In online betting, the bookmaker can update the payoffs it offers on a particular event many times before the event takes place, and the updated payoffs may depend on the bets accumulated thus far. We study the problem of bookmaking with the goal of maximizing the return in the worst-case, with respect to the gamblers' behavior and the event's outcome. We formalize this problem as the Optimal Online Bookmaking game, and provide the exact solution for the binary case. To this end, we develop the optimal bookmaking strategy, which relies on a new technique called bi-balancing trees, that assures that the house loss is the same for all decisive betting sequences, where the gambler bets all its money on a single outcome in each round.

Prediction with expert advice serves as a fundamental model in online learning and sequential decision-making. However, in many real-world settings, this classical model proves insufficient as the feedback available to the decision-maker is often subject to noise, errors, or communication constraints. This paper provides fundamental limits on performance, quantified by the regret, in the case when the feedback is corrupted by an additive noise. Our general analysis achieves sharp regret bounds for canonical examples of such additive noise as the Gaussian distribution, the uniform distribution, and a general noise with a log-concave density. This analysis demonstrates how different noise characteristics affect regret bounds and identifies how the regret fundamentally scales as a function of the properties of the noise distribution.

We devise an online learning algorithm – titled Switching via Monotone Adapted Regret Traces (SMART) – that adapts to the data and achieves regret that is instance optimal, i.e., simultaneously competitive on every input sequence compared to the performance of the follow-the-leader (FTL) policy and the worst case guarantee of any other input policy. We show that the regret of the SMART policy on any input sequence is within a multiplicative factor e/(e-1), approximately 1.58, of the smaller of: 1) the regret obtained by FTL on the sequence, and 2) the upper bound on regret guaranteed by the given worst-case policy. This implies a strictly stronger guarantee than typical ‘best-of-both-worlds’ bounds as the guarantee holds for every input sequence regardless of how it is generated. SMART is simple to implement as it begins by playing FTL and switches at most once during the time horizon to the worst-case algorithm. Our approach and results follow from a reduction of instance optimal online learning to competitive analysis for the ski-rental problem. We complement our competitive ratio upper bounds with a fundamental lower bound showing that over all input sequences, no algorithm can get better than a 1.43-fraction of the minimum regret achieved by FTL and the minimax-optimal policy. We present a modification of SMART that combines FTL with a “small-loss" algorithm to achieve instance optimality between the regret of FTL and the small loss regret bound.

Regret minimization in the problem of prediction with expert advice in the presence of noisy feedback is a fundamental challenge in online learning and sequential decision making. A general framework is proposed for designing and analyzing no-regret algorithms in this setting. This analysis, when specialized to several canonical channel models, is shown to lead to tight bounds on the regret thus characterizing how the noise level affects the regret and demonstrating that in some cases it is possible to achieve the same regret as with noiseless feedback.

Motivated by the prevalent data science applications of processing large-scale graph data such as social networks and biological networks, this paper investigates lossless compression of data in the form of a labeled graph. Particularly, we consider a widely used random graph model, stochastic block model (SBM), which captures the clustering effects in social networks. An information-theoretic universal compression framework is applied, in which one aims to design a single compressor that achieves the asymptotically optimal compression rate, for every SBM distribution, without knowing the parameters of the SBM. Such a graph compressor is proposed in this paper, which universally achieves the optimal compression rate with polynomial time complexity for a wide class of SBMs. Existing universal compression techniques are developed mostly for stationary ergodic one-dimensional sequences. However, the adjacency matrix of SBM has complex two-dimensional correlations. The challenge is alleviated through a carefully designed transform that converts two-dimensional correlated data into almost i.i.d. submatrices. The sequence of submatrices is then compressed by a Krichevsky-Trofimov compressor, whose length analysis is generalized to identically distributed but arbitrarily correlated sequences. In four benchmark graph datasets, the compressed files from competing algorithms take 2.4 to 27 times the space needed by the proposed scheme.

This paper considers the problem of constructing a confidence sequence, which is a sequence of confidence intervals that hold uniformly over time, for estimating the mean of bounded real-valued random processes. This paper revisits the gambling-based approach established in the recent literature from a natural two-horse race perspective, and demonstrates new properties of the resulting algorithm induced by Cover (1991)’s universal portfolio. The main result of this paper is a new algorithm based on a mixture of lower bounds, which closely approximates the performance of Cover’s universal portfolio with constant per-round time complexity. A higher-order generalization of a lower bound on a logarithmic function in (Fan et al., 2015), which is developed as a key technique for the proposed algorithm, may be of independent interest.

A new portfolio selection strategy that adapts to a continuous side-information sequence is presented, with a universal wealth guarantee against a class of state-constant rebalanced portfolios with respect to a state function that maps each side-information symbol to a finite set of states. In particular, given that a state function belongs to a collection of functions of finite Natarajan dimension, the proposed strategy is shown to achieve, asymptotically to first order in the exponent, the same wealth as the best state-constant rebalanced portfolio with respect to the best state function, chosen in hindsight from observed market. This result can be viewed as an extension of the seminal work of Cover and Ordentlich (1996) that assumes a single-state function.

Sequential prediction of m−ary individual sequences under the Hamming loss, where m ≥ 2, is studied. In particular, leveraging a connection to the follow-the-regularized-leader family of algorithms in online learning, the strategy of Feder, Merhav and Gutman [1] for binary universal prediction is extended to arbitrary alphabet size, and matching upper and lower bounds obtained on the regret achieved by the aforementioned strategy.

We initiate the study of smoothed analysis for the sequential probability assignment problem with contexts. We study information-theoretically optimal minmax rates as well as a framework for algorithmic reduction involving the maximum likelihood estimator oracle. Our approach establishes a general-purpose reduction from minimax rates for sequential probability assignment for smoothed adversaries to minimax rates for transductive learning. This leads to optimal (logarithmic) fast rates for parametric classes and classes with finite VC dimension. On the algorithmic front, we develop an algorithm that efficiently taps into the MLE oracle, for general classes of functions. We show that under general conditions this algorithmic approach yields sublinear regret.

A class of parameter-free online linear optimization algorithms is proposed that harnesses the structure of an adversarial sequence by adapting to some side information. These algorithms combine the reduction technique of Orabona and Pal (2016) for adapting coin betting algorithms for online linear optimization with universal compression techniques in information theory for incorporating sequential side information to coin betting. Concrete examples are studied in which the side information has a tree structure and consists of quantized values of the previous symbols of the adversarial sequence, including fixed-order and variable-order Markov cases. By modifying the context-tree weighting technique of Willems, Shtarkov, and Tjalkens (1995), the proposed algorithm is further refined to achieve the best performance over all adaptive algorithms with tree-structured side information of a given maximum order in a computationally efficient manner.

The problem of online prediction with sequential side information under logarithmic loss is studied, and general upper and lower bounds on the minimax regret incurred by the predictor is established. The upper bounds on the minimax regret are obtained by constructing and analyzing a probability assignment based on mixture probability assignments in universal compression, and the lower bounds are obtained by way of a redundancy–capacity theorem. A tight characterization of the regret is provided in some special settings.

Motivated by the prevalent data science applications of processing large-scale graph data such as social networks, web graphs, and biological networks, as well as the high I/O and communication costs of storing and transmitting such data, this paper investigates universal compression of data appearing in the form of a labeled graph. In particular, we consider a widely used random graph model, stochastic block model (SBM), which captures the clustering effects in social networks. A universal graph compressor is proposed, which achieves the optimal compression rate for a wide family of SBMs with edge probabilities from$O$(1) to Ω(1/$n$2-∊) for any 0 < ∊ < 1. Existing universal compression techniques are developed mostly for stationary ergodic one-dimensional sequences with entropy linear in the number of variables. However, the adjacency matrix of SBM has complex two-dimensional correlations and sublinear entropy in the sparse regime. These challenges are alleviated through a carefully designed transform that converts two-dimensional correlated data into almost i.i.d. blocks. The blocks are then compressed by a Krichevsky-Trofimov compressor, whose length analysis is generalized to arbitrarily correlated processes with identical marginals.

Let X and Y be dependent random variables. This paper considers the problem of designing a scalar quantizer for Y to maximize the mutual information between the quantizer's output and X, and develops fundamental properties and bounds for this form of quantization, which is connected to the log-loss distortion criterion. The main focus is the regime of low I(X;Y), where it is shown that, if X is binary, a constant fraction of the mutual information can always be preserved usingO(log(1/I(X;Y))) quantization levels, and there exist distributions for which this many quantization levels are necessary. Furthermore, for larger finite alphabets 2X|X| /I(X;Y)))η·(|X| - 1)quantization levels.

This paper proposes a new scheme that allows the measurement of traffic loads of radio networks by decoding some information about the cell, such as downlink control information (DCI) broadcast on the physical downlink control channel (PDCCH). In the proposed scheme, a client decodes the entire DCI for all user equipments (UEs) with ongoing connections in a cell and extracts some information about the cell, such as the number of active UEs, the number of radio resources occupied, as well as the modulation and coding scheme used by each UE. Based on this information, mobile network carriers can measure exact traffic loads of cells preemptively without affecting ongoing connections. Contrary to an exhaustive searching scheme that attempts to decode DCI with all radio network temporary identifiers (RNTIs), the proposed scheme derives a small set of valid RNTIs by using an inverse function and attempts to decode DCI only with the valid RNTIs instead of the entire set of RNTIs. This reduction enables the client to recover the correct DCI with marginal computational complexity, which allows for real-time decoding of DCI. The simulation results show that the proposed scheme can significantly reduce the complexity required to decode the entire DCI in a cell, compared to the exhaustive searching scheme.

The Liu-Cuff-Verdu lemma states that in estimating a source X from an observation Y, making a random guess X' from the posterior p(xly) can go wrong at most twice as often as the optimal answer. Several variations of this fundamental, yet rather arcane, result are explored for detection, decoding, and estimation problems.