I am not an expert in the field but there are a few points that I am aware of:
- There are proofs that certain quantum machine learning algorithms cannot be efficiently simulated on a classical computer even if the classical computer has analagous sampling access to the data as the quantum algorithm does (i.e. they cannot be dequantized) [1-3]. However there is no proof that these algorithms are better at learning to classify certain datasets than the best classical algorithms out there.
- There is one paper out there proving that a certain quantum machine learning algorithm cannot be simulated efficiently using a classical computer and can learn a specific classification task exponentially faster than any classical algorithm . However it is important to state that the algorithm and specific classification task mentioned in the paper are very contrived and hence it is not a useful task. To summarize, they use an SVM with a very particular quantum kernel on a dataset generated by the discrete log problem. Nonetheless it's a really neat proof-of-concept result and gives us some hope that quantum computing can bring a meaningful advantage to machine learning.
- As of today there is no example where we know that a quantum machine learning algorithm could outperform a classical machine learning algorithm in any meaningful task, but a recent paper has suggested a methodology for assessing the potential for quantum advantage in prediction on learning tasks . However the proofs in classical machine learning are also quite few and far in between, so given the above two points, people are hopeful that when quantum computers become powerful enough to try some QML algorithms out, we would've advanced in the field enough that we are likely to find some use cases where we find advantage.
Oct 2021 - a paper (still undergoing peer-review I think) came out taking the principle from  and applying it to reinforcement learning https://arxiv.org/abs/2103.05577v1
Dec 2021 - (1) the paper from the previous update (Oct 2021) has been published: https://proceedings.neurips.cc/paper/2021/hash/eec96a7f788e88184c0e713456026f3f-Abstract.html and (2) there is another pre-print (still under peer review) that proves exponential advantage in quantum machine learning if the training data that is generated comes from a quantum experiment: https://arxiv.org/abs/2112.00778. Note that if it takes exponential time to produce the initial quantum states that are used as training data, then there is no advantage: https://arxiv.org/abs/2112.00811
March 2022 - In August 2021 Ewin dequantized the quantum PCA and quantum nearest-centroid clustering algorithms. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.127.060503. The very hand-wavy intuition is that if your quantum algorithm benefits only because it can access its inputs in superposition thanks to QRAM but that the algorithm itself has nothing particularly quantum about it, then it can be dequantized. For a much better explanation of this intuition, check out Ewin's STOC 2020 talk: https://www.youtube.com/watch?v=EsJEBJ2d1UY. Also, the first algorithm that was dequantized - the recommendation algorithm - was indeed found not to have an exponential speedup, but the quantum version still has a large (power of 8) speedup compared to the dequantized version (at least for now!).
September 2022 - The paper for demonstrating exponential advantage for quantum machine learning from the Dec 2021 update has been published: https://www.science.org/doi/10.1126/science.abn7293. The crux of is as follows: let's say you have an experiment that spits out a quantum state, and you want to know something about that state (or the experimental procedure), you can (1) make a bunch of copies of the state using your experiment, measure each copy (in whichever different bases you want), take all those (classical) measurement values and put them into a classical ML algorithm. Or you can (2) again make a bunch of copies of the state using your experiment, transfer those states (somehow in principle) into a quantum computer, and perform QML methods on those input states (including entangling those states together). This second method gives you an exponential speedup in 3 different learning tasks they describe in the paper.