Knapsack problems (KPs) are common in industry, but solving KPs is known to be NP-hard and has been tractable only at a relatively small scale. This paper examines KPs in a slightly generalized form and shows that they can be solved nearly optimally at scale via distributed algorithms. The proposed approach can be implemented fairly easily with off-the-shelf distributed computing frameworks (e.g. MPI, Hadoop, Spark). As an example, our implementation leads to one of the most efficient KP solvers known to date -- capable to solve KPs at an unprecedented scale (e.g., KPs with 1 billion decision variables and 1 billion constraints can be solved within 1 hour). The system has been deployed to production and called on a daily basis, yielding significant business impacts at Ant Financial.

We consider boosting algorithms under the restriction that the weak learners come from a class of bounded VC-dimension. In this setting, we focus on two main questions: (i) \underline{Oracle Complexity:} we show that the restriction on the complexity of the weak learner significantly improves the number of calls to the weak learner. We describe a boosting procedure which makes only~$\tilde O(1/\gamma)$ calls to the weak learner, where $\gamma$ denotes the weak learner's advantage. This circumvents a lower bound of $\Omega(1/\gamma^2)$ due to Freund and Schapire ('95, '12) for the general case. Unlike previous boosting algorithms which aggregate the weak hypotheses by majority votes, our method use more complex aggregation rules, and we show this to be necessary. (ii) \underline{Expressivity:} we consider the question of what can be learned by boosting weak hypotheses of bounded VC-dimension? Towards this end we identify a combinatorial-geometric parameter called the $\gamma$-VC dimension which quantifies the expressivity of a class of weak hypotheses when used as part of a boosting procedure. We explore the limits of the $\gamma$-VC dimension and compute it for well-studied classes such as halfspaces and decision stumps. Along the way, we establish and exploit connections with {\it Discrepancy theory}.

Email: kharen@dualitygroup.com January 30, 2020 Abstract We introduce a notion of "effective dimension" of a statistical model based on the number of cubes of size 1 / n needed to cover the model space when endowed with the Fisher Information Matrix as metric, n being the number of observations. The effective dimension is then measured via the spectrum of the Fisher Information Matrix regularized using this natural scale. A very important and challenging question in statistics and machine learning is the "real" dimension of a statistical model, such as a neural network. Many definitions of effective dimension have been proposed in the literature, either based on the so-called VC dimension (see for instance [13]), or on Gardner phase-space approach [6], or also on some effective dimension based on the rank of the Jacobian matrix of the transformation between the parameters of the network and the parameters of the observable variables [2, 15] (see also [14, 1, 4, 7]). Although these notions of dimension are all very natural when the number of observations go to infinity, they do not take into account the fact that only a finite-size sample of data is available.

The agnostic PAC learning model consists of: a Hypothesis Space $\mathcal{H}$, a probability distribution $P$, a sample complexity function $m_{\mathcal{H}}(\epsilon,\delta): [0,1]^{2} \mapsto \mathbb{Z}_{+}$ of precision $\epsilon$ and confidence $1 - \delta$, a finite i.i.d. sample $\mathcal{D}_{N}$, a cost function $\ell$ and a learning algorithm $\mathbb{A}(\mathcal{H},\mathcal{D}_{N})$, which estimates $\hat{h} \in \mathcal{H}$ that approximates a target function $h^{\star} \in \mathcal{H}$ seeking to minimize out-of-sample error. In this model, prior information is represented by $\mathcal{H}$ and $\ell$, while problem solution is performed through their instantiation in several applied learning models, with specific algebraic structures for $\mathcal{H}$ and corresponding learning algorithms. However, these applied models use additional important concepts not covered by the classic PAC learning theory: model selection and regularization. This paper presents an extension of this model which covers these concepts. The main principle added is the selection, based solely on data, of a subspace of $\mathcal{H}$ with a VC-dimension compatible with the available sample. In order to formalize this principle, the concept of Learning Space $\mathbb{L}(\mathcal{H})$, which is a poset of subsets of $\mathcal{H}$ that covers $\mathcal{H}$ and satisfies a property regarding the VC dimension of related subspaces, is presented as the natural search space for model selection algorithms. A remarkable result obtained on this new framework are conditions on $\mathbb{L}(\mathcal{H})$ and $\ell$ that lead to estimated out-of-sample error surfaces, which are true U-curves on $\mathbb{L}(\mathcal{H})$ chains, enabling a more efficient search on $\mathbb{L}(\mathcal{H})$. Hence, in this new framework, the U-curve optimization problem becomes a natural component of model selection algorithms.

How can we ensure that a machine learning system produces an o utput that generalizes to the underlying distribution, rather than overfitting its train ing data? That is, how can we ensure that the hypotheses or models that are produced are reflective of t he underlying population the training data was drawn from, rather than patterns that occur only by c hance in the training data? This is perhaps the fundamental question for the science of statist ical machine learning. A vast array of methods have been proposed to answer this ques tion. Most notably, the theory of uniform convergence shows that, if the output is sufficiently "simple," then it cannot overfit too much. A more recent line of work has used distributional stability (in the form of differential privacy) to provide generalization guarantees that compose adaptivel y - that is, statistical validity is preserved even when a dataset is reused multiple times with each succes sive analysis being influenced by the outcomes of prior analyses. Other methods for proving gener alization include compression schemes and uniform stability. Unfortunately, these different methods for providing gener alization guarantees are largely disconnected from one another; it is, in general, not possible t o compare or combine techniques. In this paper, we provide a framework to reason about many of the se these differing approaches using the unifying language of information theory.

How can we enable machines to make sense of the world, and become better at learning? To approach this goal, I believe viewing intelligence in terms of many integral aspects, and also a universal two-term tradeoff between task performance and complexity, provides two feasible perspectives. In this thesis, I address several key questions in some aspects of intelligence, and study the phase transitions in the two-term tradeoff, using strategies and tools from physics and information. Firstly, how can we make the learning models more flexible and efficient, so that agents can learn quickly with fewer examples? Inspired by how physicists model the world, we introduce a paradigm and an AI Physicist agent for simultaneously learning many small specialized models (theories) and the domain they are accurate, which can then be simplified, unified and stored, facilitating few-shot learning in a continual way. Secondly, for representation learning, when can we learn a good representation, and how does learning depend on the structure of the dataset? We approach this question by studying phase transitions when tuning the tradeoff hyperparameter. In the information bottleneck, we theoretically show that these phase transitions are predictable and reveal structure in the relationships between the data, the model, the learned representation and the loss landscape. Thirdly, how can agents discover causality from observations? We address part of this question by introducing an algorithm that combines prediction and minimizing information from the input, for exploratory causal discovery from observational time series. Fourthly, to make models more robust to label noise, we introduce Rank Pruning, a robust algorithm for classification with noisy labels. I believe that building on the work of my thesis we will be one step closer to enable more intelligent machines that can make sense of the world.

We introduce the reinforcement quantum annealing (RQA) scheme in which an intelligent agent interacts with a quantum annealer that plays the stochastic environment role of learning automata and tries to iteratively find better Ising Hamiltonians for the given problem of interest. As a proof-of-concept, we propose a novel approach for reducing the NP-complete problem of Boolean satisfiability (SAT) to minimizing Ising Hamiltonians and show how to apply the RQA for increasing the probability of finding the global optimum. Our experimental results on two different benchmark SAT problems (namely factoring pseudo-prime numbers and random SAT with phase transitions), using a D-Wave 2000Q quantum processor, demonstrated that RQA finds notably better solutions with fewer samples, compared to state-of-the-art techniques in the realm of quantum annealing.

We consider learning problems where the training set consists of two types of examples: private and public. The goal is to design a learning algorithm that satisfies differential privacy only with respect to the private examples. This setting interpolates between private learning (where all examples are private) and classical learning (where all examples are public). We study the limits of learning in this setting in terms of private and public sample complexities. We show that any hypothesis class of VC-dimension $d$ can be agnostically learned up to an excess error of $\alpha$ using only (roughly) $d/\alpha$ public examples and $d/\alpha^2$ private labeled examples. This result holds even when the public examples are unlabeled. This gives a quadratic improvement over the standard $d/\alpha^2$ upper bound on the public sample complexity (where private examples can be ignored altogether if the public examples are labeled). Furthermore, we give a nearly matching lower bound, which we prove via a generic reduction from this setting to the one of private learning without public data.

Data scientists naturally use a lot of machine learning algorithms, which work well for detecting patterns, automating simple tasks, generalizing responses and other data heavy tasks. As a subfield of computer science, machine learning evolved from the study of pattern recognition and computational learning theory in artificial intelligence. Over time, machine learning has borrowed from many other fields, including statistics. Most of today's algorithms have a history in various mathematical subfields. Many of these subfields overlap but I've taken a stab at categorizing some popular algorithms.

Relational properties, e.g., the connectivity structure of nodes in a distributed system, have many applications in software design and analysis. However, such properties often have to be written manually, which can be costly and error-prone. This paper introduces the MCML approach for empirically studying the learnability of a key class of such properties that can be expressed in the well-known software design language Alloy. A key novelty of MCML is quantification of the performance of and semantic differences among trained machine learning (ML) models, specifically decision trees, with respect to entire input spaces (up to a bound on the input size), and not just for given training and test datasets (as is the common practice). MCML reduces the quantification problems to the classic complexity theory problem of model counting, and employs state-of-the-art approximate and exact model counters for high efficiency. The results show that relatively simple ML models can achieve surprisingly high performance (accuracy and F1 score) at learning relational properties when evaluated in the common setting of using training and test datasets -- even when the training dataset is much smaller than the test dataset -- indicating the seeming simplicity of learning these properties. However, the use of MCML metrics based on model counting shows that the performance can degrade substantially when tested against the whole (bounded) input space, indicating the high complexity of precisely learning these properties, and the usefulness of model counting in quantifying the true accuracy.