This monograph presents the main complexity theorems in convex optimization and their corresponding algorithms. Starting from the fundamental theory of black-box optimization, the material progresses towards recent advances in structural optimization and stochastic optimization. Our presentation of black-box optimization, strongly influenced by Nesterov's seminal book and Nemirovski's lecture notes, includes the analysis of cutting plane methods, as well as (accelerated) gradient descent schemes. We also pay special attention to non-Euclidean settings (relevant algorithms include Frank-Wolfe, mirror descent, and dual averaging) and discuss their relevance in machine learning. We provide a gentle introduction to structural optimization with FISTA (to optimize a sum of a smooth and a simple non-smooth term), saddle-point mirror prox (Nemirovski's alternative to Nesterov's smoothing), and a concise description of interior point methods. In stochastic optimization we discuss stochastic gradient descent, mini-batches, random coordinate descent, and sublinear algorithms. We also briefly touch upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods.

Modern applications and progress in deep learning research have created renewed interest for generative models of text and of images. However, even today it is unclear what objective functions one should use to train and evaluate these models. In this paper we present two contributions. Firstly, we present a critique of scheduled sampling, a state-of-the-art training method that contributed to the winning entry to the MSCOCO image captioning benchmark in 2015. Here we show that despite this impressive empirical performance, the objective function underlying scheduled sampling is improper and leads to an inconsistent learning algorithm. Secondly, we revisit the problems that scheduled sampling was meant to address, and present an alternative interpretation. We argue that maximum likelihood is an inappropriate training objective when the end-goal is to generate natural-looking samples. We go on to derive an ideal objective function to use in this situation instead. We introduce a generalisation of adversarial training, and show how such method can interpolate between maximum likelihood training and our ideal training objective. To our knowledge this is the first theoretical analysis that explains why adversarial training tends to produce samples with higher perceived quality.

Numerous applications in data mining and machine learning require recovering a matrix of minimal rank. Robust principal component analysis (RPCA) is a general framework for handling this kind of problems. Nuclear norm based convex surrogate of the rank function in RPCA is widely investigated. Under certain assumptions, it can recover the underlying true low rank matrix with high probability. However, those assumptions may not hold in real-world applications. Since the nuclear norm approximates the rank by adding all singular values together, which is essentially a $\ell_1$-norm of the singular values, the resulting approximation error is not trivial and thus the resulting matrix estimator can be significantly biased. To seek a closer approximation and to alleviate the above-mentioned limitations of the nuclear norm, we propose a nonconvex rank approximation. This approximation to the matrix rank is tighter than the nuclear norm. To solve the associated nonconvex minimization problem, we develop an efficient augmented Lagrange multiplier based optimization algorithm. Experimental results demonstrate that our method outperforms current state-of-the-art algorithms in both accuracy and efficiency.

Learning accurate probabilistic models from data is crucial in many practical tasks in data mining. In this paper we present a new non-parametric calibration method called \textit{ensemble of near isotonic regression} (ENIR). The method can be considered as an extension of BBQ, a recently proposed calibration method, as well as the commonly used calibration method based on isotonic regression. ENIR is designed to address the key limitation of isotonic regression which is the monotonicity assumption of the predictions. Similar to BBQ, the method post-processes the output of a binary classifier to obtain calibrated probabilities. Thus it can be combined with many existing classification models. We demonstrate the performance of ENIR on synthetic and real datasets for the commonly used binary classification models. Experimental results show that the method outperforms several common binary classifier calibration methods. In particular on the real data, ENIR commonly performs statistically significantly better than the other methods, and never worse. It is able to improve the calibration power of classifiers, while retaining their discrimination power. The method is also computationally tractable for large scale datasets, as it is $O(N \log N)$ time, where $N$ is the number of samples.

We propose mS2GD: a method incorporating a mini-batching scheme for improving the theoretical complexity and practical performance of semi-stochastic gradient descent (S2GD). We consider the problem of minimizing a strongly convex function represented as the sum of an average of a large number of smooth convex functions, and a simple nonsmooth convex regularizer. Our method first performs a deterministic step (computation of the gradient of the objective function at the starting point), followed by a large number of stochastic steps. The process is repeated a few times with the last iterate becoming the new starting point. The novelty of our method is in introduction of mini-batching into the computation of stochastic steps. In each step, instead of choosing a single function, we sample $b$ functions, compute their gradients, and compute the direction based on this. We analyze the complexity of the method and show that it benefits from two speedup effects. First, we prove that as long as $b$ is below a certain threshold, we can reach any predefined accuracy with less overall work than without mini-batching. Second, our mini-batching scheme admits a simple parallel implementation, and hence is suitable for further acceleration by parallelization.

Sequential Monte Carlo (SMC), or particle filtering, is a popular class of methods for sampling from an intractable target distribution using a sequence of simpler intermediate distributions. Like other importance sampling-based methods, performance is critically dependent on the proposal distribution: a bad proposal can lead to arbitrarily inaccurate estimates of the target distribution. This paper presents a new method for automatically adapting the proposal using an approximation of the Kullback-Leibler divergence between the true posterior and the proposal distribution. The method is very flexible, applicable to any parameterized proposal distribution and it supports online and batch variants. We use the new framework to adapt powerful proposal distributions with rich parameterizations based upon neural networks leading to Neural Adaptive Sequential Monte Carlo (NASMC). Experiments indicate that NASMC significantly improves inference in a non-linear state space model outperforming adaptive proposal methods including the Extended Kalman and Unscented Particle Filters. Experiments also indicate that improved inference translates into improved parameter learning when NASMC is used as a subroutine of Particle Marginal Metropolis Hastings. Finally we show that NASMC is able to train a latent variable recurrent neural network (LV-RNN) achieving results that compete with the state-of-the-art for polymorphic music modelling. NASMC can be seen as bridging the gap between adaptive SMC methods and the recent work in scalable, black-box variational inference.

We introduce normalized nonnegative models (NNM) for explorative data analysis. NNMs are partial convexifications of models from probability theory. We demonstrate their value at the example of item recommendation. We show that NNM-based recommender systems satisfy three criteria that all recommender systems should ideally satisfy: high predictive power, computational tractability, and expressive representations of users and items. Expressive user and item representations are important in practice to succinctly summarize the pool of customers and the pool of items. In NNMs, user representations are expressive because each user's preference can be regarded as normalized mixture of preferences of stereotypical users. The interpretability of item and user representations allow us to arrange properties of items (e.g., genres of movies or topics of documents) or users (e.g., personality traits) hierarchically.

Causal terminology is often introduced in the interpretation of encoding and decoding models trained on neuroimaging data. In this article, we investigate which causal statements are warranted and which ones are not supported by empirical evidence. We argue that the distinction between encoding and decoding models is not sufficient for this purpose: relevant features in encoding and decoding models carry a different meaning in stimulus- and in response-based experimental paradigms. We show that only encoding models in the stimulus-based setting support unambiguous causal interpretations. By combining encoding and decoding models trained on the same data, however, we obtain insights into causal relations beyond those that are implied by each individual model type. We illustrate the empirical relevance of our theoretical findings on EEG data recorded during a visuo-motor learning task.

We present a convex approach to probabilistic segmentation and modeling of time series data. Our approach builds upon recent advances in multivariate total variation regularization, and seeks to learn a separate set of parameters for the distribution over the observations at each time point, but with an additional penalty that encourages the parameters to remain constant over time. We propose efficient optimization methods for solving the resulting (large) optimization problems, and a two-stage procedure for estimating recurring clusters under such models, based upon kernel density estimation. Finally, we show on a number of real-world segmentation tasks, the resulting methods often perform as well or better than existing latent variable models, while being substantially easier to train.

Emotional content is a key element in user-generated videos. However, it is difficult to understand emotions conveyed in such videos due to the complex and unstructured nature of user-generated content and the sparsity of video frames that express emotion. In this paper, for the first time, we study the problem of transferring knowledge from heterogeneous external sources, including image and textual data, to facilitate three related tasks in video emotion understanding: emotion recognition, emotion attribution and emotion-oriented summarization. Specifically, our framework (1) learns a video encoding from an auxiliary emotional image dataset in order to improve supervised video emotion recognition, and (2) transfers knowledge from an auxiliary textual corpus for zero-shot \pl{recognition} of emotion classes unseen during training. The proposed technique for knowledge transfer facilitates novel applications of emotion attribution and emotion-oriented summarization. A comprehensive set of experiments on multiple datasets demonstrate the effectiveness of our framework.