# Computational Learning Theory

### Subsumption-driven clause learning with DPLL+restarts

Complete SAT solvers make deductions until they find a model or produce the empty clause. In DPLL and CDCL solvers, these deductions are produced using assumptions generally called decisions. In DPLL solvers [DLL62], the knowledge accumulated since the beginning of the search is represented by the phases of decision literals. Each new conflict induced by decisions increases the amount of information being accumulated. This amount of information can be interpreted as a proportion of search space already explored that is known not to contain a model.

### Online Matrix Completion with Side Information

We give an online algorithm and prove novel mistake and regret bounds for online binary matrix completion with side information. The bounds we prove are of the form $\tilde{\mathcal{O}}({\mathcal{D}}/{\gamma^2})$. The term ${1}/{\gamma^2}$ is analogous to the usual margin term in SVM (perceptron) bounds. More specifically, if we assume that there is some factorization of the underlying $m\times n$ matrix into $\mathbf{P} \mathbf{Q}^{\top}$ where the rows of $\mathbf{P}$ are interpreted as classifiers'' in $\Re^d$ and the rows of $\mathbf{Q}$ as instances'' in $\Re^d$, then $\gamma$ is is the maximum (normalized) margin over all factorizations $\mathbf{P} \mathbf{Q}^{\top}$ consistent with the observed matrix. The quasi-dimension term $\mathcal{D}$ measures the quality of side information. In the presence of no side information, $\mathcal{D} = m+n$. However, if the side information is predictive of the underlying factorization of the matrix, then in the best case, $\mathcal{D} \in \mathcal{O}(k + \ell)$ where $k$ is the number of distinct row factors and $\ell$ is the number of distinct column factors. We additionally provide a generalization of our algorithm to the inductive setting. In this setting, the side information is not specified in advance. The results are similar to the transductive setting but in the best case, the quasi-dimension $\mathcal{D}$ is now bounded by $\mathcal{O}(k^2 + \ell^2)$.

### Effective problem solving using SAT solvers

In this article we demonstrate how to solve a variety of problems and puzzles using the built-in SAT solver of the computer algebra system Maple. Once the problems have been encoded into Boolean logic, solutions can be found (or shown to not exist) automatically, without the need to implement any search algorithm. In particular, we describe how to solve the $n$-queens problem, how to generate and solve Sudoku puzzles, how to solve logic puzzles like the Einstein riddle, how to solve the 15-puzzle, how to solve the maximum clique problem, and finding Graeco-Latin squares.

### Lower Bounds for Adversarially Robust PAC Learning

In this work, we initiate a formal study of probably approximately correct (PAC) learning under evasion attacks, where the adversary's goal is to \emph{misclassify} the adversarially perturbed sample point $\widetilde{x}$, i.e., $h(\widetilde{x})\neq c(\widetilde{x})$, where $c$ is the ground truth concept and $h$ is the learned hypothesis. Previous work on PAC learning of adversarial examples have all modeled adversarial examples as corrupted inputs in which the goal of the adversary is to achieve $h(\widetilde{x}) \neq c(x)$, where $x$ is the original untampered instance. These two definitions of adversarial risk coincide for many natural distributions, such as images, but are incomparable in general. We first prove that for many theoretically natural input spaces of high dimension $n$ (e.g., isotropic Gaussian in dimension $n$ under $\ell_2$ perturbations), if the adversary is allowed to apply up to a sublinear $o(||x||)$ amount of perturbations on the test instances, PAC learning requires sample complexity that is exponential in $n$. This is in contrast with results proved using the corrupted-input framework, in which the sample complexity of robust learning is only polynomially more. We then formalize hybrid attacks in which the evasion attack is preceded by a poisoning attack. This is perhaps reminiscent of "trapdoor attacks" in which a poisoning phase is involved as well, but the evasion phase here uses the error-region definition of risk that aims at misclassifying the perturbed instances. In this case, we show PAC learning is sometimes impossible all together, even when it is possible without the attack (e.g., due to the bounded VC dimension).

### Graph-based Discriminators: Sample Complexity and Expressiveness

A basic question in learning theory is to identify if two distributions are identical when we have access only to examples sampled from the distributions. This basic task is considered, for example, in the context of Generative Adversarial Networks (GANs), where a discriminator is trained to distinguish between a real-life distribution and a synthetic distribution. % Classically, we use a hypothesis class $H$ and claim that the two distributions are distinct if for some $h\in H$ the expected value on the two distributions is (significantly) different. Our starting point is the following fundamental problem: "is having the hypothesis dependent on more than a single random example beneficial". To address this challenge we define $k$-ary based discriminators, which have a family of Boolean $k$-ary functions $\mathcal{G}$. Each function $g\in \mathcal{G}$ naturally defines a hyper-graph, indicating whether a given hyper-edge exists. A function $g\in \mathcal{G}$ distinguishes between two distributions, if the expected value of $g$, on a $k$-tuple of i.i.d examples, on the two distributions is (significantly) different. We study the expressiveness of families of $k$-ary functions, compared to the classical hypothesis class $H$, which is $k=1$. We show a separation in expressiveness of $k+1$-ary versus $k$-ary functions. This demonstrate the great benefit of having $k\geq 2$ as distinguishers. For $k\geq 2$ we introduce a notion similar to the VC-dimension, and show that it controls the sample complexity. We proceed and provide upper and lower bounds as a function of our extended notion of VC-dimension.

### Dependency Learning for QBF

Quantified Boolean Formulas (QBFs) can be used to succinctly encode problems from domains such as formal verification, planning, and synthesis. One of the main approaches to QBF solving is Quantified Conflict Driven Clause Learning (QCDCL). By default, QCDCL assigns variables in the order of their appearance in the quantifier prefix so as to account for dependencies among variables. Dependency schemes can be used to relax this restriction and exploit independence among variables in certain cases, but only at the cost of nontrivial interferences with the proof system underlying QCDCL. We introduce dependency learning, a new technique for exploiting variable independence within QCDCL that allows solvers to learn variable dependencies on the fly. The resulting version of QCDCL enjoys improved propagation and increased flexibility in choosing variables for branching while retaining ordinary (long-distance) Q-resolution as its underlying proof system. We show that dependency learning can achieve exponential speedups over ordinary QCDCL. Experiments on standard benchmark sets demonstrate the effectiveness of this technique.

### Regularity Normalization: Constraining Implicit Space with Minimum Description Length

Inspired by the adaptation phenomenon of biological neuronal firing, we propose regularity normalization: a reparameterization of the activation in the neural network that take into account the statistical regularity in the implicit space. By considering the neural network optimization process as a model selection problem, the implicit space is constrained by the normalizing factor, the minimum description length of the optimal universal code. We introduce an incremental version of computing this universal code as normalized maximum likelihood and demonstrated its flexibility to include data prior such as top-down attention and other oracle information and its compatibility to be incorporated into batch normalization and layer normalization. The preliminary results showed that the proposed method outperforms existing normalization methods in tackling the limited and imbalanced data from a non-stationary distribution benchmarked on computer vision tasks. As an unsupervised attention mechanism given input data, this biologically plausible normalization has the potential to deal with other complicated real-world scenarios as well as reinforcement learning setting where the rewards are sparse and non-uniform. Further research is proposed to discover these scenarios and explore the behaviors among different variants.

### Characterization of Glue Variables in CDCL SAT Solving

A state-of-the-art criterion to evaluate the importance of a given learned clause is called Literal Block Distance (LBD) score. It measures the number of distinct decision levels in a given learned clause. The lower the LBD score of a learned clause, the better is its quality. The learned clauses with LBD score of 2, called glue clauses, are known to possess high pruning power which are never deleted from the clause databases of the modern CDCL SAT solvers. In this work, we relate glue clauses to decision variables. We call the variables that appeared in at least one glue clause up to the current search state glue variables. We first show experimentally, by running the state-of-the-art CDCL SAT solver MapleL-CMDist on benchmarks from SAT Competition-2017 and 2018, that branching decisions with glue variables are categorically more inference and conflict efficient than nonglue variables. Based on this observation, we develop a structure aware CDCL variable bumping scheme, which bumps the activity score of a glue variable based on its appearance count in the glue clauses that are learned so far by the search. Empirical evaluation shows effectiveness of the new method over the main track instances from SAT Competition 2017 and 2018.

### Bounds in Query Learning

ABSTRACT.We introduce new combinatorial quantities for concept classes, and prove lower and upper bounds for learning complexity in several models of query learning in terms of various combinatorial quantities. Our approach is flexible and powerful enough to enough to give new and very short proofs of the efficient learnability of several prominent examples (e.g. In the setting of equivalence plus membership queries, we give an algorithm which learns a class in polynomially many queries whenever any such algorithm exists. We also study equivalence query learning in a randomized model, producing new bounds on the expected number of queries required to learn an arbitrary concept. Many of the techniques and notions of dimension draw inspiration from or are related to notions from model theory, and these connections are explained. We also use techniques from query learning to mildly improve a result of Laskowski regarding compression schemes. A concept class C on X is a subset of P(X).In C by means of a series of data requests called equivalence queries.