The problem of learning-to-learn (LTL) or meta-learning is gaining increasing attention due to recent empirical evidence of its effectiveness in applications. The goal addressed in LTL is to select an algorithm that works well on tasks sampled from a meta-distribution. In this work, we consider the family of algorithms given by a variant of Ridge Regression, in which the regularizer is the square distance to an unknown mean vector. We show that, in this setting, the LTL problem can be reformulated as a Least Squares (LS) problem and we exploit a novel meta- algorithm to efficiently solve it. At each iteration the meta-algorithm processes only one dataset. Specifically, it firstly estimates the stochastic LS objective function, by splitting this dataset into two subsets used to train and test the inner algorithm, respectively. Secondly, it performs a stochastic gradient step with the estimated value. Under specific assumptions, we present a bound for the generalization error of our meta-algorithm, which suggests the right splitting parameter to choose. When the hyper-parameters of the problem are fixed, this bound is consistent as the number of tasks grows, even if the sample size is kept constant. Preliminary experiments confirm our theoretical findings, highlighting the advantage of our approach, with respect to independent task learning.

We present Memory Augmented Policy Optimization (MAPO), a simple and novel way to leverage a memory buffer of promising trajectories to reduce the variance of policy gradient estimate. MAPO is applicable to deterministic environments with discrete actions, such as structured prediction and combinatorial optimization tasks. We express the expected return objective as a weighted sum of two terms: an expectation over the high-reward trajectories inside the memory buffer, and a separate expectation over trajectories outside the buffer. To make an efficient algorithm of MAPO, we propose: (1) memory weight clipping to accelerate and stabilize training; (2) systematic exploration to discover high-reward trajectories; (3) distributed sampling from inside and outside of the memory buffer to scale up training. MAPO improves the sample efficiency and robustness of policy gradient, especially on tasks with sparse rewards. We evaluate MAPO on weakly supervised program synthesis from natural language (semantic parsing). On the WikiTableQuestions benchmark, we improve the state-of-the-art by 2.6%, achieving an accuracy of 46.3%. On the WikiSQL benchmark, MAPO achieves an accuracy of 74.9% with only weak supervision, outperforming several strong baselines with full supervision. Our source code is available at https://goo.gl/TXBp4e

Adaptive gradient methods that rely on scaling gradients down by the square root of exponential moving averages of past squared gradients, such RMSProp, Adam, Adadelta have found wide application in optimizing the nonconvex problems that arise in deep learning. However, it has been recently demonstrated that such methods can fail to converge even in simple convex optimization settings. In this work, we provide a new analysis of such methods applied to nonconvex stochastic optimization problems, characterizing the effect of increasing minibatch size. Our analysis shows that under this scenario such methods do converge to stationarity up to the statistical limit of variance in the stochastic gradients (scaled by a constant factor). In particular, our result implies that increasing minibatch sizes enables convergence, thus providing a way to circumvent the non-convergence issues. Furthermore, we provide a new adaptive optimization algorithm, Yogi, which controls the increase in effective learning rate, leading to even better performance with similar theoretical guarantees on convergence. Extensive experiments show that Yogi with very little hyperparameter tuning outperforms methods such as Adam in several challenging machine learning tasks.

Meta-learning for few-shot learning entails acquiring a prior over previous tasks and experiences, such that new tasks be learned from small amounts of data. However, a critical challenge in few-shot learning is task ambiguity: even when a powerful prior can be meta-learned from a large number of prior tasks, a small dataset for a new task can simply be too ambiguous to acquire a single model (e.g., a classifier) for that task that is accurate. In this paper, we propose a probabilistic meta-learning algorithm that can sample models for a new task from a model distribution. Our approach extends model-agnostic meta-learning, which adapts to new tasks via gradient descent, to incorporate a parameter distribution that is trained via a variational lower bound. At meta-test time, our algorithm adapts via a simple procedure that injects noise into gradient descent, and at meta-training time, the model is trained such that this stochastic adaptation procedure produces samples from the approximate model posterior. Our experimental results show that our method can sample plausible classifiers and regressors in ambiguous few-shot learning problems. We also show how reasoning about ambiguity can also be used for downstream active learning problems.

We interpret meta-reinforcement learning as the problem of learning how to quickly find a good sampling distribution in a new environment. This interpretation leads to the development of two new meta-reinforcement learning algorithms: E-MAML and E-$\text{RL}^2$. Results are presented on a new environment we call `Krazy World': a difficult high-dimensional gridworld which is designed to highlight the importance of correctly differentiating through sampling distributions in meta-reinforcement learning. Further results are presented on a set of maze environments. We show E-MAML and E-$\text{RL}^2$ deliver better performance than baseline algorithms on both tasks.

We propose to solve a label ranking problem as a structured output regression task. In this view, we adopt a least square surrogate loss approach that solves a supervised learning problem in two steps: a regression step in a well-chosen feature space and a pre-image (or decoding) step. We use specific feature maps/embeddings for ranking data, which convert any ranking/permutation into a vector representation. These embeddings are all well-tailored for our approach, either by resulting in consistent estimators, or by solving trivially the pre-image problem which is often the bottleneck in structured prediction. Their extension to the case of incomplete or partial rankings is also discussed. Finally, we provide empirical results on synthetic and real-world datasets showing the relevance of our method.

Suppose we have many copies of an unknown n-qubit state $\rho$. We measure some copies of $\rho$ using a known two-outcome measurement E_1, then other copies using a measurement E_2, and so on. At each stage t, we generate a current hypothesis $\omega_t$ about the state $\rho$, using the outcomes of the previous measurements. We show that it is possible to do this in a way that guarantees that $|\trace(E_i \omega_t) - \trace(E_i\rho)|$, the error in our prediction for the next measurement, is at least $eps$ at most $O(n / eps^2) $\ times. Even in the non-realizable setting---where there could be arbitrary noise in the measurement outcomes---we show how to output hypothesis states that incur at most $O(\sqrt {Tn}) $ excess loss over the best possible state on the first $T$ measurements. These results generalize a 2007 theorem by Aaronson on the PAC-learnability of quantum states, to the online and regret-minimization settings. We give three different ways to prove our results---using convex optimization, quantum postselection, and sequential fat-shattering dimension---which have different advantages in terms of parameters and portability.

We address reinforcement learning problems with finite state and action spaces where the underlying MDP has some known structure that could be potentially exploited to minimize the exploration rates of suboptimal (state, action) pairs. For any arbitrary structure, we derive problem-specific regret lower bounds satisfied by any learning algorithm. These lower bounds are made explicit for unstructured MDPs and for those whose transition probabilities and average reward functions are Lipschitz continuous w.r.t. the state and action. For Lipschitz MDPs, the bounds are shown not to scale with the sizes S and A of the state and action spaces, i.e., they are smaller than c log T where T is the time horizon and the constant c only depends on the Lipschitz structure, the span of the bias function, and the minimal action sub-optimality gap. This contrasts with unstructured MDPs where the regret lower bound typically scales as SA log T . We devise DEL (Directed Exploration Learning), an algorithm that matches our regret lower bounds. We further simplify the algorithm for Lipschitz MDPs, and show that the simplified version is still able to efficiently exploit the structure.

We investigate 1) the rate at which refined properties of the empirical risk---in particular, gradients---converge to their population counterparts in standard non-convex learning tasks, and 2) the consequences of this convergence for optimization. Our analysis follows the tradition of norm-based capacity control. We propose vector-valued Rademacher complexities as a simple, composable, and user-friendly tool to derive dimension-free uniform convergence bounds for gradients in non-convex learning problems. As an application of our techniques, we give a new analysis of batch gradient descent methods for non-convex generalized linear models and non-convex robust regression, showing how to use any algorithm that finds approximate stationary points to obtain optimal sample complexity, even when dimension is high or possibly infinite and multiple passes over the dataset are allowed. Moving to non-smooth models we show----in contrast to the smooth case---that even for a single ReLU it is not possible to obtain dimension-independent convergence rates for gradients in the worst case. On the positive side, it is still possible to obtain dimension-independent rates under a new type of distributional assumption.

We present a neurosymbolic framework for the lifelong learning of algorithmic tasks that mix perception and procedural reasoning. Reusing high-level concepts across domains and learning complex procedures are key challenges in lifelong learning. We show that a program synthesis approach that combines gradient descent with combinatorial search over programs can be a more effective response to these challenges than purely neural methods. Our framework, called HOUDINI, represents neural networks as strongly typed, differentiable functional programs that use symbolic higher-order combinators to compose a library of neural functions. Our learning algorithm consists of: (1) a symbolic program synthesizer that performs a type-directed search over parameterized programs, and decides on the library functions to reuse, and the architectures to combine them, while learning a sequence of tasks; and (2) a neural module that trains these programs using stochastic gradient descent. We evaluate HOUDINI on three benchmarks that combine perception with the algorithmic tasks of counting, summing, and shortest-path computation. Our experiments show that HOUDINI transfers high-level concepts more effectively than traditional transfer learning and progressive neural networks, and that the typed representation of networks significantly accelerates the search.