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In modern statistical and data science applications, the probability distribution generating the data in question is unknown (or even absent) and decisions must be taken in a purely data-driven manner. Thus motivated, in this dissertation the information-theoretic approach of universal probability is revisited and expanded upon. This approach gives us general principles and guidelines for assigning sequential probabilities to data (based on which a decision can then be made), and has been used successfully over the years to problems in compression and estimation among others. The utility of this approach is then demonstrated through three example problems, motivated by the aforementioned modern statistical applications---universal compression of graphical data, sequential prediction with side information, and universal portfolio selection with side information.

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.

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.

We study the relative entropy between the empirical estimate of a discrete distribution and the true underlying distribution. If the minimum value of the probability mass function exceeds an \\( > 0\\) (i.e. when the true underlying distribution is bounded sufficiently away from the boundary of the simplex), we prove an upper bound on the moment generating function of the centered relative entropy that matches (up to logarithmic factors in the alphabet size and \\(\\)) the optimal asymptotic rates, subsequently leading to a sharp concentration inequality for the centered relative entropy. As a corollary of this result we also obtain confidence intervals and moment bounds for the centered relative entropy that are sharp up to logarithmic factors in the alphabet size and \\(\\).

Generating data from discrete distributions is important for a number of application domains including text, tabular data, and genomic data. Several groups have recently used random \\(k\\)-satisfiability (\\(k\\)-SAT) as a synthetic benchmark for new generative techniques. In this paper, we show that fundamental insights from the theory of random constraint satisfaction problems have observable implications (sometime contradicting intuition) on the behavior of generative techniques on such benchmarks. More precisely, we study the problem of generating a uniformly random solution of a given (random) \\(k\\)-SAT or \\(k\\)-XORSAT formula. Among other findings, we observe that: \\((i)\\)~Continuous diffusions outperform masked discrete diffusions; \\((ii)\\)~Learned diffusions can match the theoretical `ideal' accuracy; \\((iii)\\)~Smart ordering of the variables can significantly improve accuracy, although not following popular heuristics.

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) 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 an operational 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 also 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.

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.

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 providing and analyzing a probability assignment inspired by mixture probability assignments in universal compression, and the lower bounds are obtained by way of a redundancy-capacity theorem. The 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 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.

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.