Recommendation models mainly deal with categorical variables, such as user item ID and attributes. Besides the high-cardinality issue, the interactions among such categorical variables are usually long-tailed, with the head made up of highly frequent values and a long tail of rare ones. This phenomenon results in the data sparsity issue, making it essential to regularize the models to ensure generalization. The common practice is to employ grid search to manually tune regularization hyperparameters based on the validation data. However, it requires non-trivial efforts and large computation resources to search the whole candidate space; even so, it may not lead to the optimal choice, for which different parameters should have different regularization strengths. In this paper, we propose a hyperparameter optimization method, LambdaOpt, which automatically and adaptively enforces regularization during training. Specifically, it updates the regularization coefficients based on the performance of validation data. With LambdaOpt, the notorious tuning of regularization hyperparameters can be avoided; more importantly, it allows fine-grained regularization (i.e. each parameter can have an individualized regularization coefficient), leading to better generalized models. We show how to employ LambdaOpt on matrix factorization, a classical model that is representative of a large family of recommender models. Extensive experiments on two public benchmarks demonstrate the superiority of our method in boosting the performance of top-K recommendation.
* . The close work is SGDA @cite_31 , where adaptive regularization for rating prediction is achieved by alternating optimization for model parameters @math and regularization coefficients @math . Similar validation set based alternating optimization method has also been proposed in @cite_4 . Both work focused on the reduced computation complexity while ignoring the potential performance boost. @cite_31 only considers dimension-wise @math , which might be the reason why the algorithm does not outperform the best fixed @math algorithm in the reported experimental results. Instead, our work shows the effectiveness of incorporating fine-grained regularization. Besides, our method is more generic in terms of endowing the practitioners more freedom in optimizer choices while SGDA @cite_31 applies only to SGD optimizers.
@cite_31 @cite_4
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Despite superior training outcomes, adaptive optimization methods such as Adam, Adagrad or RMSprop have been found to generalize poorly compared to Stochastic gradient descent (SGD). These methods tend to perform well in the initial portion of training but are outperformed by SGD at later stages of training. We investigate a hybrid strategy that begins training with an adaptive method and switches to SGD when appropriate. Concretely, we propose SWATS, a simple strategy which switches from Adam to SGD when a triggering condition is satisfied. The condition we propose relates to the projection of Adam steps on the gradient subspace. By design, the monitoring process for this condition adds very little overhead and does not increase the number of hyperparameters in the optimizer. We report experiments on several standard benchmarks such as: ResNet, SENet, DenseNet and PyramidNet for the CIFAR-10 and CIFAR-100 data sets, ResNet on the tiny-ImageNet data set and language modeling with recurrent networks on the PTB and WT2 data sets. The results show that our strategy is capable of closing the generalization gap between SGD and Adam on a majority of the tasks. Matrix factorization, when the matrix has missing values, has become one of the leading techniques for recommender systems. To handle web-scale datasets with millions of users and billions of ratings, scalability becomes an important issue. Alternating least squares (ALS) and stochastic gradient descent (SGD) are two popular approaches to compute matrix factorization, and there has been a recent flurry of activity to parallelize these algorithms. However, due to the cubic time complexity in the target rank, ALS is not scalable to large-scale datasets. On the other hand, SGD conducts efficient updates but usually suffers from slow convergence that is sensitive to the parameters. Coordinate descent, a classical optimization approach, has been used for many other large-scale problems, but its application to matrix factorization for recommender systems has not been thoroughly explored. In this paper, we show that coordinate descent-based methods have a more efficient update rule compared to ALS and have faster and more stable convergence than SGD. We study different update sequences and propose the CCD++ algorithm, which updates rank-one factors one by one. In addition, CCD++ can be easily parallelized on both multi-core and distributed systems. We empirically show that CCD++ is much faster than ALS and SGD in both settings. As an example, with a synthetic dataset containing 14.6 billion ratings, on a distributed memory cluster with 64 processors, to deliver the desired test RMSE, CCD++ is 49 times faster than SGD and 20 times faster than ALS. When the number of processors is increased to 256, CCD++ takes only 16 s and is still 40 times faster than SGD and 20 times faster than ALS. Matrix factorization, when the matrix has missing values, has become one of the leading techniques for recommender systems. To handle web-scale datasets with millions of users and billions of ratings, scalability becomes an important issue. Alternating Least Squares (ALS) and Stochastic Gradient Descent (SGD) are two popular approaches to compute matrix factorization. There has been a recent flurry of activity to parallelize these algorithms. However, due to the cubic time complexity in the target rank, ALS is not scalable to large-scale datasets. On the other hand, SGD conducts efficient updates but usually suffers from slow convergence that is sensitive to the parameters. Coordinate descent, a classical optimization approach, has been used for many other large-scale problems, but its application to matrix factorization for recommender systems has not been explored thoroughly. In this paper, we show that coordinate descent based methods have a more efficient update rule compared to ALS, and are faster and have more stable convergence than SGD. We study different update sequences and propose the CCD++ algorithm, which updatesrank-one factors one by one. In addition, CCD++ can be easily parallelized on both multi-core and distributed systems. We empirically show that CCD++ is much faster than ALS and SGD in both settings. As an example, on a synthetic dataset with 2 billion ratings, CCD++ is 4 times faster than both SGD and ALS using a distributed system with 20 machines. We consider stochastic strongly convex optimization with a complex inequality constraint. This complex inequality constraint may lead to computationally expensive projections in algorithmic iterations of the stochastic gradient descent (SGD) methods. To reduce the computation costs pertaining to the projections, we propose an Epoch-Projection Stochastic Gradient Descent (Epro-SGD) method. The proposed Epro-SGD method consists of a sequence of epochs; it applies SGD to an augmented objective function at each iteration within the epoch, and then performs a projection at the end of each epoch. Given a strongly convex optimization and for a total number of @math iterations, Epro-SGD requires only @math projections, and meanwhile attains an optimal convergence rate of @math , both in expectation and with a high probability. To exploit the structure of the optimization problem, we propose a proximal variant of Epro-SGD, namely Epro-ORDA, based on the optimal regularized dual averaging method. We apply the proposed methods on real-world applications; the empirical results demonstrate the effectiveness of our methods.