We study a class of private learning problems in which the data is a join of private and public features. This is often the case in private personalization tasks such as recommendation or ad prediction, in which features related to individuals are sensitive, while features related to items (the movies or songs to be recommended, or the ads to be shown to users) are publicly available and do not require protection. A natural question is whether private algorithms can achieve higher utility in the presence of public features. We give a positive answer for multi-encoder models where one of the encoders operates on public features. We develop new algorithms that take advantage of this separation by only protecting certain sufficient statistics (instead of adding noise to the gradient). This method has a guaranteed utility improvement for linear regression, and importantly, achieves the state of the art on two standard private recommendation benchmarks, demonstrating the importance of methods that adapt to the private-public feature separation.

We study the problem of multi-task learning under user-level differential privacy, in which n users contribute data to m tasks, each involving a subset of users. One important aspect of the problem, that can significantly impact quality, is the distribution skew among tasks. Tasks that have much fewer data samples than others are more susceptible to the noise added for privacy. It is natural to ask whether algorithms can adapt to this skew to improve the overall utility. We give a systematic analysis of the problem, by studying how to optimally allocate a user’s privacy budget among tasks. We propose a generic algorithm, based on an adaptive reweighting of the empirical loss, and show that in the presence of distribution skew, this gives a quantifiable improvement of excess empirical risk. Experimental studies on recommendation problems that exhibit a long tail of small tasks, demonstrate that our methods significantly improve utility, achieving the state of the art on two standard benchmarks.

Data is pooled across entities (individuals or enterprises) to create machine learning models, and sometimes, the entities that contribute the data also benefit from the models. Consider for instance a recommender system (e.g. Spotify, Instagram or YouTube), a health care app that predicts the risk for some disease, or a service built by pooling data across enterprises. In this work we propose a framework to study this value exchange, i.e., we model and measure contributions (outflows), benefits (inflows) and the balance between contributions and benefits (the degree of reciprocity). We show theoretically, and via experiments that under certain distributional assumptions, some classes of models are approximately reciprocal. These results only scratch the surface; we conclude with several open directions.

We consider the problem of training private recommendation models with access to public item features. Training with Differential Privacy (DP) offers strong privacy guarantees, at the expense of loss in recommendation quality. We show that incorporating public item features during training can help mitigate this loss in quality. We propose a general approach based on collective matrix factorization (CMF), that works by simultaneously factorizing two matrices: the user feedback matrix (representing sensitive data) and an item feature matrix that encodes publicly available (non-sensitive) item information.

Matrix factorization learned by implicit alternating least squares (iALS) is a popular baseline in recommender system research publications. iALS is known to be one of the most computationally efficient and scalable collaborative filtering methods. However, recent studies suggest that its prediction quality is not competitive with the current state of the art, in particular autoencoders and other item-based collaborative filtering methods. In this work, we revisit four well-studied benchmarks where iALS was reported to perform poorly and show that with proper tuning, iALS is highly competitive and outperforms any method on at least half of the comparisons. We hope that these high quality results together with iALS’s known scalability spark new interest in applying and further improving this decade old technique.

We study the problem of differentially private (DP) matrix completion under user-level privacy. We design a joint differentially private variant of the popular Alternating-Least-Squares (ALS) method that achieves: i) (nearly) optimal sample complexity for matrix completion (in terms of number of items, users), and ii) the best known privacy/utility trade-off both theoretically, as well as on benchmark data sets. In particular, we provide the first global convergence analysis of ALS with noise introduced to ensure DP, and show that, in comparison to the best known alternative (the Private Frank-Wolfe algorithm by Jain et al. (2018)), our error bounds scale significantly better with respect to the number of items and users, which is critical in practical problems. Extensive validation on standard benchmarks demonstrate that the algorithm, in combination with carefully designed sampling procedures, is significantly more accurate than existing techniques, thus promising to be the first practical DP embedding model.

Recommender systems personalize content by recommending items to users. Item recommendation algorithms are evaluated by metrics that compare the positions of truly relevant items among the recommended items. To speed up the computation of metrics, recent work often uses sampled metrics where only a smaller set of random items and the relevant items are ranked. This paper investigates such sampled metrics and shows that they are inconsistent with their exact counterpart, in the sense that they do not persist relative statements, e.g., recommender A is better than B, not even in expectation. We show that it is possible to improve the quality of the sampled metrics by applying a correction. We conclude with an empirical evaluation of the naive sampled metrics and their corrected variants. Our work suggests that sampling should be avoided for metric calculation, however if an experimental study needs to sample, the proposed corrections can improve the estimates.

The task of item recommendation requires ranking a large catalogue of items given a context. Item recommendation algorithms are evaluated using ranking metrics that depend on the positions of relevant items. To speed up the computation of metrics, recent work often uses sampled metrics where only a smaller set of random items and the relevant items are ranked. This paper investigates sampled metrics in more detail and shows that they are inconsistent with their exact version, in the sense that they do not persist relative statements, e.g., recommender A is better than B, not even in expectation. Moreover, the smaller the sampling size, the less difference there is between metrics, and for very small sampling size, all metrics collapse to the AUC metric. We show that it is possible to improve the quality of the sampled metrics by applying a correction, obtained by minimizing different criteria such as bias or mean squared error. We conclude with an empirical evaluation of the naive sampled metrics and their corrected variants. To summarize, our work suggests that sampling should be avoided for metric calculation, however if an experimental study needs to sample, the proposed corrections can improve the quality of the estimate.

Several machine learning models involve mapping a score vector to a probability vector. Usually, this is done by projecting the score vector onto a probability simplex, and such projections are often characterized as Lipschitz continuous approximations of the argmax function, whose Lipschitz constant is controlled by a parameter that is similar to a softmax temperature. The aforementioned parameter has been observed to affect the quality of these models and is typically either treated as a constant or decayed over time. In this work, we propose a method that adapts this parameter to individual training examples. The resulting method exhibits desirable properties, such as sparsity of its support and numerically efficient implementation, and we find that it significantly outperforms competing non-adaptive projection methods. In our analysis, we also derive the general solution of (Bregman) projections onto the (n, k)-simplex, a result which may be of independent interest.

Embedding based models have been the state of the art in collaborative filtering for over a decade. Traditionally, the dot product or higher order equivalents have been used to combine two or more embeddings, e.g., most notably in matrix factorization. In recent years, it was suggested to replace the dot product with a learned similarity e.g. using a multilayer perceptron (MLP). This approach is often referred to as neural collaborative filtering (NCF). In this work, we revisit the experiments of the NCF paper that popularized learned similarities using MLPs. First, we show that with a proper hyperparameter selection, a simple dot product substantially outperforms the proposed learned similarities. Second, while a MLP can in theory approximate any function, we show that it is non-trivial to learn a dot product with an MLP. Finally, we discuss practical issues that arise when applying MLP based similarities and show that MLPs are too costly to use for item recommendation in production environments while dot products allow to apply very efficient retrieval algorithms. We conclude that MLPs should be used with care as embedding combiner and that dot products might be a better default choice.

The routing game models congestion in transportation networks, communication networks, and other cyber-physical systems in which agents compete for shared resources. We consider an online learning model of player dynamics: at each iteration, every player chooses a route (or a probability distribution over routes, which corresponds to a flow allocation over the physical network), then the joint decision of all players determines the costs of each path, which are then revealed to the players. We pose the following estimation problem: given a sequence of player decisions and the corresponding costs, we would like to estimate the parameters of the learning model. We consider, in particular, entropic mirror descent dynamics and reduce the problem to estimating the learning rates of each player. In order to demonstrate our methods, we developed a web application that allows players to participate in a distributed, online routing game, and we deployed the application on Amazon Mechanical Turk. When players log in, they are assigned an origin and destination on a shared network. They can choose, at each iteration, a distribution over their available routes, and each player seeks to minimize her own cost. We collect a dataset using this platform, then apply the proposed method to estimate the learning rates of each player. We observe, in particular, that after an exploration phase, the joint decision of the players remains within a small distance of the set of equilibria. We also use the estimated model parameters to predict the flow distribution over routes, and compare our predictions to the actual distributions, showing that the online learning model can be used as a predictive model over short horizons. Finally, we discuss some of the qualitative insights from the experiments, and give directions for future research.

We formulate and study a general family of (continuous-time) stochastic dynamics for accelerated first-order minimization of smooth convex functions. Building on an averaging formulation of accelerated mirror descent, we propose a stochastic variant in which the gradient is contaminated by noise, and study the resulting stochastic differential equation. We prove a bound on the rate of change of an energy function associated with the problem, then use it to derive estimates of convergence rates of the function values (almost surely and in expectation), both for persistent and asymptotically vanishing noise. We discuss the interaction between the parameters of the dynamics (learning rate and averaging rates) and the covariation of the noise process. In particular, we show how the asymptotic rate of covariation affects the choice of parameters and, ultimately, the convergence rate.

We study a general adversarial online learning problem, in which we are given a decision set X' in a reflexive Banach space X and a sequence of reward vectors in the dual space of X. At each iteration, we choose an action from X', based on the observed sequence of previous rewards. Our goal is to minimize regret, defined as the gap between the realized reward and the reward of the best fixed action in hindsight. Using results from infinite dimensional convex analysis, we generalize the method of Dual Averaging (or Follow the Regularized Leader) to our setting and obtain upper bounds on the worst-case regret that generalize many previous results. Under the assumption of uniformly continuous rewards, we obtain explicit regret bounds in a setting where the decision set is the set of probability distributions on a compact metric space S. Importantly, we make no convexity assumptions on either the set S or the reward functions. We also prove a general lower bound on the worst-case regret for any online algorithm. We then apply these results to the problem of learning in repeated two-player zero-sum games on compact metric spaces. In doing so, we first prove that if both players play a Hannan-consistent strategy, then with probability 1 the empirical distributions of play weakly converge to the set of Nash equilibria of the game. We then show that, under mild assumptions, Dual Averaging on the (infinite-dimensional) space of probability distributions indeed achieves Hannan-consistency.

We study accelerated descent dynamics for constrained convex optimization. This dynamics can be described naturally as a coupling of a dual variable accumulating gradients at a given rate $\eta(t)$, and a primal variable obtained as the weighted average of the mirrored dual trajectory, with weights $w(t)$. Using a Lyapunov argument, we give sufficient conditions on $\eta$ and $w$ to achieve a desired convergence rate. As an example, we show that the replicator dynamics (an example of mirror descent on the simplex) can be accelerated using a simple averaging scheme. We then propose an adaptive averaging heuristic which adaptively computes the weights to speed up the decrease of the Lyapunov function. We provide guarantees on adaptive averaging in continuous-time, prove that it preserves the quadratic convergence rate of accelerated first-order methods in discrete-time, and give numerical experiments to compare it with existing heuristics, such as adaptive restarting. The experiments indicate that adaptive averaging performs at least as well as adaptive restarting, with significant improvements in some cases.

We consider an online optimization problem on a subset S of R^n (not necessarily convex), in which a decision maker chooses, at each iteration t, a probability distribution x^(t) over S, and seeks to minimize a cumulative expected loss, where each loss is a Lipschitz function revealed at the end of iteration t. Building on previous work, we propose a generalized Hedge algorithm and show a O(\sqrtt \log t) bound on the regret when the losses are uniformly Lipschitz and S is uniformly fat (a weaker condition than convexity). Finally, we propose a generalization to the dual averaging method on the set of Lebesgue-continuous distributions over S.

We study accelerated mirror descent dynamics in continuous and discrete time. Combining the original continuous-time motivation of mirror descent with a recent ODE interpretation of Nesterov's accelerated method, we propose a family of continuous-time descent dynamics for convex functions with Lipschitz gradients, such that the solution trajectories are guaranteed to converge to the optimum at a $O(1/t^2)$ rate. We then show that a large family of first-order accelerated methods can be obtained as a discretization of the ODE, and these methods converge at a $O(1/k^2)$ rate. This connection between accelerated mirror descent and the ODE provides an intuitive approach to the design and analysis of accelerated first-order algorithms.

We study the repeated, non-atomic routing game, in which selfish players make a sequence of routing decisions. We consider a model in which players use regret-minimizing algorithms as the learning mechanism, and study the resulting dynamics. We are concerned in particular with the convergence to the set of Nash equilibria of the routing game. No-regret learning algorithms are known to guarantee convergence of a subsequence of population strategies. We are concerned with convergence of the actual sequence. We show that convergence holds for a large class of online learning algorithms, inspired from the continuous-time replicator dynamics. In particular, the discounted Hedge algorithm is proved to belong to this class, which guarantees its convergence.