Ah yeah I take your point. The squared error term is over the whole
user-item matrix, technically, in the implicit case. I suppose I am
used to assuming that the 0 terms in this matrix are weighted so much
less (because alpha is usually large-ish) that they're almost not
there, but they are. So I had just used the explicit formulation.
I suppose the result is kind of scale invariant, but not exactly. I
had not prioritized this property since I had generally built models
on the full data set and not a sample, and had assumed that lambda
would need to be retuned over time as the input grew anyway.
So, basically I don't know anything more than you do, sorry!
On Tue, Mar 31, 2015 at 10:41 PM, Xiangrui Meng <men...@gmail.com> wrote:
> Hey Sean,
>
> That is true for explicit model, but not for implicit. The ALS-WR
> paper doesn't cover the implicit model. In implicit formulation, a
> sub-problem (for v_j) is:
>
> min_{v_j} \sum_i c_ij (p_ij - u_i^T v_j)^2 + lambda * X * \|v_j\|_2^2
>
> This is a sum for all i but not just the users who rate item j. In
> this case, if we set X=m_j, the number of observed ratings for item j,
> it is not really scale invariant. We have #users user vectors in the
> least squares problem but only penalize lambda * #ratings. I was
> suggesting using lambda * m directly for implicit model to match the
> number of vectors in the least squares problem. Well, this is my
> theory. I don't find any public work about it.
>
> Best,
> Xiangrui
>
> On Tue, Mar 31, 2015 at 5:17 AM, Sean Owen <so...@cloudera.com> wrote:
>> I had always understood the formulation to be the first option you
>> describe. Lambda is scaled by the number of items the user has rated /
>> interacted with. I think the goal is to avoid fitting the tastes of
>> prolific users disproportionately just because they have many ratings
>> to fit. This is what's described in the ALS-WR paper we link to on the
>> Spark web site, in equation 5
>> (http://www.grappa.univ-lille3.fr/~mary/cours/stats/centrale/reco/paper/MatrixFactorizationALS.pdf)
>>
>> I think this also gets you the scale-invariance? For every additional
>> rating from user i to product j, you add one new term to the
>> squared-error sum, (r_ij - u_i . m_j)^2, but also, you'd increase the
>> regularization term by lambda * (|u_i|^2 + |m_j|^2) They are at least
>> both increasing about linearly as ratings increase. If the
>> regularization term is multiplied by the total number of users and
>> products in the model, then it's fixed.
>>
>> I might misunderstand you and/or be speaking about something slightly
>> different when it comes to invariance. But FWIW I had always
>> understood the regularization to be multiplied by the number of
>> explicit ratings.
>>
>> On Mon, Mar 30, 2015 at 5:51 PM, Xiangrui Meng <men...@gmail.com> wrote:
>>> Okay, I didn't realize that I changed the behavior of lambda in 1.3.
>>> to make it "scale-invariant", but it is worth discussing whether this
>>> is a good change. In 1.2, we multiply lambda by the number ratings in
>>> each sub-problem. This makes it "scale-invariant" for explicit
>>> feedback. However, in implicit feedback model, a user's sub-problem
>>> contains all item factors. Then the question is whether we should
>>> multiply lambda by the number of explicit ratings from this user or by
>>> the total number of items. We used the former in 1.2 but changed to
>>> the latter in 1.3. So you should try a smaller lambda to get a similar
>>> result in 1.3.
>>>
>>> Sean and Shuo, which approach do you prefer? Do you know any existing
>>> work discussing this?
>>>
>>> Best,
>>> Xiangrui
>>>
>>>
>>> On Fri, Mar 27, 2015 at 11:27 AM, Xiangrui Meng <men...@gmail.com> wrote:
>>>> This sounds like a bug ... Did you try a different lambda? It would be
>>>> great if you can share your dataset or re-produce this issue on the
>>>> public dataset. Thanks! -Xiangrui
>>>>
>>>> On Thu, Mar 26, 2015 at 7:56 AM, Ravi Mody <rmody...@gmail.com> wrote:
>>>>> After upgrading to 1.3.0, ALS.trainImplicit() has been returning vastly
>>>>> smaller factors (and hence scores). For example, the first few product's
>>>>> factor values in 1.2.0 are (0.04821, -0.00674, -0.0325). In 1.3.0, the
>>>>> first few factor values are (2.535456E-8, 1.690301E-8, 6.99245E-8). This
>>>>> difference of several orders of magnitude is consistent throughout both
>>>>> user
>>>>> and product. The recommendations from 1.2.0 are subjectively much better
>>>>> than in 1.3.0. 1.3.0 trains significantly faster than 1.2.0, and uses less
>>>>> memory.
>>>>>
>>>>> My first thought is that there is too much regularization in the 1.3.0
>>>>> results, but I'm using the same lambda parameter value. This is a snippet
>>>>> of
>>>>> my scala code:
>>>>> .....
>>>>> val rank = 75
>>>>> val numIterations = 15
>>>>> val alpha = 10
>>>>> val lambda = 0.01
>>>>> val model = ALS.trainImplicit(train_data, rank, numIterations,
>>>>> lambda=lambda, alpha=alpha)
>>>>> .....
>>>>>
>>>>> The code and input data are identical across both versions. Did anything
>>>>> change between the two versions I'm not aware of? I'd appreciate any help!
>>>>>
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