R. Srikant

Bo Tan, R. Srikant

In this paper, we propose a stochastic model to describe how search service providers charge client companies based on users' queries for the keywords related to these companies' ads by using certain advertisement assignment strategies. We formulate an optimization problem to maximize the long-term average revenue for the service provider under each client's long-term average budget constraint, and design an online algorithm which captures the stochastic properties of users' queries and click-through behaviors. We solve the optimization problem by making connections to scheduling problems in wireless networks, queueing theory and stochastic networks. Unlike prior models, we do not assume that the number of query arrivals is known. Due to the stochastic nature of the arrival process considered here, either temporary "free" service, i.e., service above the specified budget or under-utilization of the budget is unavoidable. We prove that our online algorithm can achieve a revenue that is within $O(ε)$ of the optimal revenue while ensuring that the overdraft or underdraft is $O(1/ε)$, where $ε$ can be arbitrarily small. With a view towards practice, we can show that one can always operate strictly under the budget. In addition, we extend our results to a click-through rate maximization model, and also show how our algorithm can be modified to handle non-stationary query arrival processes and clients with short-term contracts. Our algorithm allows us to quantify the effect of errors in click-through rate estimation on the achieved revenue. We also show that in the long run, an expected overdraft level of $Ω(\log(1/ε))$ is unavoidable (a universal lower bound) under any stationary ad assignment algorithm which achieves a long-term average revenue within $O(ε)$ of the offline optimum.

S. Rasoul Etesami, R. Srikant

We consider the problem of learning stable matchings with unknown preferences in a decentralized and uncoordinated manner, where "decentralized" means that players make decisions individually without the influence of a central platform, and "uncoordinated" means that players do not need to synchronize their decisions using pre-specified rules. First, we provide a game formulation for this problem with known preferences, where the set of pure Nash equilibria (NE) coincides with the set of stable matchings, and mixed NE can be rounded to a stable matching. Then, we show that for hierarchical markets, applying the exponential weight (EXP) learning algorithm to the stable matching game achieves logarithmic regret in a fully decentralized and uncoordinated fashion. Moreover, we show that EXP converges locally and exponentially fast to a stable matching in general markets. We also introduce another decentralized and uncoordinated learning algorithm that globally converges to a stable matching with arbitrarily high probability. Finally, we provide stronger feedback conditions under which it is possible to drive the market faster toward an approximate stable matching. Our proposed game-theoretic framework bridges the discrete problem of learning stable matchings with the problem of learning NE in continuous-action games.

Ameya Anjarlekar, Rasoul Etesami, R. Srikant

We address the challenge of solving machine learning tasks using data from privacy-sensitive sellers. Since the data is private, we design a data market that incentivizes sellers to provide their data in exchange for payments. Therefore our objective is to design a mechanism that optimizes a weighted combination of test loss, seller privacy, and payment, striking a balance between building a good privacy-preserving ML model and minimizing payments to the sellers. To achieve this, we first propose an approach to solve logistic regression with known heterogeneous differential privacy guarantees. Building on these results and leveraging standard mechanism design theory, we develop a two-step optimization framework. We further extend this approach to an online algorithm that handles the sequential arrival of sellers.

R. Srikant

We prove a non-asymptotic central limit theorem for vector-valued martingale differences using Stein's method, and use Poisson's equation to extend the result to functions of Markov Chains. We then show that these results can be applied to establish a non-asymptotic central limit theorem for Temporal Difference (TD) learning with averaging.

Qi Yu, Zhichen Zeng, Yuchen Yan et al.

Network alignment, which aims to find node correspondence across different networks, is the cornerstone of various downstream multi-network and Web mining tasks. Most of the embedding-based methods indirectly model cross-network node relationships by contrasting positive and negative node pairs sampled from hand-crafted strategies, which are vulnerable to graph noises and lead to potential misalignment of nodes. Another line of work based on the optimal transport (OT) theory directly models cross-network node relationships and generates noise-reduced alignments. However, OT methods heavily rely on fixed, pre-defined cost functions that prohibit end-to-end training and are hard to generalize. In this paper, we aim to unify the embedding and OT-based methods in a mutually beneficial manner and propose a joint optimal transport and embedding framework for network alignment named JOENA. For one thing (OT for embedding), through a simple yet effective transformation, the noise-reduced OT mapping serves as an adaptive sampling strategy directly modeling all cross-network node pairs for robust embedding learning.For another (embedding for OT), on top of the learned embeddings, the OT cost can be gradually trained in an end-to-end fashion, which further enhances the alignment quality. With a unified objective, the mutual benefits of both methods can be achieved by an alternating optimization schema with guaranteed convergence. Extensive experiments on real-world networks validate the effectiveness and scalability of JOENA, achieving up to 16% improvement in MRR and 20x speedup compared with the state-of-the-art alignment methods.

Yihan Du, R. Srikant, Wei Chen

Cascading bandits have gained popularity in recent years due to their applicability to recommendation systems and online advertising. In the cascading bandit model, at each timestep, an agent recommends an ordered subset of items (called an item list) from a pool of items, each associated with an unknown attraction probability. Then, the user examines the list, and clicks the first attractive item (if any), and after that, the agent receives a reward. The goal of the agent is to maximize the expected cumulative reward. However, the prior literature on cascading bandits ignores the influences of user states (e.g., historical behaviors) on recommendations and the change of states as the session proceeds. Motivated by this fact, we propose a generalized cascading RL framework, which considers the impact of user states and state transition into decisions. In cascading RL, we need to select items not only with large attraction probabilities but also leading to good successor states. This imposes a huge computational challenge due to the combinatorial action space. To tackle this challenge, we delve into the properties of value functions, and design an oracle BestPerm to efficiently find the optimal item list. Equipped with BestPerm, we develop two algorithms CascadingVI and CascadingBPI, which are both computationally-efficient and sample-efficient, and provide near-optimal regret and sample complexity guarantees. Furthermore, we present experiments to show the improved computational and sample efficiencies of our algorithms compared to straightforward adaptations of existing RL algorithms in practice.

Isaac Grosof, Siva Theja Maguluri, R. Srikant

A wide variety of queueing systems can be naturally modeled as infinite-state Markov Decision Processes (MDPs). In the reinforcement learning (RL) context, a variety of algorithms have been developed to learn and optimize these MDPs. At the heart of many popular policy-gradient based learning algorithms, such as natural actor-critic, TRPO, and PPO, lies the Natural Policy Gradient (NPG) policy optimization algorithm. Convergence results for these RL algorithms rest on convergence results for the NPG algorithm. However, all existing results on the convergence of the NPG algorithm are limited to finite-state settings. We study a general class of queueing MDPs, and prove a $O(1/\sqrt{T})$ convergence rate for the NPG algorithm, if the NPG algorithm is initialized with the MaxWeight policy. This is the first convergence rate bound for the NPG algorithm for a general class of infinite-state average-reward MDPs. Moreover, our result applies to a beyond the queueing setting to any countably-infinite MDP satisfying certain mild structural assumptions, given a sufficiently good initial policy. Key to our result are state-dependent bounds on the relative value function achieved by the iterate policies of the NPG algorithm.

Ameya Anjarlekar, Rasoul Etesami, R Srikant

The continuous nature of belief states in POMDPs presents significant computational challenges in learning the optimal policy. In this paper, we consider an approach that solves a Partially Observable Reinforcement Learning (PORL) problem by approximating the corresponding POMDP model into a finite-state Markov Decision Process (MDP) (called Superstate MDP). We first derive theoretical guarantees that improve upon prior work that relate the optimal value function of the transformed Superstate MDP to the optimal value function of the original POMDP. Next, we propose a policy-based learning approach with linear function approximation to learn the optimal policy for the Superstate MDP. Consequently, our approach shows that a POMDP can be approximately solved using TD-learning followed by Policy Optimization by treating it as an MDP, where the MDP state corresponds to a finite history. We show that the approximation error decreases exponentially with the length of this history. To the best of our knowledge, our finite-time bounds are the first to explicitly quantify the error introduced when applying standard TD learning to a setting where the true dynamics are not Markovian.

Yihan Du, Seo Taek Kong, R. Srikant

The widespread application of Large Language Models (LLMs) imposes increasing demands on safety, such as reducing harmful content and fake information, and avoiding certain forbidden tokens due to rules and laws. While there have been several recent works studying safe alignment of LLMs, these works either require the training of reward and cost models and incur high memory and computational costs, or need prior knowledge about the optimal solution. Motivated by this fact, we study the problem of constrained alignment in LLMs, i.e., maximizing the output reward while restricting the cost due to potentially unsafe content to stay below a threshold. For this problem, we propose a novel primal-dual DPO approach, which first trains a model using standard DPO on reward preference data to provide reward information, and then adopts a rearranged Lagrangian DPO objective utilizing the provided reward information to fine-tune LLMs on cost preference data. Our approach significantly reduces memory and computational costs, and does not require extra prior knowledge. Moreover, we establish rigorous theoretical guarantees on the suboptimality and constraint violation of the output policy. We also extend our approach to an online data setting by incorporating exploration bonuses, which enables our approach to explore uncovered prompt-response space, and then provide theoretical results that get rid of the dependence on preference data coverage. Experimental results on the widely-used preference dataset PKU-SafeRLHF demonstrate the effectiveness of our approach.

Ronshee Chawla, Daniel Vial, Sanjay Shakkottai et al.

The study of collaborative multi-agent bandits has attracted significant attention recently. In light of this, we initiate the study of a new collaborative setting, consisting of $N$ agents such that each agent is learning one of $M$ stochastic multi-armed bandits to minimize their group cumulative regret. We develop decentralized algorithms which facilitate collaboration between the agents under two scenarios. We characterize the performance of these algorithms by deriving the per agent cumulative regret and group regret upper bounds. We also prove lower bounds for the group regret in this setting, which demonstrates the near-optimal behavior of the proposed algorithms.

Daniel Vial, Advait Parulekar, Sanjay Shakkottai et al.

For the misspecified linear Markov decision process (MLMDP) model of Jin et al. [2020], we propose an algorithm with three desirable properties. (P1) Its regret after $K$ episodes scales as $K \max \{ \varepsilon_{\text{mis}}, \varepsilon_{\text{tol}} \}$, where $\varepsilon_{\text{mis}}$ is the degree of misspecification and $\varepsilon_{\text{tol}}$ is a user-specified error tolerance. (P2) Its space and per-episode time complexities remain bounded as $K \rightarrow \infty$. (P3) It does not require $\varepsilon_{\text{mis}}$ as input. To our knowledge, this is the first algorithm satisfying all three properties. For concrete choices of $\varepsilon_{\text{tol}}$, we also improve existing regret bounds (up to log factors) while achieving either (P2) or (P3) (existing algorithms satisfy neither). At a high level, our algorithm generalizes (to MLMDPs) and refines the Sup-Lin-UCB algorithm, which Takemura et al. [2021] recently showed satisfies (P3) for contextual bandits. We also provide an intuitive interpretation of their result, which informs the design of our algorithm.

Semih Cayci, Niao He, R. Srikant

Natural policy gradient (NPG) methods with entropy regularization achieve impressive empirical success in reinforcement learning problems with large state-action spaces. However, their convergence properties and the impact of entropy regularization remain elusive in the function approximation regime. In this paper, we establish finite-time convergence analyses of entropy-regularized NPG with linear function approximation under softmax parameterization. In particular, we prove that entropy-regularized NPG with averaging satisfies the \emph{persistence of excitation} condition, and achieves a fast convergence rate of $\tilde{O}(1/T)$ up to a function approximation error in regularized Markov decision processes. This convergence result does not require any a priori assumptions on the policies. Furthermore, under mild regularity conditions on the concentrability coefficient and basis vectors, we prove that entropy-regularized NPG exhibits \emph{linear convergence} up to a function approximation error.

Siddhartha Satpathi, R Srikant

We consider the dynamics of gradient descent (GD) in overparameterized single hidden layer neural networks with a squared loss function. Recently, it has been shown that, under some conditions, the parameter values obtained using GD achieve zero training error and generalize well if the initial conditions are chosen appropriately. Here, through a Lyapunov analysis, we show that the dynamics of neural network weights under GD converge to a point which is close to the minimum norm solution subject to the condition that there is no training error when using the linear approximation to the neural network. To illustrate the application of this result, we show that the GD converges to a prediction function that generalizes well, thereby providing an alternative proof of the generalization results in Arora et al. (2019).

Daniel Vial, Advait Parulekar, Sanjay Shakkottai et al.

We propose an algorithm that uses linear function approximation (LFA) for stochastic shortest path (SSP). Under minimal assumptions, it obtains sublinear regret, is computationally efficient, and uses stationary policies. To our knowledge, this is the first such algorithm in the LFA literature (for SSP or other formulations). Our algorithm is a special case of a more general one, which achieves regret square root in the number of episodes given access to a certain computation oracle.

Shiyu Liang, Ruoyu Sun, R. Srikant

Recent theoretical works on over-parameterized neural nets have focused on two aspects: optimization and generalization. Many existing works that study optimization and generalization together are based on neural tangent kernel and require a very large width. In this work, we are interested in the following question: for a binary classification problem with two-layer mildly over-parameterized ReLU network, can we find a point with small test error in polynomial time? We first show that the landscape of loss functions with explicit regularization has the following property: all local minima and certain other points which are only stationary in certain directions achieve small test error. We then prove that for convolutional neural nets, there is an algorithm which finds one of these points in polynomial time (in the input dimension and the number of data points). In addition, we prove that for a fully connected neural net, with an additional assumption on the data distribution, there is a polynomial time algorithm.

Semih Cayci, Siddhartha Satpathi, Niao He et al.

In this paper, we study the dynamics of temporal difference learning with neural network-based value function approximation over a general state space, namely, \emph{Neural TD learning}. We consider two practically used algorithms, projection-free and max-norm regularized Neural TD learning, and establish the first convergence bounds for these algorithms. An interesting observation from our results is that max-norm regularization can dramatically improve the performance of TD learning algorithms, both in terms of sample complexity and overparameterization. In particular, we prove that max-norm regularization improves state-of-the-art sample complexity and overparameterization bounds. The results in this work rely on a novel Lyapunov drift analysis of the network parameters as a stopped and controlled random process.

Joseph Lubars, Anna Winnicki, Michael Livesay et al.

We consider Markov Decision Processes (MDPs) in which every stationary policy induces the same graph structure for the underlying Markov chain and further, the graph has the following property: if we replace each recurrent class by a node, then the resulting graph is acyclic. For such MDPs, we prove the convergence of the stochastic dynamics associated with a version of optimistic policy iteration (OPI), suggested in Tsitsiklis (2002), in which the values associated with all the nodes visited during each iteration of the OPI are updated.

Daniel Vial, Sanjay Shakkottai, R. Srikant

We consider a variant of the traditional multi-armed bandit problem in which each arm is only able to provide one-bit feedback during each pull based on its past history of rewards. Our main result is the following: given an upper confidence bound policy which uses full-reward feedback, there exists a coding scheme for generating one-bit feedback, and a corresponding decoding scheme and arm selection policy, such that the ratio of the regret achieved by our policy and the regret of the full-reward feedback policy asymptotically approaches one.

Joseph Lubars, Harsh Gupta, Sandeep Chinchali et al.

We consider the problem of designing an algorithm to allow a car to autonomously merge on to a highway from an on-ramp. Two broad classes of techniques have been proposed to solve motion planning problems in autonomous driving: Model Predictive Control (MPC) and Reinforcement Learning (RL). In this paper, we first establish the strengths and weaknesses of state-of-the-art MPC and RL-based techniques through simulations. We show that the performance of the RL agent is worse than that of the MPC solution from the perspective of safety and robustness to out-of-distribution traffic patterns, i.e., traffic patterns which were not seen by the RL agent during training. On the other hand, the performance of the RL agent is better than that of the MPC solution when it comes to efficiency and passenger comfort. We subsequently present an algorithm which blends the model-free RL agent with the MPC solution and show that it provides better trade-offs between all metrics -- passenger comfort, efficiency, crash rate and robustness.

Xiaotian Xie, Dimitrios Katselis, Carolyn L. Beck et al.

We consider the problem of identifying parameters of a particular class of Markov chains, called Bernoulli Autoregressive (BAR) processes. The structure of any BAR model is encoded by a directed graph. Incoming edges to a node in the graph indicate that the state of the node at a particular time instant is influenced by the states of the corresponding parental nodes in the previous time instant. The associated edge weights determine the corresponding level of influence from each parental node. In the simplest setup, the Bernoulli parameter of a particular node's state variable is a convex combination of the parental node states in the previous time instant and an additional Bernoulli noise random variable. This paper focuses on the problem of edge weight identification using Maximum Likelihood (ML) estimation and proves that the ML estimator is strongly consistent for two variants of the BAR model. We additionally derive closed-form estimators for the aforementioned two variants and prove their strong consistency.

Arghyadip Roy, Sanjay Shakkottai, R. Srikant

In the regret-based formulation of Multi-armed Bandit (MAB) problems, except in rare instances, much of the literature focuses on arms with i.i.d. rewards. In this paper, we consider the problem of obtaining regret guarantees for MAB problems in which the rewards of each arm form a Markov chain which may not belong to a single parameter exponential family. To achieve a logarithmic regret in such problems is not difficult: a variation of standard Kullback-Leibler Upper Confidence Bound (KL-UCB) does the job. However, the constants obtained from such an analysis are poor for the following reason: i.i.d. rewards are a special case of Markov rewards and it is difficult to design an algorithm that works well independent of whether the underlying model is truly Markovian or i.i.d. To overcome this issue, we introduce a novel algorithm that identifies whether the rewards from each arm are truly Markovian or i.i.d. using a total variation distance-based test. Our algorithm then switches from using a standard KL-UCB to a specialized version of KL-UCB when it determines that the arm reward is Markovian, thus resulting in low regrets for both i.i.d. and Markovian settings.

Wentao Weng, Harsh Gupta, Niao He et al.

In this paper, we establish a theoretical comparison between the asymptotic mean-squared error of Double Q-learning and Q-learning. Our result builds upon an analysis for linear stochastic approximation based on Lyapunov equations and applies to both tabular setting and with linear function approximation, provided that the optimal policy is unique and the algorithms converge. We show that the asymptotic mean-squared error of Double Q-learning is exactly equal to that of Q-learning if Double Q-learning uses twice the learning rate of Q-learning and outputs the average of its two estimators. We also present some practical implications of this theoretical observation using simulations.

Daniel Vial, Sanjay Shakkottai, R. Srikant

Recent works have shown that agents facing independent instances of a stochastic $K$-armed bandit can collaborate to decrease regret. However, these works assume that each agent always recommends their individual best-arm estimates to other agents, which is unrealistic in envisioned applications (machine faults in distributed computing or spam in social recommendation systems). Hence, we generalize the setting to include $n$ honest and $m$ malicious agents who recommend best-arm estimates and arbitrary arms, respectively. We first show that even with a single malicious agent, existing collaboration-based algorithms fail to improve regret guarantees over a single-agent baseline. We propose a scheme where honest agents learn who is malicious and dynamically reduce communication with (i.e., "block") them. We show that collaboration indeed decreases regret for this algorithm, assuming $m$ is small compared to $K$ but without assumptions on malicious agents' behavior, thus ensuring that our algorithm is robust against any malicious recommendation strategy.

Ruoyu Sun, Dawei Li, Shiyu Liang et al.

One of the major concerns for neural network training is that the non-convexity of the associated loss functions may cause bad landscape. The recent success of neural networks suggests that their loss landscape is not too bad, but what specific results do we know about the landscape? In this article, we review recent findings and results on the global landscape of neural networks. First, we point out that wide neural nets may have sub-optimal local minima under certain assumptions. Second, we discuss a few rigorous results on the geometric properties of wide networks such as "no bad basin", and some modifications that eliminate sub-optimal local minima and/or decreasing paths to infinity. Third, we discuss visualization and empirical explorations of the landscape for practical neural nets. Finally, we briefly discuss some convergence results and their relation to landscape results.

Semih Cayci, Atilla Eryilmaz, R. Srikant

Time-constrained decision processes have been ubiquitous in many fundamental applications in physics, biology and computer science. Recently, restart strategies have gained significant attention for boosting the efficiency of time-constrained processes by expediting the completion times. In this work, we investigate the bandit problem with controlled restarts for time-constrained decision processes, and develop provably good learning algorithms. In particular, we consider a bandit setting where each decision takes a random completion time, and yields a random and correlated reward at the end, with unknown values at the time of decision. The goal of the decision-maker is to maximize the expected total reward subject to a time constraint $τ$. As an additional control, we allow the decision-maker to interrupt an ongoing task and forgo its reward for a potentially more rewarding alternative. For this problem, we develop efficient online learning algorithms with $O(\log(τ))$ and $O(\sqrt{τ\log(τ)})$ regret in a finite and continuous action space of restart strategies, respectively. We demonstrate an applicability of our algorithm by using it to boost the performance of SAT solvers.

Semih Cayci, Atilla Eryilmaz, R. Srikant

We consider a budget-constrained bandit problem where each arm pull incurs a random cost, and yields a random reward in return. The objective is to maximize the total expected reward under a budget constraint on the total cost. The model is general in the sense that it allows correlated and potentially heavy-tailed cost-reward pairs that can take on negative values as required by many applications. We show that if moments of order $(2+γ)$ for some $γ> 0$ exist for all cost-reward pairs, $O(\log B)$ regret is achievable for a budget $B>0$. In order to achieve tight regret bounds, we propose algorithms that exploit the correlation between the cost and reward of each arm by extracting the common information via linear minimum mean-square error estimation. We prove a regret lower bound for this problem, and show that the proposed algorithms achieve tight problem-dependent regret bounds, which are optimal up to a universal constant factor in the case of jointly Gaussian cost and reward pairs.

Shiyu Liang, Ruoyu Sun, R. Srikant

Traditional landscape analysis of deep neural networks aims to show that no sub-optimal local minima exist in some appropriate sense. From this, one may be tempted to conclude that descent algorithms which escape saddle points will reach a good local minimum. However, basic optimization theory tell us that it is also possible for a descent algorithm to diverge to infinity if there are paths leading to infinity, along which the loss function decreases. It is not clear whether for non-linear neural networks there exists one setting that no bad local-min and no decreasing paths to infinity can be simultaneously achieved. In this paper, we give the first positive answer to this question. More specifically, for a large class of over-parameterized deep neural networks with appropriate regularizers, the loss function has no bad local minima and no decreasing paths to infinity. The key mathematical trick is to show that the set of regularizers which may be undesirable can be viewed as the image of a Lipschitz continuous mapping from a lower-dimensional Euclidean space to a higher-dimensional Euclidean space, and thus has zero measure.

Harsh Gupta, R. Srikant, Lei Ying

We study two time-scale linear stochastic approximation algorithms, which can be used to model well-known reinforcement learning algorithms such as GTD, GTD2, and TDC. We present finite-time performance bounds for the case where the learning rate is fixed. The key idea in obtaining these bounds is to use a Lyapunov function motivated by singular perturbation theory for linear differential equations. We use the bound to design an adaptive learning rate scheme which significantly improves the convergence rate over the known optimal polynomial decay rule in our experiments, and can be used to potentially improve the performance of any other schedule where the learning rate is changed at pre-determined time instants.

R. Srikant, Lei Ying

We consider the dynamics of a linear stochastic approximation algorithm driven by Markovian noise, and derive finite-time bounds on the moments of the error, i.e., deviation of the output of the algorithm from the equilibrium point of an associated ordinary differential equation (ODE). We obtain finite-time bounds on the mean-square error in the case of constant step-size algorithms by considering the drift of an appropriately chosen Lyapunov function. The Lyapunov function can be interpreted either in terms of Stein's method to obtain bounds on steady-state performance or in terms of Lyapunov stability theory for linear ODEs. We also provide a comprehensive treatment of the moments of the square of the 2-norm of the approximation error. Our analysis yields the following results: (i) for a given step-size, we show that the lower-order moments can be made small as a function of the step-size and can be upper-bounded by the moments of a Gaussian random variable; (ii) we show that the higher-order moments beyond a threshold may be infinite in steady-state; and (iii) we characterize the number of samples needed for the finite-time bounds to be of the same order as the steady-state bounds. As a by-product of our analysis, we also solve the open problem of obtaining finite-time bounds for the performance of temporal difference learning algorithms with linear function approximation and a constant step-size, without requiring a projection step or an i.i.d. noise assumption.

Harsh Gupta, Seo Taek Kong, R. Srikant et al.

Boltzmann exploration is widely used in reinforcement learning to provide a trade-off between exploration and exploitation. Recently, in (Cesa-Bianchi et al., 2017) it has been shown that pure Boltzmann exploration does not perform well from a regret perspective, even in the simplest setting of stochastic multi-armed bandit (MAB) problems. In this paper, we show that a simple modification to Boltzmann exploration, motivated by a variation of the standard doubling trick, achieves $O(K\log^{1+α} T)$ regret for a stochastic MAB problem with $K$ arms, where $α>0$ is a parameter of the algorithm. This improves on the result in (Cesa-Bianchi et al., 2017), where an algorithm inspired by the Gumbel-softmax trick achieves $O(K\log^2 T)$ regret. We also show that our algorithm achieves $O(β(G) \log^{1+α} T)$ regret in stochastic MAB problems with graph-structured feedback, without knowledge of the graph structure, where $β(G)$ is the independence number of the feedback graph. Additionally, we present extensive experimental results on real datasets and applications for multi-armed bandits with both traditional bandit feedback and graph-structured feedback. In all cases, our algorithm performs as well or better than the state-of-the-art.

Shiyu Liang, Ruoyu Sun, Jason D. Lee et al.

One of the main difficulties in analyzing neural networks is the non-convexity of the loss function which may have many bad local minima. In this paper, we study the landscape of neural networks for binary classification tasks. Under mild assumptions, we prove that after adding one special neuron with a skip connection to the output, or one special neuron per layer, every local minimum is a global minimum.

Siddhartha Satpathi, Supratim Deb, R Srikant et al.

We consider the problem of separating error messages generated in large distributed data center networks into error events. In such networks, each error event leads to a stream of messages generated by hardware and software components affected by the event. These messages are stored in a giant message log. We consider the unsupervised learning problem of identifying the signatures of events that generated these messages; here, the signature of an error event refers to the mixture of messages generated by the event. One of the main contributions of the paper is a novel mapping of our problem which transforms it into a problem of topic discovery in documents. Events in our problem correspond to topics and messages in our problem correspond to words in the topic discovery problem. However, there is no direct analog of documents. Therefore, we use a non-parametric change-point detection algorithm, which has linear computational complexity in the number of messages, to divide the message log into smaller subsets called episodes, which serve as the equivalents of documents. After this mapping has been done, we use a well-known algorithm for topic discovery, called LDA, to solve our problem. We theoretically analyze the change-point detection algorithm, and show that it is consistent and has low sample complexity. We also demonstrate the scalability of our algorithm on a real data set consisting of $97$ million messages collected over a period of $15$ days, from a distributed data center network which supports the operations of a large wireless service provider.

Shiyu Liang, Ruoyu Sun, Yixuan Li et al.

It is widely conjectured that the reason that training algorithms for neural networks are successful because all local minima lead to similar performance, for example, see (LeCun et al., 2015, Choromanska et al., 2015, Dauphin et al., 2014). Performance is typically measured in terms of two metrics: training performance and generalization performance. Here we focus on the training performance of single-layered neural networks for binary classification, and provide conditions under which the training error is zero at all local minima of a smooth hinge loss function. Our conditions are roughly in the following form: the neurons have to be strictly convex and the surrogate loss function should be a smooth version of hinge loss. We also provide counterexamples to show that when the loss function is replaced with quadratic loss or logistic loss, the result may not hold.

Shiyu Liang, Yixuan Li, R. Srikant

We consider the problem of detecting out-of-distribution images in neural networks. We propose ODIN, a simple and effective method that does not require any change to a pre-trained neural network. Our method is based on the observation that using temperature scaling and adding small perturbations to the input can separate the softmax score distributions between in- and out-of-distribution images, allowing for more effective detection. We show in a series of experiments that ODIN is compatible with diverse network architectures and datasets. It consistently outperforms the baseline approach by a large margin, establishing a new state-of-the-art performance on this task. For example, ODIN reduces the false positive rate from the baseline 34.7% to 4.3% on the DenseNet (applied to CIFAR-10) when the true positive rate is 95%.

Dimitrios Katselis, Carolyn L. Beck, R. Srikant

We introduce a novel multivariate random process producing Bernoulli outputs per dimension, that can possibly formalize binary interactions in various graphical structures and can be used to model opinion dynamics, epidemics, financial and biological time series data, etc. We call this a Bernoulli Autoregressive Process (BAR). A BAR process models a discrete-time vector random sequence of $p$ scalar Bernoulli processes with autoregressive dynamics and corresponds to a particular Markov Chain. The benefit from the autoregressive dynamics is the description of a $2^p\times 2^p$ transition matrix by at most $pd$ effective parameters for some $d\ll p$ or by two sparse matrices of dimensions $p\times p^2$ and $p\times p$, respectively, parameterizing the transitions. Additionally, we show that the BAR process mixes rapidly, by proving that the mixing time is $O(\log p)$. The hidden constant in the previous mixing time bound depends explicitly on the values of the chain parameters and implicitly on the maximum allowed in-degree of a node in the corresponding graph. For a network with $p$ nodes, where each node has in-degree at most $d$ and corresponds to a scalar Bernoulli process generated by a BAR, we provide a greedy algorithm that can efficiently learn the structure of the underlying directed graph with a sample complexity proportional to the mixing time of the BAR process. The sample complexity of the proposed algorithm is nearly order-optimal as it is only a $\log p$ factor away from an information-theoretic lower bound. We present simulation results illustrating the performance of our algorithm in various setups, including a model for a biological signaling network.

Shiyu Liang, R. Srikant

Recently there has been much interest in understanding why deep neural networks are preferred to shallow networks. We show that, for a large class of piecewise smooth functions, the number of neurons needed by a shallow network to approximate a function is exponentially larger than the corresponding number of neurons needed by a deep network for a given degree of function approximation. First, we consider univariate functions on a bounded interval and require a neural network to achieve an approximation error of $\varepsilon$ uniformly over the interval. We show that shallow networks (i.e., networks whose depth does not depend on $\varepsilon$) require $Ω(\text{poly}(1/\varepsilon))$ neurons while deep networks (i.e., networks whose depth grows with $1/\varepsilon$) require $\mathcal{O}(\text{polylog}(1/\varepsilon))$ neurons. We then extend these results to certain classes of important multivariate functions. Our results are derived for neural networks which use a combination of rectifier linear units (ReLUs) and binary step units, two of the most popular type of activation functions. Our analysis builds on a simple observation: the multiplication of two bits can be represented by a ReLU.

Kobi Cohen, Angelia Nedic, R. Srikant

The problem of least squares regression of a $d$-dimensional unknown parameter is considered. A stochastic gradient descent based algorithm with weighted iterate-averaging that uses a single pass over the data is studied and its convergence rate is analyzed. We first consider a bounded constraint set of the unknown parameter. Under some standard regularity assumptions, we provide an explicit $O(1/k)$ upper bound on the convergence rate, depending on the variance (due to the additive noise in the measurements) and the size of the constraint set. We show that the variance term dominates the error and decreases with rate $1/k$, while the term which is related to the size of the constraint set decreases with rate $\log k/k^2$. We then compare the asymptotic ratio $ρ$ between the convergence rate of the proposed scheme and the empirical risk minimizer (ERM) as the number of iterations approaches infinity. We show that $ρ\leq 4$ under some mild conditions for all $d\geq 1$. We further improve the upper bound by showing that $ρ\leq 4/3$ for the case of $d=1$ and unbounded parameter set. Simulation results demonstrate strong performance of the algorithm as compared to existing methods, and coincide with $ρ\leq 4/3$ even for large $d$ in practice.

Huasen Wu, R. Srikant, Xin Liu et al.

We study contextual bandits with budget and time constraints, referred to as constrained contextual bandits.The time and budget constraints significantly complicate the exploration and exploitation tradeoff because they introduce complex coupling among contexts over time.Such coupling effects make it difficult to obtain oracle solutions that assume known statistics of bandits. To gain insight, we first study unit-cost systems with known context distribution. When the expected rewards are known, we develop an approximation of the oracle, referred to Adaptive-Linear-Programming (ALP), which achieves near-optimality and only requires the ordering of expected rewards. With these highly desirable features, we then combine ALP with the upper-confidence-bound (UCB) method in the general case where the expected rewards are unknown {\it a priori}. We show that the proposed UCB-ALP algorithm achieves logarithmic regret except for certain boundary cases. Further, we design algorithms and obtain similar regret analysis results for more general systems with unknown context distribution and heterogeneous costs. To the best of our knowledge, this is the first work that shows how to achieve logarithmic regret in constrained contextual bandits. Moreover, this work also sheds light on the study of computationally efficient algorithms for general constrained contextual bandits.

Rui Wu, Jiaming Xu, R. Srikant et al.

Given a set of pairwise comparisons, the classical ranking problem computes a single ranking that best represents the preferences of all users. In this paper, we study the problem of inferring individual preferences, arising in the context of making personalized recommendations. In particular, we assume that there are $n$ users of $r$ types; users of the same type provide similar pairwise comparisons for $m$ items according to the Bradley-Terry model. We propose an efficient algorithm that accurately estimates the individual preferences for almost all users, if there are $r \max \{m, n\}\log m \log^2 n$ pairwise comparisons per type, which is near optimal in sample complexity when $r$ only grows logarithmically with $m$ or $n$. Our algorithm has three steps: first, for each user, compute the \emph{net-win} vector which is a projection of its $\binom{m}{2}$-dimensional vector of pairwise comparisons onto an $m$-dimensional linear subspace; second, cluster the users based on the net-win vectors; third, estimate a single preference for each cluster separately. The net-win vectors are much less noisy than the high dimensional vectors of pairwise comparisons and clustering is more accurate after the projection as confirmed by numerical experiments. Moreover, we show that, when a cluster is only approximately correct, the maximum likelihood estimation for the Bradley-Terry model is still close to the true preference.

Kai Zhu, Rui Wu, Lei Ying et al.

In this paper, we consider a popular model for collaborative filtering in recommender systems where some users of a website rate some items, such as movies, and the goal is to recover the ratings of some or all of the unrated items of each user. In particular, we consider both the clustering model, where only users (or items) are clustered, and the co-clustering model, where both users and items are clustered, and further, we assume that some users rate many items (information-rich users) and some users rate only a few items (information-sparse users). When users (or items) are clustered, our algorithm can recover the rating matrix with $ω(MK \log M)$ noisy entries while $MK$ entries are necessary, where $K$ is the number of clusters and $M$ is the number of items. In the case of co-clustering, we prove that $K^2$ entries are necessary for recovering the rating matrix, and our algorithm achieves this lower bound within a logarithmic factor when $K$ is sufficiently large. We compare our algorithms with a well-known algorithms called alternating minimization (AM), and a similarity score-based algorithm known as the popularity-among-friends (PAF) algorithm by applying all three to the MovieLens and Netflix data sets. Our co-clustering algorithm and AM have similar overall error rates when recovering the rating matrix, both of which are lower than the error rate under PAF. But more importantly, the error rate of our co-clustering algorithm is significantly lower than AM and PAF in the scenarios of interest in recommender systems: when recommending a few items to each user or when recommending items to users who only rated a few items (these users are the majority of the total user population). The performance difference increases even more when noise is added to the datasets.

Jiaming Xu, Rui Wu, Kai Zhu et al.

In standard clustering problems, data points are represented by vectors, and by stacking them together, one forms a data matrix with row or column cluster structure. In this paper, we consider a class of binary matrices, arising in many applications, which exhibit both row and column cluster structure, and our goal is to exactly recover the underlying row and column clusters by observing only a small fraction of noisy entries. We first derive a lower bound on the minimum number of observations needed for exact cluster recovery. Then, we propose three algorithms with different running time and compare the number of observations needed by them for successful cluster recovery. Our analytical results show smooth time-data trade-offs: one can gradually reduce the computational complexity when increasingly more observations are available.

Joohwan Kim, R. Srikant

In earlier work, we showed that it is possible to achieve $O(\log N)$ streaming delay with high probability in a peer-to-peer network, where each peer has as little as four neighbors, while achieving any arbitrary fraction of the maximum possible streaming rate. However, the constant in the $O(log N)$ delay term becomes rather large as we get closer to the maximum streaming rate. In this paper, we design an alternative pairing and chunk dissemination algorithm that allows us to transmit at the maximum streaming rate while ensuring that all, but a negligible fraction of the peers, receive the data stream with $O(\log N)$ delay with high probability. The result is established by examining the properties of graph formed by the union of two or more random 1-regular digraphs, i.e., directed graphs in which each node has an incoming and an outgoing node degree both equal to one.

Joohwan Kim, R. Srikant

We are motivated by the problem of designing a simple distributed algorithm for Peer-to-Peer streaming applications that can achieve high throughput and low delay, while allowing the neighbor set maintained by each peer to be small. While previous works have mostly used tree structures, our algorithm constructs multiple random directed Hamiltonian cycles and disseminates content over the superposed graph of the cycles. We show that it is possible to achieve the maximum streaming capacity even when each peer only transmits to and receives from Theta(1) neighbors. Further, we show that the proposed algorithm achieves the streaming delay of Theta(log N) when the streaming rate is less than (1-1/K) of the maximum capacity for any fixed integer K>1, where N denotes the number of peers in the network. The key theoretical contribution is to characterize the distance between peers in a graph formed by the superposition of directed random Hamiltonian cycles, in which edges from one of the cycles may be dropped at random. We use Doob martingales and graph expansion ideas to characterize this distance as a function of N, with high probability.

Rui Wu, R. Srikant, Jian Ni

We consider the structure learning problem for graphical models that we call loosely connected Markov random fields, in which the number of short paths between any pair of nodes is small, and present a new conditional independence test based algorithm for learning the underlying graph structure. The novel maximization step in our algorithm ensures that the true edges are detected correctly even when there are short cycles in the graph. The number of samples required by our algorithm is C*log p, where p is the size of the graph and the constant C depends on the parameters of the model. We show that several previously studied models are examples of loosely connected Markov random fields, and our algorithm achieves the same or lower computational complexity than the previously designed algorithms for individual cases. We also get new results for more general graphical models, in particular, our algorithm learns general Ising models on the Erdos-Renyi random graph G(p, c/p) correctly with running time O(np^5).