We use the lens of weak signal asymptotics to study a class of sequentially randomized experiments, including those that arise in solving multi-armed bandit problems. In an experiment with $n$ time steps, we let the mean reward gaps between actions scale to the order $1/\sqrt{n}$ so as to preserve the difficulty of the learning task as $n$ grows. In this regime, we show that the sample paths of a class of sequentially randomized experiments -- adapted to this scaling regime and with arm selection probabilities that vary continuously with state -- converge weakly to a diffusion limit, given as the solution to a stochastic differential equation. The diffusion limit enables us to derive refined, instance-specific characterization of stochastic dynamics, and to obtain several insights on the regret and belief evolution of a number of sequential experiments including Thompson sampling (but not UCB, which does not satisfy our continuity assumption). We show that all sequential experiments whose randomization probabilities have a Lipschitz-continuous dependence on the observed data suffer from sub-optimal regret performance when the reward gaps are relatively large. Conversely, we find that a version of Thompson sampling with an asymptotically uninformative prior variance achieves near-optimal instance-specific regret scaling, including with large reward gaps, but these good regret properties come at the cost of highly unstable posterior beliefs.
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讓 iOS 8 和 OS X Yosemite 無縫切換的一個新特性。
> Apple products have always been designed to work together beautifully. But now they may really surprise you. With iOS 8 and OS X Yosemite, you’ll be able to do more wonderful things than ever before.
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Two combined numerical methods for solving time-varying semilinear differential-algebraic equations (DAEs) are obtained. The convergence and correctness of the methods are proved. When constructing the methods, time-varying spectral projectors which can be found numerically are used. This enables to numerically solve the DAE in the original form without additional analytical transformations. To improve the accuracy of the second method, recalculation is used. The developed methods are applicable to the DAEs with the continuous nonlinear part which may not be differentiable in time, and the restrictions of the type of the global Lipschitz condition are not used in the presented theorems on the DAE global solvability and the convergence of the methods. This extends the scope of methods. The fulfillment of the conditions of the global solvability theorem ensures the existence of a unique exact solution on any given time interval, which enables to seek an approximate solution also on any time interval. Numerical examples illustrating the capabilities of the methods and their effectiveness in various situations are provided. To demonstrate this, mathematical models of the dynamics of electrical circuits are considered. It is shown that the results of the theoretical and numerical analyses of these models are consistent.
Chaotic dynamical systems (DS) are ubiquitous in nature and society. Often we are interested in reconstructing such systems from observed time series for prediction or mechanistic insight, where by reconstruction we mean learning geometrical and invariant temporal properties of the system in question (like attractors). However, training reconstruction algorithms like recurrent neural networks (RNNs) on such systems by gradient-descent based techniques faces severe challenges. This is mainly due to exploding gradients caused by the exponential divergence of trajectories in chaotic systems. Moreover, for (scientific) interpretability we wish to have as low dimensional reconstructions as possible, preferably in a model which is mathematically tractable. Here we report that a surprisingly simple modification of teacher forcing leads to provably strictly all-time bounded gradients in training on chaotic systems, and, when paired with a simple architectural rearrangement of a tractable RNN design, piecewise-linear RNNs (PLRNNs), allows for faithful reconstruction in spaces of at most the dimensionality of the observed system. We show on several DS that with these amendments we can reconstruct DS better than current SOTA algorithms, in much lower dimensions. Performance differences were particularly compelling on real world data with which most other methods severely struggled. This work thus led to a simple yet powerful DS reconstruction algorithm which is highly interpretable at the same time.
The Fokker-Planck equation describes the evolution of the probability density associated with a stochastic differential equation. As the dimension of the system grows, solving this partial differential equation (PDE) using conventional numerical methods becomes computationally prohibitive. Here, we introduce a fast, scalable, and interpretable method for solving the Fokker-Planck equation which is applicable in higher dimensions. This method approximates the solution as a linear combination of shape-morphing Gaussians with time-dependent means and covariances. These parameters evolve according to the method of reduced-order nonlinear solutions (RONS) which ensures that the approximate solution stays close to the true solution of the PDE for all times. As such, the proposed method approximates the transient dynamics as well as the equilibrium density, when the latter exists. Our approximate solutions can be viewed as an evolution on a finite-dimensional statistical manifold embedded in the space of probability densities. We show that the metric tensor in RONS coincides with the Fisher information matrix on this manifold. We also discuss the interpretation of our method as a shallow neural network with Gaussian activation functions and time-varying parameters. In contrast to existing deep learning methods, our method is interpretable, requires no training, and automatically ensures that the approximate solution satisfies all properties of a probability density.
Prompt-tuning is an emerging strategy to adapt large language models (LLM) to downstream tasks by learning a (soft-)prompt parameter from data. Despite its success in LLMs, there is limited theoretical understanding of the power of prompt-tuning and the role of the attention mechanism in prompting. In this work, we explore prompt-tuning for one-layer attention architectures and study contextual mixture-models where each input token belongs to a context-relevant or -irrelevant set. We isolate the role of prompt-tuning through a self-contained prompt-attention model. Our contributions are as follows: (1) We show that softmax-prompt-attention is provably more expressive than softmax-self-attention and linear-prompt-attention under our contextual data model. (2) We analyze the initial trajectory of gradient descent and show that it learns the prompt and prediction head with near-optimal sample complexity and demonstrate how prompt can provably attend to sparse context-relevant tokens. (3) Assuming a known prompt but an unknown prediction head, we characterize the exact finite sample performance of prompt-attention which reveals the fundamental performance limits and the precise benefit of the context information. We also provide experiments that verify our theoretical insights on real datasets and demonstrate how prompt-tuning enables the model to attend to context-relevant information.
Predictive dynamical models for marine ecosystems are used for a variety of needs. Due to sparse measurements and limited understanding of the myriad of ocean processes, there is however significant uncertainty. There is model uncertainty in the parameter values, functional forms with diverse parameterizations, level of complexity needed, and thus in the state fields. We develop a Bayesian model learning methodology that allows interpolation in the space of candidate models and discovery of new models from noisy, sparse, and indirect observations, all while estimating state fields and parameter values, as well as the joint PDFs of all learned quantities. We address the challenges of high-dimensional and multidisciplinary dynamics governed by PDEs by using state augmentation and the computationally efficient GMM-DO filter. Our innovations include stochastic formulation and complexity parameters to unify candidate models into a single general model as well as stochastic expansion parameters within piecewise function approximations to generate dense candidate model spaces. These innovations allow handling many compatible and embedded candidate models, possibly none of which are accurate, and learning elusive unknown functional forms. Our new methodology is generalizable, interpretable, and extrapolates out of the space of models to discover new ones. We perform a series of twin experiments based on flows past a ridge coupled with three-to-five component ecosystem models, including flows with chaotic advection. The probabilities of known, uncertain, and unknown model formulations, and of state fields and parameters, are updated jointly using Bayes' law. Non-Gaussian statistics, ambiguity, and biases are captured. The parameter values and model formulations that best explain the data are identified. When observations are sufficiently informative, model complexity and functions are discovered.
Recent developments in neural speech synthesis and vocoding have sparked a renewed interest in voice conversion (VC). Beyond timbre transfer, achieving controllability on para-linguistic parameters such as pitch and Speed is critical in deploying VC systems in many application scenarios. Existing studies, however, either only provide utterance-level global control or lack interpretability on the controls. In this paper, we propose ControlVC, the first neural voice conversion system that achieves time-varying controls on pitch and speed. ControlVC uses pre-trained encoders to compute pitch and linguistic embeddings from the source utterance and speaker embeddings from the target utterance. These embeddings are then concatenated and converted to speech using a vocoder. It achieves speed control through TD-PSOLA pre-processing on the source utterance, and achieves pitch control by manipulating the pitch contour before feeding it to the pitch encoder. Systematic subjective and objective evaluations are conducted to assess the speech quality and controllability. Results show that, on non-parallel and zero-shot conversion tasks, ControlVC significantly outperforms two other self-constructed baselines on speech quality, and it can successfully achieve time-varying pitch and speed control.
DIFF2: Differential Private Optimization via Gradient Differences for Nonconvex Distributed Learning
Differential private optimization for nonconvex smooth objective is considered. In the previous work, the best known utility bound is $\widetilde O(\sqrt{d}/(n\varepsilon_\mathrm{DP}))$ in terms of the squared full gradient norm, which is achieved by Differential Private Gradient Descent (DP-GD) as an instance, where $n$ is the sample size, $d$ is the problem dimensionality and $\varepsilon_\mathrm{DP}$ is the differential privacy parameter. To improve the best known utility bound, we propose a new differential private optimization framework called \emph{DIFF2 (DIFFerential private optimization via gradient DIFFerences)} that constructs a differential private global gradient estimator with possibly quite small variance based on communicated \emph{gradient differences} rather than gradients themselves. It is shown that DIFF2 with a gradient descent subroutine achieves the utility of $\widetilde O(d^{2/3}/(n\varepsilon_\mathrm{DP})^{4/3})$, which can be significantly better than the previous one in terms of the dependence on the sample size $n$. To the best of our knowledge, this is the first fundamental result to improve the standard utility $\widetilde O(\sqrt{d}/(n\varepsilon_\mathrm{DP}))$ for nonconvex objectives. Additionally, a more computational and communication efficient subroutine is combined with DIFF2 and its theoretical analysis is also given. Numerical experiments are conducted to validate the superiority of DIFF2 framework.
Transfer learning - i.e., further fine-tuning a pre-trained model on a downstream task - can confer significant advantages, including improved downstream performance, faster convergence, and better sample efficiency. These advantages have led to a proliferation of task-specific fine-tuned models, which typically can only perform a single task and do not benefit from one another. Recently, model merging techniques have emerged as a solution to combine multiple task-specific models into a single multitask model without performing additional training. However, existing merging methods often ignore the interference between parameters of different models, resulting in large performance drops when merging multiple models. In this paper, we demonstrate that prior merging techniques inadvertently lose valuable information due to two major sources of interference: (a) interference due to redundant parameter values and (b) disagreement on the sign of a given parameter's values across models. To address this, we propose our method, TrIm, Elect Sign & Merge (TIES-Merging), which introduces three novel steps when merging models: (1) resetting parameters that only changed a small amount during fine-tuning, (2) resolving sign conflicts, and (3) merging only the parameters that are in alignment with the final agreed-upon sign. We find that TIES-Merging outperforms several existing methods in diverse settings covering a range of modalities, domains, number of tasks, model sizes, architectures, and fine-tuning settings. We further analyze the impact of different types of interference on model parameters, highlight the importance of resolving sign interference. Our code is available at //github.com/prateeky2806/ties-merging
We introduce an adaptive method with formal quality guarantees for weak supervision in a non-stationary setting. Our goal is to infer the unknown labels of a sequence of data by using weak supervision sources that provide independent noisy signals of the correct classification for each data point. This setting includes crowdsourcing and programmatic weak supervision. We focus on the non-stationary case, where the accuracy of the weak supervision sources can drift over time, e.g., because of changes in the underlying data distribution. Due to the drift, older data could provide misleading information to infer the label of the current data point. Previous work relied on a priori assumptions on the magnitude of the drift to decide how much data to use from the past. Comparatively, our algorithm does not require any assumptions on the drift, and it adapts based on the input. In particular, at each step, our algorithm guarantees an estimation of the current accuracies of the weak supervision sources over a window of past observations that minimizes a trade-off between the error due to the variance of the estimation and the error due to the drift. Experiments on synthetic and real-world labelers show that our approach indeed adapts to the drift. Unlike fixed-window-size strategies, it dynamically chooses a window size that allows it to consistently maintain good performance.
A tenet of open source software development is to accept contributions from users-developers (typically after appropriate vetting). But should this also include interventions done as part of research on open source development? Following an incident in which buggy code was submitted to the Linux kernel to see whether it would be caught, we conduct a survey among open source developers and empirical software engineering researchers to see what behaviors they think are acceptable. This covers two main issues: the use of publicly accessible information, and conducting active experimentation. The survey had 224 respondents. The results indicate that open-source developers are largely open to research, provided it is done transparently. In other words, many would agree to experiments on open-source projects if the subjects were notified and provided informed consent, and in special cases also if only the project leaders agree. While researchers generally hold similar opinions, they sometimes fail to appreciate certain nuances that are important to developers. Examples include observing license restrictions on publishing open-source code and safeguarding the code. Conversely, researchers seem to be more concerned than developers about privacy issues. Based on these results, it is recommended that open source repositories and projects address use for research in their access guidelines, and that researchers take care to ask permission also when not formally required to do so. We note too that the open source community wants to be heard, so professional societies and IRBs should consult with them when formulating ethics codes.
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