Creating impact in real-world settings requires artificial intelligence techniques to span the full pipeline from data, to predictive models, to decisions. These components are typically approached separately: a machine learning model is first trained via a measure of predictive accuracy, and then its predictions are used as input into an optimization algorithm which produces a decision. However, the loss function used to train the model may easily be misaligned with the end goal, which is to make the best decisions possible. Hand-tuning the loss function to align with optimization is a difficult and error-prone process (which is often skipped entirely). We focus on combinatorial optimization problems and introduce a general framework for decision-focused learning, where the machine learning model is directly trained in conjunction with the optimization algorithm to produce high-quality decisions. Technically, our contribution is a means of integrating discrete optimization problems into deep learning or other predictive models, which are typically trained via gradient descent. The main idea is to use a continuous relaxation of the discrete problem to propagate gradients through the optimization procedure. We instantiate this framework for two broad classes of combinatorial problems: linear programs and submodular maximization. Experimental results across a variety of domains show that decision-focused learning often leads to improved optimization performance compared to traditional methods. We find that standard measures of accuracy are not a reliable proxy for a predictive model's utility in optimization, and our method's ability to specify the true goal as the model's training objective yields substantial dividends across a range of decision problems.

In this paper, we analyze discrete gradient descent and $\epsilon$-noisy gradient descent on a special but important class of functions. We find that when used to minimize a function $L: \mathbb{R}^n \rightarrow \mathbb{R}$ in this class we consider, discrete gradient descent can exhibit strikingly different behavior from continuous gradient descent. On long time scales, discrete gradient descent and continuous gradient descent tend toward different global minima of $L$. Discrete gradient descent preferentially finds global minima at which the graph of the function $L$ is shallowest, while gradient flow shows no such preference.

Variational Optimization forms a differentiable upper bound on an objective. We show that approaches such as Natural Evolution Strategies and Gaussian Perturbation, are special cases of Variational Optimization in which the expectations are approximated by Gaussian sampling. These approaches are of particular interest because they are parallelizable. We calculate the approximate bias and variance of the corresponding gradient estimators and demonstrate that using antithetic sampling or a baseline is crucial to mitigate their problems. We contrast these methods with an alternative parallelizable method, namely Directional Derivatives. We conclude that, for differentiable objectives, using Directional Derivatives is preferable to using Variational Optimization to perform parallel Stochastic Gradient Descent.

We explore a new research direction in Bayesian variational inference with discrete latent variable priors where we exploit Kronecker matrix algebra for efficient and exact computations of the evidence lower bound (ELBO). The proposed "DIRECT" approach has several advantages over its predecessors; (i) it can exactly compute ELBO gradients (i.e. unbiased, zero-variance gradient estimates), eliminating the need for high-variance stochastic gradient estimators and enabling the use of quasi-Newton optimization methods; (ii) its training complexity is independent of the number of training points, permitting inference on large datasets; and (iii) its posterior samples consist of sparse and low-precision quantized integers which permit fast inference on hardware limited devices. In addition, our DIRECT models can exactly compute statistical moments of the parameterized predictive posterior without relying on Monte Carlo sampling. Our numerical studies demonstrate accurate inference using latent variables discretized as extremely low-precision 4-bit quantized integers. While the ELBO computations considered require over $10^{2352}$ log-likelihood evaluations, we train on datasets with over two-million points in just seconds.

While machine learning has achieved remarkable results in a wide variety of domains, the training of models often requires large datasets that may need to be collected from different individuals. As sensitive information may be contained in the individual's dataset, sharing training data may lead to severe privacy concerns. Therefore, there is a compelling need to develop privacy-aware machine learning methods, for which one effective approach is to leverage the generic framework of differential privacy. Considering that stochastic gradient descent (SGD) is one of the mostly adopted methods for large-scale machine learning problems, two decentralized differentially private SGD algorithms are proposed in this work. Particularly, we focus on SGD without replacement due to its favorable structure for practical implementation. In addition, both privacy and convergence analysis are provided for the proposed algorithms. Finally, extensive experiments are performed to verify the theoretical results and demonstrate the effectiveness of the proposed algorithms.

While momentum-based methods, in conjunction with the stochastic gradient descent, are widely used when training machine learning models, there is little theoretical understanding on the generalization error of such methods. In practice, the momentum parameter is often chosen in a heuristic fashion with little theoretical guidance. In the first part of this paper, for the case of general loss functions, we analyze a modified momentum-based update rule, i.e., the method of early momentum, and develop an upper-bound on the generalization error using the framework of algorithmic stability. Our results show that machine learning models can be trained for multiple epochs of this method while their generalization errors are bounded. We also study the convergence of the method of early momentum by establishing an upper-bound on the expected norm of the gradient. In the second part of the paper, we focus on the case of strongly convex loss functions and the classical heavy-ball momentum update rule. We use the framework of algorithmic stability to provide an upper-bound on the generalization error of the stochastic gradient method with momentum. We also develop an upper-bound on the expected true risk, in terms of the number of training steps, the size of the training set, and the momentum parameter. Experimental evaluations verify the consistency between the numerical results and our theoretical bounds and the effectiveness of the method of early momentum for the case of non-convex loss functions.

Differentially private learning has recently emerged as the leading approach for privacy-preserving machine learning. Differential privacy can complicate learning procedures because each access to the data needs to be carefully designed and carries a privacy cost. For example, standard parameter tuning with a validation set cannot be easily applied. In this paper, we propose a differentially private algorithm for the adaptation of the learning rate for differentially private stochastic gradient descent (SGD) that avoids the need for validation set use. The idea for the adaptiveness comes from the technique of extrapolation in classical numerical analysis: to get an estimate for the error against the gradient flow which underlies SGD, we compare the result obtained by one full step and two half-steps. We prove the privacy of the method using the moments accountant mechanism. This allows us to compute tight privacy bounds. Empirically we show that our method is competitive with manually tuned commonly used optimisation methods for training deep neural networks and differentially private variational inference.

Stochastic gradient methods are the workhorse (algorithms) of large-scale optimization problems in machine learning, signal processing, and other computational sciences and engineering. This paper studies Markov chain gradient descent, a variant of stochastic gradient descent where the random samples are taken on the trajectory of a Markov chain. Existing results of this method assume convex objectives and a reversible Markov chain and thus have their limitations. We establish new non-ergodic convergence under wider step sizes, for nonconvex problems, and for non-reversible finite-state Markov chains. Nonconvexity makes our method applicable to broader problem classes. Non-reversible finite-state Markov chains, on the other hand, can mix substatially faster. To obtain these results, we introduce a new technique that varies the mixing levels of the Markov chains. The reported numerical results validate our contributions.

We show that many machine learning goals, such as improved fairness metrics, can be expressed as constraints on the model's predictions, which we call rate constraints. We study the problem of training non-convex models subject to these rate constraints (or any non-convex and non-differentiable constraints). In the non-convex setting, the standard approach of Lagrange multipliers may fail. Furthermore, if the constraints are non-differentiable, then one cannot optimize the Lagrangian with gradient-based methods. To solve these issues, we introduce the proxy-Lagrangian formulation. This new formulation leads to an algorithm that produces a stochastic classifier by playing a two-player non-zero-sum game solving for what we call a semi-coarse correlated equilibrium, which in turn corresponds to an approximately optimal and feasible solution to the constrained optimization problem. We then give a procedure which shrinks the randomized solution down to one that is a mixture of at most $m+1$ deterministic solutions, given $m$ constraints. This culminates in algorithms that can solve non-convex constrained optimization problems with possibly non-differentiable and non-convex constraints with theoretical guarantees. We provide extensive experimental results enforcing a wide range of policy goals including different fairness metrics, and other goals on accuracy, coverage, recall, and churn.

Finding tight bounds on the optimal solution is a critical element of practical solution methods for discrete optimization problems. In the last decade, decision diagrams (DDs) have brought a new perspective on obtaining upper and lower bounds that can be significantly better than classical bounding mechanisms, such as linear relaxations. It is well known that the quality of the bound achieved through this flexible bounding method is highly reliant on the ordering of variables chosen for building the diagram, and finding an ordering that optimizes standard metrics, or even improving one, is an NP-hard problem. In this paper, we propose an innovative and generic approach based on deep reinforcement learning for obtaining an ordering for tightening the bounds obtained with relaxed and restricted DDs. We apply the approach to both the Maximum Independent Set Problem and the Maximum Cut Problem. Experimental results on synthetic instances show that the deep reinforcement learning approach, by achieving tighter objective function bounds, generally outperforms ordering methods commonly used in the literature when the distribution of instances is known. To the best knowledge of the authors, this is the first paper to apply machine learning to directly improve relaxation bounds obtained by general-purpose bounding mechanisms for combinatorial optimization problems.