serves as a flexible hyperparameter that can be tailored to meet specific image quality requirements. Secondly, SAG uses the symplectic adjoint method to obtain the gradients accurately and efficiently in terms of the memory requirements. Extensive experiments demonstrate that SAG generates images with higher qualities compared to the baselines in both guided image and video generation tasks.

Fixed-Budget Best-Arm Identification in Sparse Linear Bandits

We study the best-arm identification problem in sparse linear bandits under the fixed-budget setting. In sparse linear bandits, the unknown feature vector $\theta^*$ may be of large dimension $d$, but only a few, say $s \ll d$ of these features have non-zero values. We design a two-phase algorithm, Lasso and Optimal-Design- (Lasso-OD) based linear best-arm identification. The first phase of Lasso-OD leverages the sparsity of the feature vector by applying the thresholded Lasso introduced by Zhou (2009), which estimates the support of $\theta^*$ correctly with high probability using rewards from the selected arms and a judicious choice of the design matrix. The second phase of Lasso-OD applies the OD-LinBAI algorithm by Yang and Tan (2022) on that estimated support. We derive a non-asymptotic upper bound on the error probability of Lasso-OD by carefully choosing hyperparameters (such as Lasso's regularization parameter) and balancing the error probabilities of both phases. For fixed sparsity $s$ and budget $T$, the exponent in the error probability of Lasso-OD depends on $s$ but not on the dimension $d$, yielding a significant performance improvement for sparse and high-dimensional linear bandits. Furthermore, we show that Lasso-OD is almost minimax optimal in the exponent. Finally, we provide numerical examples to demonstrate the significant performance improvement over the existing algorithms for non-sparse linear bandits such as OD-LinBAI, BayesGap, Peace, LinearExploration, and GSE.

Learning Regularized Graphon Mean-Field Games with Unknown Graphons

We design and analyze reinforcement learning algorithms for Graphon Mean-Field Games (GMFGs). In contrast to previous works that require the precise values of the graphons, we aim to learn the Nash Equilibrium (NE) of the regularized GMFGs when the graphons are unknown. Our contributions are threefold. First, we propose the Proximal Policy Optimization for GMFG (GMFG-PPO) algorithm and show that it converges at a rate of $O(T^{-1/3})$ after $T$ iterations with an estimation oracle, improving on a previous work by Xie et al. (ICML, 2021). Second, using kernel embedding of distributions, we design efficient algorithms to estimate the transition kernels, reward functions, and graphons from sampled agents. Convergence rates are then derived when the positions of the agents are either known or unknown. Results for the combination of the optimization algorithm GMFG-PPO and the estimation algorithm are then provided. These algorithms are the first specifically designed for learning graphons from sampled agents. Finally, the efficacy of the proposed algorithms are corroborated through simulations. These simulations demonstrate that learning the unknown graphons reduces the exploitability effectively.

Optimal Best Arm Identification with Fixed Confidence in Restless Bandits

We study best arm identification in a restless multi-armed bandit setting with finitely many arms. The discrete-time data generated by each arm forms a homogeneous Markov chain taking values in a common, finite state space. The state transitions in each arm are captured by an ergodic transition probability matrix (TPM) that is a member of a single-parameter exponential family of TPMs. The real-valued parameters of the arm TPMs are unknown and belong to a given space. Given a function $f$ defined on the common state space of the arms, the goal is to identify the best arm -- the arm with the largest average value of $f$ evaluated under the arm's stationary distribution -- with the fewest number of samples, subject to an upper bound on the decision's error probability (i.e., the fixed-confidence regime). A lower bound on the growth rate of the expected stopping time is established in the asymptote of a vanishing error probability. Furthermore, a policy for best arm identification is proposed, and its expected stopping time is proved to have an asymptotic growth rate that matches the lower bound. It is demonstrated that tracking the long-term behavior of a certain Markov decision process and its state-action visitation proportions are the key ingredients in analyzing the converse and achievability bounds. It is shown that under every policy, the state-action visitation proportions satisfy a specific approximate flow conservation constraint and that these proportions match the optimal proportions dictated by the lower bound under any asymptotically optimal policy. The prior studies on best arm identification in restless bandits focus on independent observations from the arms, rested Markov arms, and restless Markov arms with known arm TPMs. In contrast, this work is the first to study best arm identification in restless bandits with unknown arm TPMs.

Learning Regularized Monotone Graphon Mean-Field Games

This paper studies two fundamental problems in regularized Graphon Mean-Field Games (GMFGs). First, we establish the existence of a Nash Equilibrium (NE) of any $\lambda$-regularized GMFG (for $\lambda\geq 0$). This result relies on weaker conditions than those in previous works for analyzing both unregularized GMFGs ($\lambda=0$) and $\lambda$-regularized MFGs, which are special cases of GMFGs. Second, we propose provably efficient algorithms to learn the NE in weakly monotone GMFGs, motivated by Lasry and Lions [2007]. Previous literature either only analyzed continuous-time algorithms or required extra conditions to analyze discrete-time algorithms. In contrast, we design a discrete-time algorithm and derive its convergence rate solely under weakly monotone conditions. Furthermore, we develop and analyze the action-value function estimation procedure during the online learning process, which is absent from algorithms for monotone GMFGs. This serves as a sub-module in our optimization algorithm. The efficiency of the designed algorithm is corroborated by empirical evaluations.

Blink: Link Local Differential Privacy in Graph Neural Networks via Bayesian Estimation

Graph neural networks (GNNs) have gained an increasing amount of popularity due to their superior capability in learning node embeddings for various graph inference tasks, but training them can raise privacy concerns. To address this, we propose using link local differential privacy over decentralized nodes, enabling collaboration with an untrusted server to train GNNs without revealing the existence of any link. Our approach spends the privacy budget separately on links and degrees of the graph for the server to better denoise the graph topology using Bayesian estimation, alleviating the negative impact of LDP on the accuracy of the trained GNNs. We bound the mean absolute error of the inferred link probabilities against the ground truth graph topology. We then propose two variants of our LDP mechanism complementing each other in different privacy settings, one of which estimates fewer links under lower privacy budgets to avoid false positive link estimates when the uncertainty is high, while the other utilizes more information and performs better given relatively higher privacy budgets. Furthermore, we propose a hybrid variant that combines both strategies and is able to perform better across different privacy budgets. Extensive experiments show that our approach outperforms existing methods in terms of accuracy under varying privacy budgets.

AdjointDPM: Adjoint Sensitivity Method for Gradient Backpropagation of Diffusion Probabilistic Models

Existing customization methods require access to multiple reference examples to align pre-trained diffusion probabilistic models (DPMs) with user-provided concepts. This paper aims to address the challenge of DPM customization when the only available supervision is a differentiable metric defined on the generated contents. Since the sampling procedure of DPMs involves recursive calls to the denoising UNet, na\"ive gradient backpropagation requires storing the intermediate states of all iterations, resulting in extremely high memory consumption. To overcome this issue, we propose a novel method AdjointDPM, which first generates new samples from diffusion models by solving the corresponding probability-flow ODEs. It then uses the adjoint sensitivity method to backpropagate the gradients of the loss to the models' parameters (including conditioning signals, network weights, and initial noises) by solving another augmented ODE. To reduce numerical errors in both the forward generation and gradient backpropagation processes, we further reparameterize the probability-flow ODE and augmented ODE as simple non-stiff ODEs using exponential integration. Finally, we demonstrate the effectiveness of AdjointDPM on three interesting tasks: converting visual effects into identification text embeddings, finetuning DPMs for specific types of stylization, and optimizing initial noise to generate adversarial samples for security auditing.