The performance of policy gradient methods is sensitive to hyperparameter settings that must be tuned for any new application. Widely used grid search methods for tuning hyperparameters are sample inefficient and computationally expensive. More advanced methods like Population Based Training that learn optimal schedules for hyperparameters instead of fixed settings can yield better results, but are also sample inefficient and computationally expensive. In this paper, we propose Hyperparameter Optimisation on the Fly (HOOF), a gradient-free meta-learning algorithm that can automatically learn an optimal schedule for hyperparameters that affect the policy update directly through the gradient. The main idea is to use existing trajectories sampled by the policy gradient method to optimise a one-step improvement objective, yielding a sample and computationally efficient algorithm that is easy to implement. Our experimental results across multiple domains and algorithms show that using HOOF to learn these hyperparameter schedules leads to faster learning with improved performance.
Like HOOF, gradient based methods @cite_22 @cite_17 @cite_19 @cite_2 @cite_37 are highly sample efficient and require only one training run to optimise hyperparameters. They perform gradient descent on some suitably chosen loss function with respect to the hyperparameters. Hence, they are even more restricted than HOOF in the hyperparameters that they can optimise. For example, the approach of optimises only the learning rate. Meta-gradients @cite_37 optimise only the discount rate and TD( @math ) hyperparameters and, unlike HOOF, cannot optimise, e.g., the learning rate and the policy entropy coefficient in the A2C objective function.
@cite_37 @cite_22 @cite_19 @cite_2 @cite_17
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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. Maximization of mutual information of voxel intensities has been demonstrated to be a very powerful criterion for three-dimensional medical image registration, allowing robust and accurate fully automated affine registration of multimodal images in a variety of applications, without the need for segmentation or other preprocessing of the images. In this paper, we investigate the performance of various optimization methods and multiresolution strategies for maximization of mutual information, aiming at increasing registration speed when matching large high-resolution images. We show that mutual information is a continuous function of the affine registration parameters when appropriate interpolation is used and we derive analytic expressions of its derivatives that allow numerically exact evaluation of its gradient. Various multiresolution gradient- and non-gradient-based optimization strategies, such as Powell, simplex, steepest-descent, conjugate-gradient, quasi-Newton and Levenberg—Marquardt methods, are evaluated for registration of computed tomography (CT) and magnetic resonance images of the brain. Speed-ups of a factor of 3 on average compared to Powell's method at full resolution are achieved with similar precision and without a loss of robustness with the simplex, conjugate-gradient and Levenberg—Marquardt method using a two-level multiresolution scheme. Large data sets such as 2562 × 128 MR and 5122 × 48 CT images can be registered with subvoxel precision in < 5 min CPU time on current workstations. In this paper, we present new stochastic methods for solving two important classes of nonconvex optimization problems. We first introduce a randomized accelerated proximal gradient (RapGrad) method for solving a class of nonconvex optimization problems consisting of the sum of @math component functions, and show that it can significantly reduce the number of gradient computations especially when the condition number @math (i.e., the ratio between the Lipschitz constant and negative curvature) is large. More specifically, RapGrad can save up to @math gradient computations than existing deterministic nonconvex accelerated gradient methods. Moreover, the number of gradient computations required by RapGrad can be @math (at least @math ) times smaller than the best-known randomized nonconvex gradient methods when @math . Inspired by RapGrad, we also develop a new randomized accelerated proximal dual (RapDual) method for solving a class of multi-block nonconvex optimization problems coupled with linear constraints. We demonstrate that RapDual can also save up to a factor of @math projection subproblems than its deterministic counterpart, where @math denotes the number of blocks. To the best of our knowledge, all these complexity results associated with RapGrad and RapDual seem to be new in the literature. We also illustrate potential advantages of these algorithms through our preliminary numerical experiments. Many machine learning algorithms can be formulated as the minimization of a training criterion that involves a hyperparameter. This hyperparameter is usually chosen by trial and error with a model selection criterion. In this article we present a methodology to optimize several hyperparameters, based on the computation of the gradient of a model selection criterion with respect to the hyperparameters. In the case of a quadratic training criterion, the gradient of the selection criterion with respect to the hyperparameters is efficiently computed by backpropagating through a Cholesky decomposition. In the more general case, we show that the implicit function theorem can be used to derive a formula for the hyperparameter gradient involving second derivatives of the training criterion. We consider the problem of learning a structured multi-task regression, where the output consists of multiple responses that are related by a graph and the correlated response variables are dependent on the common inputs in a sparse but synergistic manner. Previous methods such as l1 l2-regularized multi-task regression assume that all of the output variables are equally related to the inputs, although in many real-world problems, outputs are related in a complex manner. In this paper, we propose graph-guided fused lasso (GFlasso) for structured multi-task regression that exploits the graph structure over the output variables. We introduce a novel penalty function based on fusion penalty to encourage highly correlated outputs to share a common set of relevant inputs. In addition, we propose a simple yet efficient proximal-gradient method for optimizing GFlasso that can also be applied to any optimization problems with a convex smooth loss and the general class of fusion penalty defined on arbitrary graph structures. By exploiting the structure of the non-smooth ''fusion penalty'', our method achieves a faster convergence rate than the standard first-order method, sub-gradient method, and is significantly more scalable than the widely adopted second-order cone-programming and quadratic-programming formulations. In addition, we provide an analysis of the consistency property of the GFlasso model. Experimental results not only demonstrate the superiority of GFlasso over the standard lasso but also show the efficiency and scalability of our proximal-gradient method. We formulate gradient-based Markov chain Monte Carlo (MCMC) sampling as optimization on the space of probability measures, with Kullback-Leibler (KL) divergence as the objective function. We show that an underdamped form of the Langevin algorithm perform accelerated gradient descent in this metric. To characterize the convergence of the algorithm, we construct a Lyapunov functional and exploit hypocoercivity of the underdamped Langevin algorithm. As an application, we show that accelerated rates can be obtained for a class of nonconvex functions with the Langevin algorithm. Nesterov's accelerated gradient descent (AGD), an instance of the general family of momentum methods,'' provably achieves faster convergence rate than gradient descent (GD) in the convex setting. While these methods are widely used in modern nonconvex applications, including training of deep neural networks, whether they are provably superior to GD in the nonconvex setting remains open. This paper studies a simple variant of Nesterov's AGD, and shows that it escapes saddle points and finds a second-order stationary point in ( O (1 ^ 7 4 ) ) iterations, matching the best known convergence rate, which is faster than the ( O (1 ^ 2 ) ) iterations required by GD. To the best of our knowledge, this is the first direct acceleration (single-loop) algorithm that is provably faster than GD in general nonconvex setting---all previous nonconvex accelerated algorithms rely on more complex mechanisms such as nested loops and proximal terms. Our analysis is based on two key ideas: (1) the use of a simple Hamiltonian function, inspired by a continuous-time perspective, which AGD monotonically decreases on each step even for nonconvex functions, and (2) a novel framework called improve or localize, which is useful for tracking the long-term behavior of gradient-based optimization algorithms. We believe that these techniques may deepen our understanding of both acceleration algorithms and nonconvex optimization. Hyperparameter optimization is often done manually or by using a grid search. However, recent research has shown that automatic optimization techniques are able to accelerate this optimization process and find hyperparameter configurations that lead to better models. Currently, transferring knowledge from previous experiments to a new experiment is of particular interest because it has been shown that it allows to further improve the hyperparameter optimization. We propose to transfer knowledge by means of an initialization strategy for hyperparameter optimization. In contrast to the current state of the art initialization strategies, our strategy is neither limited to hyperparameter configurations that have been evaluated on previous experiments nor does it need meta-features. The initial hyperparameter configurations are derived by optimizing for a meta-loss formally defined in this paper. This loss depends on the hyperparameter response function of the data sets that were investigated in past experiments. Since this function is unknown and only few observations are given, the meta-loss is not differentiable. We propose to approximate the response function by a differentiable plug-in estimator. Then, we are able to learn the initial hyperparameter configuration sequence by applying gradient-based optimization techniques. Extensive experiments are conducted on two meta-data sets. Our initialization strategy is compared to the state of the art for initialization strategies and further methods that are able to transfer knowledge between data sets. We give empirical evidence that our work provides an improvement over the state of the art. In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We nonasymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings. Tuning hyperparameters of learning algorithms is hard because gradients are usually unavailable. We compute exact gradients of cross-validation performance with respect to all hyperparameters by chaining derivatives backwards through the entire training procedure. These gradients allow us to optimize thousands of hyperparameters, including step-size and momentum schedules, weight initialization distributions, richly parameterized regularization schemes, and neural network architectures. We compute hyperparameter gradients by exactly reversing the dynamics of stochastic gradient descent with momentum. Tuning hyperparameters of learning algorithms is hard because gradients are usually unavailable. We compute exact gradients of cross-validation performance with respect to all hyperparameters by chaining derivatives backwards through the entire training procedure. These gradients allow us to optimize thousands of hyperparameters, including step-size and momentum schedules, weight initialization distributions, richly parameterized regularization schemes, and neural network architectures. We compute hyperparameter gradients by exactly reversing the dynamics of stochastic gradient descent with momentum. Nesterov's accelerated gradient descent (AGD), an instance of the general family of "momentum methods", provably achieves faster convergence rate than gradient descent (GD) in the convex setting. However, whether these methods are superior to GD in the nonconvex setting remains open. This paper studies a simple variant of AGD, and shows that it escapes saddle points and finds a second-order stationary point in @math iterations, faster than the @math iterations required by GD. To the best of our knowledge, this is the first Hessian-free algorithm to find a second-order stationary point faster than GD, and also the first single-loop algorithm with a faster rate than GD even in the setting of finding a first-order stationary point. Our analysis is based on two key ideas: (1) the use of a simple Hamiltonian function, inspired by a continuous-time perspective, which AGD monotonically decreases per step even for nonconvex functions, and (2) a novel framework called improve or localize, which is useful for tracking the long-term behavior of gradient-based optimization algorithms. We believe that these techniques may deepen our understanding of both acceleration algorithms and nonconvex optimization. We examine gradient descent on unregularized logistic regression problems, with homogeneous linear predictors on linearly separable datasets. We show the predictor converges to the direction of the max-margin (hard margin SVM) solution. The result also generalizes to other monotone decreasing loss functions with an infimum at infinity, to multi-class problems, and to training a weight layer in a deep network in a certain restricted setting. Furthermore, we show this convergence is very slow, and only logarithmic in the convergence of the loss itself. This can help explain the benefit of continuing to optimize the logistic or cross-entropy loss even after the training error is zero and the training loss is extremely small, and, as we show, even if the validation loss increases. Our methodology can also aid in understanding implicit regularization n more complex models and with other optimization methods. Neural networks dominate the modern machine learning landscape, but their training and success still suffer from sensitivity to empirical choices of hyperparameters such as model architecture, loss function, and optimisation algorithm. In this work we present , a simple asynchronous optimisation algorithm which effectively utilises a fixed computational budget to jointly optimise a population of models and their hyperparameters to maximise performance. Importantly, PBT discovers a schedule of hyperparameter settings rather than following the generally sub-optimal strategy of trying to find a single fixed set to use for the whole course of training. With just a small modification to a typical distributed hyperparameter training framework, our method allows robust and reliable training of models. We demonstrate the effectiveness of PBT on deep reinforcement learning problems, showing faster wall-clock convergence and higher final performance of agents by optimising over a suite of hyperparameters. In addition, we show the same method can be applied to supervised learning for machine translation, where PBT is used to maximise the BLEU score directly, and also to training of Generative Adversarial Networks to maximise the Inception score of generated images. In all cases PBT results in the automatic discovery of hyperparameter schedules and model selection which results in stable training and better final performance. We study the implicit bias of gradient descent methods in solving a binary classification problem over a linearly separable dataset. The classifier is described by a nonlinear ReLU model and the objective function adopts the exponential loss function. We first characterize the landscape of the loss function and show that there can exist spurious asymptotic local minima besides asymptotic global minima. We then show that gradient descent (GD) can converge to either a global or a local max-margin direction, or may diverge from the desired max-margin direction in a general context. For stochastic gradient descent (SGD), we show that it converges in expectation to either the global or the local max-margin direction if SGD converges. We further explore the implicit bias of these algorithms in learning a multi-neuron network under certain stationary conditions, and show that the learned classifier maximizes the margins of each sample pattern partition under the ReLU activation. A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for these local methods to find the global minimum is the proliferation of local minima with much higher error than the global minimum. Here we argue, based on results from statistical physics, random matrix theory, neural network theory, and empirical evidence, that a deeper and more profound difficulty originates from the proliferation of saddle points, not local minima, especially in high dimensional problems of practical interest. Such saddle points are surrounded by high error plateaus that can dramatically slow down learning, and give the illusory impression of the existence of a local minimum. Motivated by these arguments, we propose a new approach to second-order optimization, the saddle-free Newton method, that can rapidly escape high dimensional saddle points, unlike gradient descent and quasi-Newton methods. We apply this algorithm to deep or recurrent neural network training, and provide numerical evidence for its superior optimization performance. We consider parallel global optimization of derivative-free expensive-to-evaluate functions, and propose an efficient method based on stochastic approximation for implementing a conceptual Bayesian optimization algorithm proposed by (2007). To accomplish this, we use infinitessimal perturbation analysis (IPA) to construct a stochastic gradient estimator and show that this estimator is unbiased. We also show that the stochastic gradient ascent algorithm using the constructed gradient estimator converges to a stationary point of the q-EI surface, and therefore, as the number of multiple starts of the gradient ascent algorithm and the number of steps for each start grow large, the one-step Bayes optimal set of points is recovered. We show in numerical experiments that our method for maximizing the q-EI is faster than methods based on closed-form evaluation using high-dimensional integration, when considering many parallel function evaluations, and is comparable in speed when considering few. We also show that the resulting one-step Bayes optimal algorithm for parallel global optimization finds high quality solutions with fewer evaluations that a heuristic based on approximately maximizing the q-EI. A high quality open source implementation of this algorithm is available in the open source Metrics Optimization Engine (MOE). The performance of many machine learning algorithms depends crucially on the hyperparameter settings, especially in Deep Learning. Manually tuning the hyperparameters is laborious and time consuming. To address this issue, Bayesian optimization (BO) methods and their extensions have been proposed to optimize the hyperparameters automatically. However, they still suffer from highly computational expense when applying to deep generative models (DGMs) due to their strategy of the black-box function optimization. This paper provides a new hyperparameter optimization procedure at the pre-training phase of the DGMs, where we avoid combining all layers as one black-box function by taking advantage of the layer-by-layer learning strategy. Following this procedure, we are able to optimize multiple hyperparameters in an adaptive way by using Gaussian process. In contrast to the traditional BO methods, which mainly focus on the supervised models, the pre-training procedure is unsupervised where there is no validation error can be used. To alleviate this problem, this paper proposes a new holdout loss, the free energy gap, which takes into account both factors of the model fitting and over-fitting. The empirical evaluations demonstrate that our method not only speeds up the process of hyperparameter optimization, but also improves the performances of DGMs significantly in both the supervised and unsupervised learning tasks. We consider the minimization of a smooth loss function regularized by the trace norm of the matrix variable. Such formulation finds applications in many machine learning tasks including multi-task learning, matrix classification, and matrix completion. The standard semidefinite programming formulation for this problem is computationally expensive. In addition, due to the non-smooth nature of the trace norm, the optimal first-order black-box method for solving such class of problems converges as O(1 √k), where k is the iteration counter. In this paper, we exploit the special structure of the trace norm, based on which we propose an extended gradient algorithm that converges as O(1 k). We further propose an accelerated gradient algorithm, which achieves the optimal convergence rate of O(1 k2) for smooth problems. Experiments on multi-task learning problems demonstrate the efficiency of the proposed algorithms. Accelerated gradient (AG) methods are breakthroughs in convex optimization, improving the convergence rate of the gradient descent method for optimization with smooth functions. However, the analysis of AG methods for non-convex optimization is still limited. It remains an open question whether AG methods from convex optimization can accelerate the convergence of the gradient descent method for finding local minimum of non-convex optimization problems. This paper provides an affirmative answer to this question. In particular, we analyze two renowned variants of AG methods (namely Polyak's Heavy Ball method and Nesterov's Accelerated Gradient method) for extracting the negative curvature from random noise, which is central to escaping from saddle points. By leveraging the proposed AG methods for extracting the negative curvature, we present a new AG algorithm with double loops for non-convex optimization this is in contrast to a single-loop AG algorithm proposed in a recent manuscript , which directly analyzed the Nesterov's AG method for non-convex optimization and appeared online on November 29, 2017. However, we emphasize that our work is an independent work, which is inspired by our earlier work and is based on a different novel analysis. , which converges to second-order stationary point @math such that @math and @math with @math iteration complexity, improving that of gradient descent method by a factor of @math and matching the best iteration complexity of second-order Hessian-free methods for non-convex optimization.